This is a modern-English version of The Meaning of Relativity: Four lectures delivered at Princeton University, May, 1921, originally written by Einstein, Albert. It has been thoroughly updated, including changes to sentence structure, words, spelling, and grammar—to ensure clarity for contemporary readers, while preserving the original spirit and nuance. If you click on a paragraph, you will see the original text that we modified, and you can toggle between the two versions.

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THE MEANING OF
RELATIVITY

FOUR LECTURES DELIVERED AT PRINCETON UNIVERSITY, MAY, 1921

ALBERT EINSTEIN

WITH FOUR DIAGRAMS

PRINCETON
PRINCETON UNIVERSITY PRESS
1923

NOTE.—The translation of these lectures into English was made by EDWIN PLIMPTON ADAMS, Professor of Physics in Princeton University

NOTE.—The translation of these lectures into English was done by EDWIN PLIMPTON ADAMS, Professor of Physics at Princeton University.

LECTURE I
SPACE AND TIME IN PRE-RELATIVITY PHYSICS
LECTURE II
THE THEORY OF SPECIAL RELATIVITY
LECTURE III
THE GENERAL THEORY OF RELATIVITY
LECTURE IV
THE GENERAL THEORY OF RELATIVITY (continued)
INDEX

LECTURE I
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LECTURE II
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LECTURE III
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LECTURE IV
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THE MEANING OF RELATIVITY

LECTURE I

SPACE AND TIME IN PRE-RELATIVITY PHYSICS

THE theory of relativity is intimately connected with the theory of space and time. I shall therefore begin with a brief investigation of the origin of our ideas of space and time, although in doing so I know that I introduce a controversial subject. The object of all science, whether natural science or psychology, is to co-ordinate our experiences and to bring them into a logical system. How are our customary ideas of space and time related to the character of our experiences?

THE theory of relativity is closely linked to the theory of space and time. So, I'll start with a quick look at the origins of our concepts of space and time, even though I realize this is a controversial topic. The goal of all science, whether it's natural science or psychology, is to organize our experiences and create a logical system. How are our usual ideas of space and time connected to the nature of our experiences?

The experiences of an individual appear to us arranged in a series of events; in this series the single events which we remember appear to be ordered according to the criterion of "earlier" and "later," which cannot be analysed further. There exists, therefore, for the individual, an I-time, or subjective time. This in itself is not measurable. I can, indeed, associate numbers with the events, in such a way that a greater number is associated with the later event than with an earlier one; but the nature of this association may be quite arbitrary. This association I can define by means of a clock by comparing the order of events furnished by the clock with the order of the given series of events. We understand by a clock something which provides a series of events which can be counted, and which has other properties of which we shall speak later. [Pg 1]

The experiences of a person seem to us to be arranged in a sequence of events; in this sequence, the individual events we remember appear to be ordered by the criteria of "earlier" and "later," which can't be broken down any further. Thus, for the individual, there exists an I-time, or subjective time. This, in itself, can't be measured. I can, however, link numbers to the events, so that a larger number corresponds to a later event than to an earlier one; but the nature of this connection can be quite random. I can define this relationship using a clock by comparing the order of events indicated by the clock with the order of the specific series of events. We understand a clock as something that provides a sequence of events that can be counted, along with other characteristics that we will discuss later. [Pg 1]

By the aid of speech different individuals can, to a certain extent, compare their experiences. In this way it is shown that certain sense perceptions of different individuals correspond to each other, while for other sense perceptions no such correspondence can be established. We are accustomed to regard as real those sense perceptions which are common to different individuals, and which therefore are, in a measure, impersonal. The natural sciences, and in particular, the most fundamental of them, physics, deal with such sense perceptions. The conception of physical bodies, in particular of rigid bodies, is a relatively constant complex of such sense perceptions. A clock is also a body, or a system, in the same sense, with the additional property that the series of events which it counts is formed of elements all of which can be regarded as equal.

Through communication, different people can, to some degree, share their experiences. This reveals that certain sensory perceptions among different individuals align, while for other sensory perceptions, no such alignment can be found. We tend to consider as real those sensory perceptions that are shared by various people, making them somewhat impersonal. The natural sciences, especially the most basic one, physics, focus on these shared sensory perceptions. The idea of physical objects, particularly rigid ones, is a relatively stable group of these sensory perceptions. A clock is also an object, or a system, in the same way, with the added feature that the sequence of events it measures is made up of elements that can all be seen as equal.

The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori. For even if it should appear that the universe of ideas cannot be deduced from experience by logical means, but is, in a sense, a creation of the human mind, without which no science is possible, nevertheless this universe of ideas is just as little independent of the nature of our experiences as clothes are of the form of the human body. This is particularly true of our concepts of time and space, which physicists have been obliged by the facts to bring down from the Olympus of the a priori in order to adjust them and put them in a serviceable condition. [Pg 2]

The only reason for our concepts and system of concepts is that they help us make sense of our experiences; beyond that, they hold no real authority. I believe that philosophers have negatively impacted the advancement of scientific thinking by taking certain fundamental concepts away from empiricism, where we can actually control them, and pushing them into the abstract realm of the a priori. Even if it seems that the universe of ideas can't be derived from experience through logical reasoning, but is, in a sense, a product of the human mind—without which no science can exist—this universe of ideas is just as dependent on the nature of our experiences as clothing is on the shape of the human body. This is especially true for our concepts of time and space, which physicists have had to pull down from the lofty heights of the a priori to refine and make them functional. [Pg 2]

We now come to our concepts and judgments of space. It is essential here also to pay strict attention to the relation of experience to our concepts. It seems to me that Poincaré clearly recognized the truth in the account he gave in his book, "La Science et l'Hypothèse." Among all the changes which we can perceive in a rigid body those are marked by their simplicity which can be made reversibly by an arbitrary motion of the body; Poincaré calls these, changes in position. By means of simple changes in position we can bring two bodies into contact. The theorems of congruence, fundamental in geometry, have to do with the laws that govern such changes in position. For the concept of space the following seems essential. We can form new bodies by bringing bodies $B$ , $C$ , ... up to body $A$ ; we say that we continue body $A$ . We can continue body $A$ in such a way that it comes into contact with any other body, $X$ . The ensemble of all continuations of body $A$ we can designate as the "space of the body $A$ ." Then it is true that all bodies are in the "space of the (arbitrarily chosen) body $A$ ." In this sense we cannot speak of space in the abstract, but only of the "space belonging to a body $A$ ." The earth's crust plays such a dominant rôle in our daily life in judging the relative positions of bodies that it has led to an abstract conception of space which certainly cannot be defended. In order to free ourselves from this fatal error we shall speak only of "bodies of reference," or "space of reference." It was only through the theory of general relativity that refinement of these concepts became necessary, as we shall see later.

We now turn to our ideas and perceptions of space. It's crucial to closely examine how experience relates to our concepts. I believe Poincaré clearly identified the truth in what he explained in his book, "La Science et l'Hypothèse." Among all the changes we can notice in a solid object, those marked by their simplicity can be reversed by its motion; Poincaré refers to these as changes in position. By making simple positional changes, we can bring two objects together. The theorems of congruence, which are fundamental in geometry, relate to the rules governing these positional changes. For the concept of space, the following seems essential: we can create new objects by bringing objects $B$ , $C$ , ... up to the object $A$ ; we say that we continue object $A$ . We can extend object $A$ in a way that it connects with any other object, $X$ . The collection of all continuations of object $A$ can be referred to as the "space of the object $A$

I shall not go into detail concerning those properties of the space of reference which lead to our conceiving points as elements of space, and space as a continuum. Nor shall I attempt to analyse further the properties of space which justify the conception [Pg 3] of continuous series of points, or lines. If these concepts are assumed, together with their relation to the solid bodies of experience, then it is easy to say what we mean by the three-dimensionality of space; to each point three numbers, $x_{1}$ , $x_{2}$ , $x_{3}$ (co-ordinates), may be associated, in such a way that this association is uniquely reciprocal, and that $x_{1}$ , $x_{2}$ and $x_{3}$ vary continuously when the point describes a continuous series of points (a line).

I won't go into detail about the properties of the reference space that lead us to understand points as elements of space, and space as a continuum. I also won't further analyze the properties of space that justify the idea of continuous series of points, or lines. If we accept these concepts along with their connection to the solid bodies we experience, then it's straightforward to explain what we mean by the three-dimensionality of space; we can assign three numbers, $x_{1}$ , $x_{2}$ , $x_{3}$ (coordinates), to each point in such a way that this assignment is uniquely reciprocal, and that $x_{1}$ , $x_{2}$ and $x_{3}$ change continuously as the point moves along a continuous series of points (a line).

It is assumed in pre-relativity physics that the laws of the orientation of ideal rigid bodies are consistent with Euclidean geometry. What this means may be expressed as follows: Two points marked on a rigid body form an interval. Such an interval can be oriented at rest, relatively to our space of reference, in a multiplicity of ways. If, now, the points of this space can be referred to co-ordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , in such a way that the differences of the co-ordinates, $\Delta x_{1}$ , $\Delta x_{2}$ , $\Delta x_{3}$ , of the two ends of the interval furnish the same sum of squares, $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}, \qquad\text{(1)}$ for every orientation of the interval, then the space of reference is called Euclidean, and the co-ordinates Cartesian.[1] It is sufficient, indeed, to make this assumption in the limit for an infinitely small interval. Involved in this assumption there are some which are rather less special, to which we must call attention on account of their fundamental significance. In the first place, it is assumed that one can move an ideal rigid body in an arbitrary manner. In the second place, it is assumed that the behaviour of ideal rigid bodies towards orientation is independent [Pg 4] of the material of the bodies and their changes of position, in the sense that if two intervals can once be brought into coincidence, they can always and everywhere be brought into coincidence. Both of these assumptions, which are of fundamental importance for geometry and especially for physical measurements, naturally arise from experience; in the theory of general relativity their validity needs to be assumed only for bodies and spaces of reference which are infinitely small compared to astronomical dimensions.

In pre-relativity physics, it's assumed that the rules for positioning ideal rigid bodies align with Euclidean geometry. This means that two points on a rigid body create an interval. This interval can be aligned at rest, relative to our reference space, in many different ways. If the points in this space can be related to coordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , so that the differences in coordinates, $\Delta x_{1}$ , $\Delta x_{2}$ , $\Delta x_{3}$ , at both ends of the interval provide the same sum of squares, $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2}, \qquad\text{(1)}$ for every orientation of the interval, then this reference space is called Euclidean, and the coordinates are Cartesian.[1] It's actually sufficient to make this assumption at the limit for an infinitely small interval. This assumption involves some less specific ones that are important to note due to their fundamental nature. First, it’s assumed that an ideal rigid body can be moved in any way. Second, it’s assumed that the behavior of ideal rigid bodies concerning orientation is independent of the materials of the bodies and their position changes, in the sense that if two intervals can be matched up once, they can always be matched up again anywhere. Both of these assumptions are fundamentally important for geometry and especially for physical measurements, and they naturally come from experience; in the theory of general relativity, their validity only needs to be assumed for bodies and reference spaces that are infinitely small compared to astronomical dimensions.

[1]This relation must hold for an arbitrary choice of the origin and of the direction (ratios $\Delta x_{1}: \Delta x_{2}: \Delta x_{3}$ ) of the interval.

[1]This relationship must be true for any choice of origin and direction (ratios $\Delta x_{1}: \Delta x_{2}: \Delta x_{3}$ ) of the interval.

The quantity $s$ we call the length of the interval. In order that this may be uniquely determined it is necessary to fix arbitrarily the length of a definite interval; for example, we can put it equal to 1 (unit of length). Then the lengths of all other intervals may be determined. If we make the $x_{\nu}$ linearly dependent upon a parameter $\lambda$ , $x_{\nu} = a_{\nu} + \lambda b_{\nu},$ we obtain a line which has all the properties of the straight lines of the Euclidean geometry. In particular, it easily follows that by laying off $n$ times the interval $s$ upon a straight line, an interval of length $n·s$ is obtained. A length, therefore, means the result of a measurement carried out along a straight line by means of a unit measuring rod. It has a significance which is as independent of the system of co-ordinates as that of a straight line, as will appear in the sequel.

The quantity $s$ is what we refer to as the length of the interval. To uniquely determine this, we need to arbitrarily set the length of a specific interval; for instance, we can define it to be 1 (one unit of length). From there, we can determine the lengths of all other intervals. If we make the $x_{\nu}$ dependent on a parameter $\lambda$ , $x_{\nu} = a_{\nu} + \lambda b_{\nu},$ we get a line that has all the characteristics of straight lines in Euclidean geometry. This means that by measuring $n$ times the interval $s$ along a straight line, we get an interval that is $n·s$ long. Thus, a length represents the result of a measurement taken along a straight line using a unit measuring rod. Its significance is as independent of the coordinate system as that of a straight line, as will be demonstrated later.

We come now to a train of thought which plays an analogous role in the theories of special and general relativity. We ask the question: besides the Cartesian co-ordinates which we have used are there other equivalent co-ordinates? An interval has [Pg 5] a physical meaning which is independent of the choice of co-ordinates; and so has the spherical surface which we obtain as the locus of the end points of all equal intervals that we lay off from an arbitrary point of our space of reference. If $x_{\nu}$ as well as $x'_{\nu}$ ( $\nu$ from 1 to 3) are Cartesian co-ordinates of our space of reference, then the spherical surface will be expressed in our two systems of co-ordinates by the equations $\begin{align*} \sum {\Delta x_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2)} \\ \sum {{\Delta x'}_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2a)} \end{align*}$ How must the $x'_{\nu}$ be expressed in terms of the $x_{\nu}$ in order that equations (2) and (2a) may be equivalent to each other? Regarding the $x'_{\nu}$ expressed as functions of the $x_{\nu}$ , we can write, by Taylor's theorem, for small values of the $\Delta$ $x_{\nu}$ , ${\Delta x'}_{\nu} = \sum_{\alpha} \frac{\partial {x'}_{\nu}}{\partial x_{\alpha}}\, \Delta x_{\alpha} + \frac{1}{2} \sum_{\alpha \text{,} \beta} \frac{\partial^{2} {x'}_{\nu}}{\partial x_{\alpha}\, \partial x_{\beta}}\, \Delta x_{\alpha}\, \Delta x_{\beta}.\ldots$ If we substitute (2a) in this equation and compare with (1), we see that the $x'_{\nu}$ must be linear functions of the $x_{\nu}$ . If we therefore put $\begin{align*} & & {x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}, \qquad & &\text{(3)}\\ &\text{or} & \Delta {x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}, \qquad & &\text{(3a)} \end{align*}$ [Pg 6] then the equivalence of equations (2) and (2a) is expressed in the form $\sum{\Delta x'_{\nu}}^{2} = \lambda \sum{\Delta x_{\nu}}^{2} \qquad\text{($\lambda$ independent of $\Delta x_{\nu}$).} \qquad\text{(2b)}$ It therefore follows that $\lambda$ must be a constant. If we put $\lambda$ = 1, (2b) and (3a) furnish the conditions $\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}, \qquad\text{(4)}$ in which $\delta_{\alpha\beta}$ = 1, or $\delta_{\alpha\beta}$ = 0, according $\alpha$ = $\beta$ or $\alpha$ ≠ $\beta$ . The conditions (4) are called the conditions of orthogonality, and the transformations (3), (4), linear orthogonal transformations. If we stipulate that $s^2 = \sum {\Delta x_{\nu}}^2$ shall be equal to the square of the length in every system of co-ordinates, and if we always measure with the same unit scale, then $\lambda$ must be equal to 1. Therefore the linear orthogonal transformations are the only ones by means of which we can pass from one Cartesian system of co-ordinates in our space of reference to another. We see that in applying such transformations the equations of a straight line become equations of a straight line. Reversing equations (3a) by multiplying both sides by $b_{\nu\beta}$ and summing for all the $\nu$ 's, we obtain $\sum_{\nu} b_{\nu\beta}\, {\Delta x'}_{\nu} = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} = \Delta x_{\beta} \qquad\text{(5)}$ The same coefficients, $b$ , also determine the inverse substitution of $\Delta$ $x_{\nu}$ . Geometrically, $b_{\nu\alpha}$ is the cosine of the angle between the $x'_{\nu}$ axis and the $x_{\alpha}$ axis. [Pg 7]

We now move on to a line of thinking that plays a similar role in the theories of special and general relativity. We ask: besides the Cartesian coordinates we've been using, are there other equivalent coordinates? An interval has a physical meaning that doesn’t depend on the choice of coordinates, and so does the spherical surface, which represents the collection of endpoints of all equal intervals measured from an arbitrary point of our reference space. If $x_{\nu}$ and $x'_{\nu}$ (where $\nu$ ranges from 1 to 3) are Cartesian coordinates of our reference space, then the spherical surface will be expressed in our two coordinate systems by the equations $\begin{align*} \sum {\Delta x_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2)} \\ \sum {{\Delta x'}_{\nu}}^{2} &= \text{const.} \qquad & &\text{(2a)} \end{align*}$ How should $x'_{\nu}$ be expressed in terms of $x_{\nu}$ so that equations (2) and (2a) can be equivalent? Regarding $x'_{\nu}$ expressed as functions of $x_{\nu}$ , we can write, using Taylor's theorem, for small values of the $\Delta$ $x_{\nu}$ : ${\Delta x'}_{\nu} = \sum_{\alpha} \frac{\partial {x'}_{\nu}}{\partial x_{\alpha}}\, \Delta x_{\alpha} + \frac{1}{2} \sum_{\alpha \text{,} \beta} \frac{\partial^{2} {x'}_{\nu}}{\partial x_{\alpha}\, \partial x_{\beta}}\, \Delta x_{\alpha}\, \Delta x_{\beta}.\ldots$ If we substitute (2a) into this equation and compare it with (1), we find that $x'_{\nu}$ must be linear functions of the $x_{\nu}$ . Therefore, if we set $\begin{align*} & & {x'}_{\nu} &= a_{\nu} + \sum_{\alpha} b_{\nu\alpha} x_{\alpha}, \qquad & &\text{(3)}\\ &\text{or} & \Delta {x'}_{\nu} &= \sum_{\alpha} b_{\nu\alpha}\, \Delta x_{\alpha}, \qquad & &\text{(3a)} \end{align*}$ [Pg 6] then the equivalence of equations (2) and (2a) is expressed as $\sum{\Delta x'_{\nu}}^{2} = \lambda \sum{\Delta x_{\nu}}^{2} \qquad\text{($\lambda$ independent of $\Delta x_{\nu}$).} \qquad\text{(2b)}$ This means that $\lambda$ must be a constant. If we set $\lambda$ = 1, then (2b) and (3a) provide the conditions $\sum_{\nu} b_{\nu\alpha} b_{\nu\beta} = \delta_{\alpha\beta}, \qquad\text{(4)}$ where $\delta_{\alpha\beta}$ = 1, or $\delta_{\alpha\beta}$ = 0, depending on whether $\alpha$ = $\beta$ or $\alpha$ ≠ $\beta$ . These conditions (4) are called orthogonality conditions, and the transformations (3), (4) are termed linear orthogonal transformations. If we require that $s^2 = \sum {\Delta x_{\nu}}^2$ equals the square of the length in every coordinate system, and if we always measure with the same units, then $\lambda$ must equal 1. Therefore, linear orthogonal transformations are the only transformations through which we can transition from one Cartesian coordinate system in our reference space to another. We observe that when applying such transformations, the equations of a straight line remain equations of a straight line. To reverse equations (3a), we multiply both sides by $b_{\nu\beta}$ and sum over all $\nu$ 's, yielding $\sum_{\nu} b_{\nu\beta}\, {\Delta x'}_{\nu} = \sum_{\nu, \alpha} b_{\nu\alpha} b_{\nu\beta}\, \Delta x_{\alpha} = \sum_{\alpha} \delta_{\alpha\beta}\, \Delta x_{\alpha} = \Delta x_{\beta} \qquad\text{(5)}$ The same coefficients, $b$ , also define the inverse substitution of $\Delta$ $x_{\nu}$ . Geometrically, $b_{\nu\alpha}$ represents the cosine of the angle between the $x'_{\nu}$ axis and the $x_{\alpha}$ axis. [Pg 7]

To sum up, we can say that in the Euclidean geometry there are (in a given space of reference) preferred systems of co-ordinates, the Cartesian systems, which transform into each other by linear orthogonal transformations. The distance $s$ between two points of our space of reference, measured by a measuring rod, is expressed in such co-ordinates in a particularly simple manner. The whole of geometry may be founded upon this conception of distance. In the present treatment, geometry is related to actual things (rigid bodies), and its theorems are statements concerning the behaviour of these things, which may prove to be true or false.

To summarize, we can say that in Euclidean geometry there are preferred coordinate systems in a given reference space, specifically the Cartesian systems, which can be transformed into one another through linear orthogonal transformations. The distance $s$ between two points in our reference space, measured by a measuring rod, is expressed in these coordinates in a straightforward way. The entire field of geometry can be based on this concept of distance. In this discussion, geometry is tied to real objects (rigid bodies), and its theorems are statements about the behavior of these objects, which may turn out to be true or false.

One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. This is satisfactory to the pure mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The question as to whether Euclidean geometry is true or not does not concern him. But for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not. That Euclidean geometry, from this point of view, affirms something more than the mere deductions derived logically from definitions may be seen from the following simple consideration.

One is usually used to studying geometry as if it's separate from real-world experience. There are benefits to isolating what's purely logical and independent of, in a way, incomplete empirical evidence. This approach satisfies the pure mathematician. They are happy as long as they can correctly deduce their theorems from axioms, meaning without logical errors. The issue of whether Euclidean geometry is true or not doesn’t interest them. However, for our purposes, it's essential to connect the fundamental concepts of geometry to natural objects; without this connection, geometry is useless to the physicist. The physicist cares about whether the theorems of geometry are true or not. The fact that Euclidean geometry, from this perspective, asserts something beyond just the logical deductions derived from definitions can be seen through the following simple consideration.

Between $n$ points of space there are $\dfrac{n(n - 1)}{2}$ distances, $s_{\mu\nu}$ ; between these and the $3n$ co-ordinates we have the relations $s_{\mu\nu}^{2} = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + .\dots$ [Pg 8]

Between $n$ points in space, there are $\dfrac{n(n - 1)}{2}$ distances, $s_{\mu\nu}$ ; between these and the $3n$ coordinates, we have the relationships $s_{\mu\nu}^{2} = \bigl(x_{1(\mu)} - x_{1(\nu)}\bigr)^{2} + \bigl(x_{2(\mu)} - x_{2(\nu)}\bigr)^{2} + .\dots$ [Pg 8]

From these $\dfrac{n(n - 1)}{2}$ equations the $3n$ co-ordinates may be eliminated, and from this elimination at least $\dfrac{n(n - 1)}{2} - 3n$ equations in the $s_{\mu\nu}$ , will result.[2] Since the $s_{\mu\nu}$ are measurable quantities, and by definition are independent of each other, these relations between the $s_{\mu\nu}$ are not necessary a priori.

From these $\dfrac{n(n - 1)}{2}$ equations, the $3n$ coordinates can be eliminated, and from this elimination, at least $\dfrac{n(n - 1)}{2} - 3n$ equations in the $s_{\mu\nu}$ will result.[2] Since the $s_{\mu\nu}$ are measurable quantities and are defined to be independent of each other, these relationships between the $s_{\mu\nu}$ are not necessary a priori.

[2]In reality there are $\dfrac{n(n - 1)}{2} - 3n + 6$ equations.

In reality, there are $\dfrac{n(n - 1)}{2} - 3n + 6$ equations.

From the foregoing it is evident that the equations of transformation (3), (4) have a fundamental significance in Euclidean geometry, in that they govern the transformation from one Cartesian system of co-ordinates to another. The Cartesian systems of co-ordinates are characterized by the property that in them the measurable distance between two points, $s$ , is expressed by the equation $s^{2} = \sum {\Delta x_{\nu}}^{2}.$

From the above, it's clear that the transformation equations (3) and (4) are fundamentally important in Euclidean geometry because they govern the conversion from one Cartesian coordinate system to another. Cartesian coordinate systems are defined by the fact that the measurable distance between two points, $s$ , is described by the equation $s^{2} = \sum {\Delta x_{\nu}}^{2}.$

If $K_{x_{\nu}}$ and $K'_{x_{\nu}}$ are two Cartesian systems of co-ordinates, then $\sum {\Delta x_{\nu}}^{2} = \sum {\Delta x'_{\nu}}^{2}.$

If $K_{x_{\nu}}$ and $K'_{x_{\nu}}$ are two Cartesian coordinate systems, then $\sum {\Delta x_{\nu}}^{2} = \sum {\Delta x'_{\nu}}^{2}.$

The right-hand side is identically equal to the left-hand side on account of the equations of the linear orthogonal transformation, and the right-hand side differs from the left-hand side only in that the $x_{\nu}$ are replaced by the $x'_{\nu}$ . This is expressed by the statement that $\sum {\Delta x_{\nu}}^{2}$ is an invariant with respect to linear orthogonal transformations. It is evident that in the Euclidean geometry only such, and all such, quantities have an objective significance, independent of the particular choice of [Pg 9] the Cartesian co-ordinates, as can be expressed by an invariant with respect to linear orthogonal transformations. This is the reason that the theory of invariants, which has to do with the laws that govern the form of invariants, is so important for analytical geometry.

The right-hand side is exactly the same as the left-hand side because of the equations involved in linear orthogonal transformations, and the right-hand side only differs from the left-hand side by the fact that the $x_{\nu}$ are replaced by the $x'_{\nu}$ . This is shown by stating that $\sum {\Delta x_{\nu}}^{2}$ is invariant under linear orthogonal transformations. It's clear that in Euclidean geometry, only these quantities, and all of them, have objective significance that is independent of the specific choice of [Pg 9] Cartesian coordinates, which can be represented by an invariant in relation to linear orthogonal transformations. This makes the theory of invariants, which relates to the laws governing the form of invariants, very important for analytical geometry.

As a second example of a geometrical invariant, consider a volume. This is expressed by $V = \iiint dx_{1}\, dx_{2}\, dx_{3}.$ By means of Jacobi's theorem we may write $\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} = \iiint \frac{\partial({x'}_{1}, {x'}_{2}, {x'}_{3})}{\partial(x_{1}, x_{2}, x_{3})}\, dx_{1}\, dx_{2}\, dx_{3}$ where the integrand in the last integral is the functional determinant of the $x'_{\nu}$ with respect to the $x_{\nu}$ , and this by (3) is equal to the determinant $|b_{\mu\nu}|$ of the coefficients of substitution, $b_{\nu\alpha}$ . If we form the determinant of the $\delta_{\mu\alpha}$ from equation (4), we obtain, by means of the theorem of multiplication of determinants, $1 = |\delta_{\alpha\beta}| = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| = |b_{\mu\nu}|^{2};\quad |b_{\mu\nu}| = ±1. \qquad\text{(6)}$ If we limit ourselves to those transformations which have the determinant +1,[3] and only these arise from continuous variations of the systems of co-ordinates, then $V$ is an invariant.

As a second example of a geometric invariant, let's look at volume. This is represented by $V = \iiint dx_{1}\, dx_{2}\, dx_{3}.$ Using Jacobi's theorem, we can express it as $\iiint {dx'}_{1}\, {dx'}_{2}\, {dx'}_{3} = \iiint \frac{\partial({x'}_{1}, {x'}_{2}, {x'}_{3})}{\partial(x_{1}, x_{2}, x_{3})}\, dx_{1}\, dx_{2}\, dx_{3}$ where the integrand in the last integral is the functional determinant of the $x'_{\nu}$ with respect to the $x_{\nu}$ , and this, according to (3), is equal to the determinant $|b_{\mu\nu}|$ of the coefficients of substitution, $b_{\nu\alpha}$ . If we take the determinant of the $\delta_{\mu\alpha}$ from equation (4), we get, using the theorem for multiplying determinants, $1 = |\delta_{\alpha\beta}| = \left| \sum_{\nu} b_{\nu\alpha} b_{\nu\beta}\right| = |b_{\mu\nu}|^{2};\quad |b_{\mu\nu}| = ±1. \qquad\text{(6)}$ If we restrict ourselves to transformations that have a determinant of +1,[3] and only those arise from continuous variations of the coordinate systems, then $V$ is an invariant.

[3]There are thus two kinds of Cartesian systems which are designated as "right-handed" and "left-handed" systems. The difference between these is familiar to every physicist and engineer. It is interesting to note that these two kinds of systems cannot be defined geometrically, but only the contrast between them.

[3]There are two types of Cartesian systems known as "right-handed" and "left-handed" systems. Every physicist and engineer is familiar with the difference between them. It's interesting to point out that these two types of systems can't be defined geometrically; only their contrast can be defined.

[Pg 10]

Invariants, however, are not the only forms by means of which we can give expression to the independence of the particular choice of the Cartesian co-ordinates. Vectors and tensors are other forms of expression. Let us express the fact that the point with the current co-ordinates $x_{\nu}$ lies upon a straight line. We have $x_{\nu} - A_{\nu} = \lambda B_{\nu} \quad (\nu \text{ from 1 to 3}).$ Without limiting the generality we can put $\sum {B_{\nu}}^{2} = 1.$

Invariants aren't the only ways to express the independence of the specific choice of Cartesian coordinates. Vectors and tensors are other ways to do this. Let's show that the point with the current coordinates $x_{\nu}$ is on a straight line. We have $x_{\nu} - A_{\nu} = \lambda B_{\nu} \quad (\nu \text{ from 1 to 3}).$ Without losing generality, we can say $\sum {B_{\nu}}^{2} = 1.$

If we multiply the equations by $b_{\beta\nu}$ (compare (3a) and (5)) and sum for all the $\nu$ 's, we get $x'_{\beta} - A'_{\beta} = \lambda B'_{\beta},$ where we have written $B'_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad A'_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}.$

If we multiply the equations by $b_{\beta\nu}$ (see (3a) and (5)) and sum over all the $\nu$ 's, we get $x'_{\beta} - A'_{\beta} = \lambda B'_{\beta},$ where we have written $B'_{\beta} = \sum_{\nu} b_{\beta\nu} B_{\nu}; \quad A'_{\beta} = \sum_{\nu} b_{\beta\nu} A_{\nu}.$

These are the equations of straight lines with respect to a second Cartesian system of co-ordinates $K$ '. They have the same form as the equations with respect to the original system of co-ordinates. It is therefore evident that straight lines have a significance which is independent of the system of co-ordinates. Formally, this depends upon the fact that the quantities ( $x_{\nu} - A_{\nu}$ ) - $\lambda B_{\nu}$ are transformed as the components of an interval, $\Delta x_{\nu}$ . The ensemble of three quantities, defined for every system of Cartesian co-ordinates, and which transform as the components of an interval, is called a vector. If the three [Pg 11] components of a vector vanish for one system of Cartesian co-ordinates, they vanish for all systems, because the equations of transformation are homogeneous. We can thus get the meaning of the concept of a vector without referring to a geometrical representation. This behaviour of the equations of a straight line can be expressed by saying that the equation of a straight line is co-variant with respect to linear orthogonal transformations.

These are the equations of straight lines related to a second Cartesian coordinate system $K$ . They are in the same format as the equations for the original coordinate system. Therefore, it’s clear that straight lines have a meaning that is independent of the coordinate system. Formally, this is because the quantities ( $x_{\nu} - A_{\nu}$ ) - $\lambda B_{\nu}$ are transformed like the components of an interval, $\Delta x_{\nu}$ . The set of three quantities, defined for every Cartesian coordinate system, which transform as the components of an interval, is called a vector. If the three components of a vector are zero for one Cartesian coordinate system, they will be zero for all systems, because the transformation equations are homogeneous. Thus, we can understand the concept of a vector without needing to refer to a geometric representation. This behavior of the equations of a straight line can be summed up by saying that the equation of a straight line is co-variant with respect to linear orthogonal transformations.

We shall now show briefly that there are geometrical entities which lead to the concept of tensors. Let $P_{0}$ be the centre of a surface of the second degree, $P$ any point on the surface, and $\xi_{\nu}$ the projections of the interval $P_{0}P$ upon the co-ordinate axes. Then the equation of the surface is $\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1.$ In this, and in analogous cases, we shall omit the sign of summation, and understand that the summation is to be carried out for those indices that appear twice. We thus write the equation of the surface $a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1.$ The quantities $a_{\mu\nu}$ determine the surface completely, for a given position of the centre, with respect to the chosen system of Cartesian co-ordinates. From the known law of transformation for the $\xi_{\nu}$ (3a) for linear orthogonal transformations, we easily find the law of transformation for the $a_{\mu\nu}$ :[4] ${a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}.$

We will now briefly demonstrate that there are geometric entities that lead to the idea of tensors. Let $P_{0}$ be the center of a second-degree surface, $P$ any point on that surface, and $\xi_{\nu}$ the projections of the interval $P_{0}P$ onto the coordinate axes. Then the equation of the surface is $\sum a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1.$ In this and similar cases, we will omit the summation sign and understand that the summation is to be performed for the indices that appear twice. Thus, we express the equation of the surface as $a_{\mu\nu} \xi_{\mu} \xi_{\nu} = 1.$ The quantities $a_{\mu\nu}$ completely define the surface for a given position of the center with respect to the chosen Cartesian coordinate system. From the known transformation law for the $\xi_{\nu}$ (3a) for linear orthogonal transformations, we can easily find the transformation law for the $a_{\mu\nu}$ :[4] ${a'}_{\sigma\tau} = b_{\sigma\mu} b_{\tau\nu} a_{\mu\nu}.$

[4]The equation ${a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1$ may, by (5), be replaced by $a'_{\sigma\tau}$ $b_{\mu\sigma}$ $b_{\nu\tau}$ $\xi_{\mu}$ $\xi_{\nu}$ = 1, from which the result stated immediately follows.

[4]The equation ${a'}_{\sigma\tau} {\xi'}_{\sigma} {\xi'}_{\tau} = 1$ can be replaced by (5) $a'_{\sigma\tau}$ $b_{\mu\sigma}$ $b_{\nu\tau}$ $\xi_{\mu}$ $\xi_{\nu}$ = 1, which leads directly to the stated result.

[Pg 12]

This transformation is homogeneous and of the first degree in the $a_{\mu\nu}$ . On account of this transformation, the $a_{\mu\nu}$ , are called components of a tensor of the second rank (the latter on account of the double index). If all the components, $a_{\mu\nu}$ , of a tensor with respect to any system of Cartesian co-ordinates vanish, they vanish with respect to every other Cartesian system. The form and the position of the surface of the second degree is described by this tensor ( $a$ ).

This transformation is uniform and of the first degree in the $a_{\mu\nu}$ . Because of this transformation, the $a_{\mu\nu}$ , are referred to as components of a second-rank tensor (the latter due to the double index). If all the components, $a_{\mu\nu}$ , of a tensor with respect to any system of Cartesian coordinates are zero, they will also be zero for every other Cartesian system. The shape and position of the surface of the second degree is described by this tensor ( $a$ ).

Analytic tensors of higher rank (number of indices) may be defined. It is possible and advantageous to regard vectors as tensors of rank 1, and invariants (scalars) as tensors of rank 0. In this respect, the problem of the theory of invariants may be so formulated: according to what laws may new tensors be formed from given tensors? We shall consider these laws now, in order to be able to apply them later. We shall deal first only with the properties of tensors with respect to the transformation from one Cartesian system to another in the same space of reference, by means of linear orthogonal transformations. As the laws are wholly independent of the number of dimensions, we shall leave this number, $n$ , indefinite at first.

Analytic tensors of higher rank (the number of indices) can be defined. It’s useful to think of vectors as tensors of rank 1 and invariants (scalars) as tensors of rank 0. In this way, we can frame the problem of the theory of invariants like this: what laws allow for the creation of new tensors from existing ones? We’ll examine these laws now so we can use them later. We'll start by focusing only on the properties of tensors regarding the transformation from one Cartesian system to another within the same reference space, using linear orthogonal transformations. Since the laws are completely independent of the number of dimensions, we will initially leave this number, $n$ , unspecified.

Definition. If a figure is defined with respect to every system of Cartesian co-ordinates in a space of reference of $n$ dimensions by the $n^{\alpha}$ numbers $A_{\mu\nu\rho\ldots}$ ( $\alpha$ = number of indices), then these numbers are the components of a tensor of rank $\alpha$ if the transformation law is ${A'}_{\mu'\nu'\rho'}\ldots = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho}\ldots A_{\mu\nu\rho}.\ldots \qquad\text{(7)}$ [Pg 13]

Definition. If a figure is defined with respect to every Cartesian coordinate system in a reference space with $n$ dimensions by the $n^{\alpha}$ numbers $A_{\mu\nu\rho\ldots}$ ( $\alpha$ = number of indices), then these numbers are the components of a tensor of rank $\alpha$ ${A'}_{\mu'\nu'\rho'}\ldots = b_{\mu'\mu} b_{\nu'\nu} b_{\rho'\rho}\ldots A_{\mu\nu\rho}.\ldots \qquad\text{(7)}$ [Pg 13]

Remark. From this definition it follows that $A_{\mu\nu\rho}\ldots = B_{\mu} C_{\nu} D_{\rho}\dots \qquad\text{(8)}$ is an invariant, provided that ( $B$ ), ( $C$ ), ( $D$ ) ... are vectors. Conversely, the tensor character of ( $A$ ) may be inferred, if it is known that the expression (8) leads to an invariant for an arbitrary choice of the vectors ( $B$ ), ( $C$ ), etc.

Note. From this definition, it follows that $A_{\mu\nu\rho}\ldots = B_{\mu} C_{\nu} D_{\rho}\dots \qquad\text{(8)}$ is an invariant, as long as ( $B$ ), ( $C$ ), ( $D$ ) ... are vectors. On the other hand, the tensor nature of ( $A$ ) can be deduced if it is known that the expression (8) results in an invariant for any choice of the vectors ( $B$ ), ( $C$ ), etc.

Addition and Subtraction. By addition and subtraction of the corresponding components of tensors of the same rank, a tensor of equal rank results: $A_{\mu\nu\rho}\ldots ± B_{\mu\nu\rho}\ldots = C_{\mu\nu\rho}\ldots. \qquad\text{(9)}$ The proof follows from the definition of a tensor given above.

Addition and Subtraction. When you add or subtract the corresponding elements of tensors of the same rank, you get a tensor of the same rank: $A_{\mu\nu\rho}\ldots ± B_{\mu\nu\rho}\ldots = C_{\mu\nu\rho}\ldots. \qquad\text{(9)}$ This is proven by the tensor definition provided above.

Multiplication. From a tensor of rank $\alpha$ and a tensor of rank $\beta$ we may obtain a tensor of rank $\alpha$ + $\beta$ by multiplying all the components of the first tensor by all the components of the second tensor: $T_{\mu\nu\rho}\ldots_{\alpha\beta}\ldots = A_{\mu\nu\rho}\ldots B_{\alpha\beta\gamma}.\ldots \qquad\text{(10)}$

Multiplication. From a tensor of rank $\alpha$ and a tensor of rank $\beta$ we can get a tensor of rank $\alpha$ + $\beta$ by multiplying all the components of the first tensor with all the components of the second tensor: $T_{\mu\nu\rho}\ldots_{\alpha\beta}\ldots = A_{\mu\nu\rho}\ldots B_{\alpha\beta\gamma}.\ldots \qquad\text{(10)}$

Contraction. A tensor of rank $\alpha$ - 2 may be obtained from one of rank $\alpha$ by putting two definite indices equal to each other and then summing for this single index: $T_{\rho}\ldots = A_{\mu\mu\rho}\ldots ( = \sum_{\mu} A_{\mu\mu\rho}\ldots). \qquad\text{(11)}$

Contraction. A tensor of rank $\alpha$ - 2 can be derived from one of rank $\alpha$ by setting two specific indices equal to each other and then summing over this single index: $T_{\rho}\ldots = A_{\mu\mu\rho}\ldots ( = \sum_{\mu} A_{\mu\mu\rho}\ldots). \qquad\text{(11)}$

[Pg 14]

The proof is $\begin{align*} {A'}_{\mu\mu\rho\cdots} = b_{\mu\alpha} b_{\mu\beta} b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} = \delta_{\alpha\beta} & b_{\rho\gamma\cdots} A_{\alpha\beta\gamma\cdots} \\ {} ={} & b_{\rho\gamma\cdots} A_{\alpha\alpha\gamma\cdots}. \end{align*}$

The proof is

In addition to these elementary rules of operation there is also the formation of tensors by differentiation ("erweiterung"): $T_{\mu\nu\rho\cdots\alpha} = \frac{\partial A_{\mu\nu\rho\cdots}}{\partial x_{\alpha}}. \qquad\text{(12)}$

In addition to these basic operational rules, there is also the creation of tensors through differentiation ("extension"): $T_{\mu\nu\rho\cdots\alpha} = \frac{\partial A_{\mu\nu\rho\cdots}}{\partial x_{\alpha}}. \qquad\text{(12)}$

New tensors, in respect to linear orthogonal transformations, may be formed from tensors according to these rules of operation.

New tensors, regarding linear orthogonal transformations, can be created from tensors based on these operational rules.

Symmetrical Properties of Tensors. Tensors are called symmetrical or skew-symmetrical in respect to two of their indices, $\mu$ and $\nu$ , if both the components which result from interchanging the indices $\mu$ and $\nu$ are equal to each other or equal with opposite signs. $\begin{alignat*}{2} &\text{Condition for symmetry:} & A_{\mu\nu\rho} &= A_{{\nu\mu}\rho}. \\ &\text{Condition for skew-symmetry:}\quad & A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ \end{alignat*}$

Symmetrical Properties of Tensors. Tensors are described as symmetrical or skew-symmetrical regarding two of their indices, $\mu$ and $\nu$ , if the components obtained by swapping the indices $\mu$ and $\nu$ are either equal or equal in magnitude but opposite in sign. $\begin{alignat*}{2} &\text{Condition for symmetry:} & A_{\mu\nu\rho} &= A_{{\nu\mu}\rho}. \\ &\text{Condition for skew-symmetry:}\quad & A_{\mu\nu\rho} &= -A_{\nu\mu\rho}. \\ \end{alignat*}$

Theorem. The character of symmetry or skew-symmetry exists independently of the choice of co-ordinates, and in this lies its importance. The proof follows from the equation defining tensors.

Theorem. The quality of symmetry or skew-symmetry exists regardless of the choice of coordinates, and that’s what makes it significant. The proof is derived from the equation that defines tensors.

Special Tensors.

Special Tensors.

I. The quantities $\delta_{\rho\sigma}$ (4) are tensor components (fundamental tensor). [Pg 15]

I. The quantities $\delta_{\rho\sigma}$ (4) are components of a tensor (fundamental tensor). [Pg 15]

Proof. If in the right-hand side of the equation of transformation $A'_{\mu\nu}$ = $b_{\mu\alpha}$ $b_{\nu\beta}$ $A_{\alpha\beta}$ , we substitute for $A_{\alpha\beta}$ the quantities $\delta_{\alpha\beta}$ (which are equal to 1 or 0 according as $\alpha$ = $\beta$ or $\alpha$ ≠ $\beta$ ), we get ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}.$ The justification for the last sign of equality becomes evident if one applies (4) to the inverse substitution (5).

Proof. If on the right side of the transformation equation $A'_{\mu\nu}$ = $b_{\mu\alpha}$ $b_{\nu\beta}$ $A_{\alpha\beta}$ , we replace $A_{\alpha\beta}$ with the quantities $\delta_{\alpha\beta}$ (which equal 1 or 0 depending on whether $\alpha$ = $\beta$ or $\alpha$ ≠ $\beta$ ), we obtain ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\alpha} = \delta_{\mu\nu}.$ The reason for this final equality is clear when one applies (4) to the reverse substitution (5).

II. There is a tensor ( $\delta_{\mu\nu\rho\ldots}$ ) skew-symmetrical with respect to all pairs of indices, whose rank is equal to the number of dimensions, $n$ , and whose components are equal to +1 or -1 according as $\mu\nu\rho\ldots$ is an even or odd permutation of 1 2 3....

II. There is a tensor ( $\delta_{\mu\nu\rho\ldots}$ ) that is skew-symmetrical with respect to all pairs of indices, with a rank equal to the number of dimensions, $n$ , and its components are either +1 or -1 depending on whether $\mu\nu\rho\ldots$ is an even or odd permutation of 1 2 3....

The proof follows with the aid of the theorem proved above $|b_{\rho\sigma}| = 1$

The proof follows with the help of the theorem demonstrated above $|b_{\rho\sigma}| = 1$

These few simple theorems form the apparatus from the theory of invariants for building the equations of pre-relativity physics and the theory of special relativity.

These few simple theorems are the tools from the theory of invariants for developing the equations of pre-relativity physics and the theory of special relativity.

We have seen that in pre-relativity physics, in order to specify relations in space, a body of reference, or a space of reference, is required, and, in addition, a Cartesian system of co-ordinates. We can fuse both these concepts into a single one by thinking of a Cartesian system of co-ordinates as a cubical frame-work formed of rods each of unit length. The co-ordinates of the lattice points of this frame are integral numbers. It follows from the fundamental relation $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} \qquad\text{(13)}$ that the members of such a space-lattice are all of unit length. To specify relations in time, we require in addition a standard clock placed at the origin of our Cartesian system of co-ordinates [Pg 16] or frame of reference. If an event takes place anywhere we can assign to it three co-ordinates, $x_{\nu}$ , and a time $t$ , as soon as we have specified the time of the clock at the origin which is simultaneous with the event. We therefore give an objective significance to the statement of the simultaneity of distant events, while previously we have been concerned only with the simultaneity of two experiences of an individual. The time so specified is at all events independent of the position of the system of co-ordinates in our space of reference, and is therefore an invariant with respect to the transformation (3).

We’ve seen that in pre-relativity physics, to define relationships in space, we need a reference body or reference space, along with a Cartesian coordinate system. We can combine these two ideas by imagining a Cartesian coordinate system as a cubic framework made of rods, each one unit long. The coordinates of the points in this framework are whole numbers. From the fundamental relationship $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} \qquad\text{(13)}$ it follows that the points in such a space-lattice are all unit length. To describe relationships in time, we also need a standard clock set at the origin of our Cartesian coordinate system [Pg 16] or reference frame. When an event occurs anywhere, we can assign it three coordinates, $x_{\nu}$ , and a time $t$ , as soon as we define the time on the clock at the origin that is simultaneous with the event. Thus, we give an objective meaning to the statement of simultaneity of distant events, whereas before we were only concerned with the simultaneity of two experiences of a single individual. The time we specify is independent of the position of the coordinate system in our reference space, making it an invariant with respect to transformation (3).

It is postulated that the system of equations expressing the laws of pre-relativity physics is co-variant with respect to the transformation (3), as are the relations of Euclidean geometry. The isotropy and homogeneity of space is expressed in this way.[5] We shall now consider some of the more important equations of physics from this point of view.

It is suggested that the system of equations representing the laws of pre-relativity physics is invariant under the transformation (3), just like the principles of Euclidean geometry. This shows the uniformity and consistency of space.[5] Now, we will look at some of the key equations in physics from this perspective.

[5]The laws of physics could be expressed, even in case there were a unique direction in space, in such a way as to be co-variant with respect to the transformation (3); but such an expression would in this case be unsuitable. If there were a unique direction in space it would simplify the description of natural phenomena to orient the system of co-ordinates in a definite way in this direction. But if, on the other hand, there is no unique direction in space it is not logical to formulate the laws of nature in such a way as to conceal the equivalence of systems of co-ordinates that are oriented differently. We shall meet with this point of view again in the theories of special and general relativity.

[5]The laws of physics can be described, even if there is a specific direction in space, in a way that is consistent with the transformation (3); however, that would not be appropriate in this scenario. If there were a specific direction in space, it would make it easier to describe natural phenomena by aligning the coordinate system in that direction. On the other hand, if there is no specific direction in space, it doesn’t make sense to establish the laws of nature in a manner that hides the equivalence of coordinate systems aligned differently. We will encounter this perspective again in the theories of special and general relativity.

The equations of motion of a material particle are $m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}; \qquad\text{(14)}$ ( $dx_{\nu}$ ) is a vector; $dt$ , and therefore also $\dfrac{1}{dt}$ , an invariant; thus [Pg 17] ( $\dfrac{dx_{\nu}}{dt}$ ) is a vector; in the same way it may be shown that ( $\dfrac{d^{2} x_{\nu}}{dt^{2}}$ ) is a vector. In general, the operation of differentiation with respect to time does not alter the tensor character. Since $m$ is an invariant (tensor of rank 0), $(m\dfrac{d^{2} x_{\nu}}{dt^{2}}$ ) is a vector, or tensor of rank 1 (by the theorem of the multiplication of tensors). If the force ( $X_{\nu}$ ) has a vector character, the same holds for the difference ( $m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu})$ . These equations of motion are therefore valid in every other system of Cartesian co-ordinates in the space of reference. In the case where the forces are conservative we can easily recognize the vector character of ( $X_{\nu}$ ). For a potential energy, $\Phi$ , exists, which depends only upon the mutual distances of the particles, and is therefore an invariant. The vector character of the force, $X_{\nu}$ = $-\dfrac{\partial \Phi}{\partial x_{\nu}}$ , is then a consequence of our general theorem about the derivative of a tensor of rank 0.

The equations of motion for a material particle are $m \frac{d^{2} x_{\nu}}{dt^{2}} = X_{\nu}; \qquad\text{(14)}$ ( $dx_{\nu}$ ) is a vector; $dt$ , and so is $\dfrac{1}{dt}$ , which is an invariant; thus [Pg 17] ( $\dfrac{dx_{\nu}}{dt}$ ) is a vector; similarly, it can be shown that ( $\dfrac{d^{2} x_{\nu}}{dt^{2}}$ ) is also a vector. In general, differentiating with respect to time doesn't change the tensor nature. Since $m$ is an invariant (a tensor of rank 0), $(m\dfrac{d^{2} x_{\nu}}{dt^{2}}$ is a vector, or a tensor of rank 1 (by the tensor multiplication theorem). If the force ( $X_{\nu}$ ) has a vector nature, the same applies to the difference ( $m\dfrac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu})$ . Therefore, these equations of motion are valid in any other system of Cartesian coordinates in the reference space. In cases where the forces are conservative, we can easily see the vector nature of ( $X_{\nu}$ ). For potential energy, $\Phi$ , exists, which only depends on the distances between the particles and is therefore an invariant. The vector nature of the force, $X_{\nu}$ = $-\dfrac{\partial \Phi}{\partial x_{\nu}}$ , is then a result of our general theorem about the derivative of a tensor of rank 0.

Multiplying by the velocity, a tensor of rank 1, we obtain the tensor equation $\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0.$ By contraction and multiplication by the scalar $dt$ we obtain the equation of kinetic energy $d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\ dx_{\nu}.$ [Pg 18]

Multiplying by the velocity, which is a rank 1 tensor, we get the tensor equation $\left(m\frac{d^{2} x_{\nu}}{dt^{2}} - X_{\nu}\right) \frac{dx_{\nu}}{dt} = 0.$ By contracting and multiplying by the scalar $dt$ we derive the kinetic energy equation $d\left(\frac{mq^{2}}{2}\right) = X_{\nu}\ dx_{\nu}.$ [Pg 18]

If $\xi_{\nu}$ denotes the difference of the co-ordinates of the material particle and a point fixed in space, then the $\xi_{\nu}$ have the character of vectors. We evidently have $\dfrac{d^{2} x_{\nu}}{dt^{2}}$ = $\dfrac{d^{2} \xi_{\nu}}{dt^{2}}$ , so that the equations of motion of the particle may be written $m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0.$

If $\xi_{\nu}$ represents the difference between the coordinates of a material particle and a fixed point in space, then the $\xi_{\nu}$ act like vectors. Clearly, we have $\dfrac{d^{2} x_{\nu}}{dt^{2}}$ = $\dfrac{d^{2} \xi_{\nu}}{dt^{2}}$ so that the equations of motion for the particle can be written as $m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu} = 0.$

Multiplying this equation by $\xi_{\mu}$ we obtain a tensor equation $\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0.$

Multiplying this equation by $\xi_{\mu}$ we get a tensor equation $\left(m\frac{d^{2} \xi_{\nu}}{dt^{2}} - X_{\nu}\right) \xi_{\mu} = 0.$

Contracting the tensor on the left and taking the time average we obtain the virial theorem, which we shall not consider further. By interchanging the indices and subsequent subtraction, we obtain, after a simple transformation, the theorem of moments, $\frac{d}{dt} \biggl[ m \biggl(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\biggr)\biggr] = \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}. \qquad\text{(15)}$

Contracting the tensor on the left and averaging over time gives us the virial theorem, which we won't discuss further. By swapping the indices and then subtracting, we get, after a straightforward transformation, the theorem of moments, $\frac{d}{dt} \biggl[ m \biggl(\xi_{\mu} \frac{d\xi_{\nu}}{dt} - \xi_{\nu} \frac{d\xi_{\mu}}{dt}\biggr)\biggr] = \xi_{\mu} X_{\nu} - \xi_{\nu} X_{\mu}. \qquad\text{(15)}$

It is evident in this way that the moment of a vector is not a vector but a tensor. On account of their skew-symmetrical character there are not nine, but only three independent equations of this system. The possibility of replacing skew-symmetrical tensors of the second rank in space of three dimensions by vectors depends upon the formation of the vector $A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}.$ [Pg 19]

It is clear that the moment of a vector is not a vector but a tensor. Because of their skew-symmetrical nature, there are not nine, but only three independent equations in this system. The ability to replace skew-symmetrical tensors of the second rank in three-dimensional space with vectors is based on the formation of the vector $A_{\mu} = \frac{1}{2} A_{\sigma\tau} \delta_{\sigma\tau\mu}.$ [Pg 19]

If we multiply the skew-symmetrical tensor of rank 2 by the special skew-symmetrical tensor $\delta$ introduced above, and contract twice, a vector results whose components are numerically equal to those of the tensor. These are the so-called axial vectors which transform differently, from a right-handed system to a left-handed system, from the $\Delta x_{\nu}$ . There is a gain in picturesqueness in regarding a skew-symmetrical tensor of rank 2 as a vector in space of three dimensions, but it does not represent the exact nature of the corresponding quantity so well as considering it a tensor.

If we multiply the skew-symmetric tensor of rank 2 by the special skew-symmetric tensor $\delta$ mentioned earlier, and contract it twice, we get a vector whose components are numerically equal to those of the tensor. These are known as axial vectors, which transform differently from a right-handed system to a left-handed system, from the $\Delta x_{\nu}$ . Viewing a skew-symmetric tensor of rank 2 as a vector in three-dimensional space adds a touch of vividness, but it doesn't capture the true nature of the corresponding quantity as accurately as considering it a tensor.

We consider next the equations of motion of a continuous medium. Let $\rho$ be the density, $u_{\nu}$ the velocity components considered as functions of the co-ordinates and the time, $X_{\nu}$ the volume forces per unit of mass, and $p_{\nu\sigma}$ the stresses upon a surface perpendicular to the a-axis in the direction of increasing $x_{\nu}$ . Then the equations of motion are, by Newton's law, $\rho \frac{du_{\nu}}{dt} = -\frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + \rho X_{\nu},$ in which $\dfrac{du_{\nu}}{dt}$ is the acceleration of the particle which at time $t$ has the co-ordinates $x_{\nu}$ . If we express this acceleration by partial differential coefficients, we obtain, after dividing by $\rho$ , $\frac{\partial u_{\nu}}{dt} + \frac{\partial u_{\nu}}{dx_{\sigma}} u_{\sigma} = -\frac{1}{\rho}\, \frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + X_{\nu}. \qquad\text{(16)}$

We will now look at the equations of motion for a continuous medium. Let $\rho$ represent the density, $u_{\nu}$ the velocity components as functions of the coordinates and time, $X_{\nu}$ the volume forces per unit mass, and $p_{\nu\sigma}$ the stresses on a surface perpendicular to the a-axis in the direction of increasing $x_{\nu}$ . According to Newton's law, the equations of motion are given by: $\rho \frac{du_{\nu}}{dt} = -\frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + \rho X_{\nu},$ where $\dfrac{du_{\nu}}{dt}$ is the acceleration of the particle that, at time $t$ has the coordinates $x_{\nu}$ . If we express this acceleration using partial differential coefficients, we get, after dividing by $\rho$ , $\frac{\partial u_{\nu}}{dt} + \frac{\partial u_{\nu}}{dx_{\sigma}} u_{\sigma} = -\frac{1}{\rho}\, \frac{\partial p_{\nu\sigma}}{\partial x_{\sigma}} + X_{\nu}. \qquad\text{(16)}$

We must show that this equation holds independently of the special choice of the Cartesian system of co-ordinates. ( $u_{\nu}$ ) is a vector, and therefore $\dfrac{\partial u_{\nu}}{\partial t}$ is also a vector. $\dfrac{\partial u_{\nu}}{\partial x_{\sigma}}$ [Pg 20] is a tensor of rank 2, $\dfrac{\partial u_{\nu}}{\partial x_{\sigma}} u_{\tau}$ is a tensor of rank 3. The second term on the left results from contraction in the indices $\sigma$ , $\tau$ . The vector character of the second term on the right is obvious. In order that the first term on the right may also be a vector it is necessary for $p_{\nu\sigma}$ to be a tensor. Then by differentiation and contraction $\dfrac{\partial p_{\nu\sigma}}{\partial x_{\sigma}}$ results, and is therefore a vector, as it also is after multiplication by the reciprocal scalar $\dfrac{1}{\rho}$ . That $p_{\nu\sigma}$ is a tensor, and therefore transforms according to the equation ${p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta},$ is proved in mechanics by integrating this equation over an infinitely small tetrahedron. It is also proved there, by application of the theorem of moments to an infinitely small parallelopipedon, that $p_{\nu\sigma} = p_{\sigma\nu}$ , and hence that the tensor of the stress is a symmetrical tensor. From what has been said it follows that, with the aid of the rules given above, the equation is co-variant with respect to orthogonal transformations in space (rotational transformations); and the rules according to which the quantities in the equation must be transformed in order that the equation may be co-variant also become evident.

We need to demonstrate that this equation is valid regardless of the specific Cartesian coordinate system we choose. ( $u_{\nu}$ ) is a vector, which means $\dfrac{\partial u_{\nu}}{\partial t}$ is also a vector. $\dfrac{\partial u_{\nu}}{\partial x_{\sigma}}$ [Pg 20] is a rank 2 tensor, and $\dfrac{\partial u_{\nu}}{\partial x_{\sigma}} u_{\tau}$ is a rank 3 tensor. The second term on the left is derived from the contraction of the indices $\sigma$ , $\tau$ . The vector nature of the second term on the right is clear. For the first term on the right to also be a vector, $p_{\nu\sigma}$ must be a tensor. Thus, through differentiation and contraction, we get $\dfrac{\partial p_{\nu\sigma}}{\partial x_{\sigma}}$ , which is a vector, and it remains a vector when multiplied by the reciprocal scalar $\dfrac{1}{\rho}$ . The fact that $p_{\nu\sigma}$ is a tensor, which transforms according to the equation ${p'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} p_{\alpha\beta},$ is demonstrated in mechanics by integrating this equation over an infinitesimally small tetrahedron. It is also proven there, using the moment theorem on an infinitesimally small parallelepiped, that $p_{\nu\sigma} = p_{\sigma\nu}$ , which shows that the stress tensor is symmetric. From this, we can conclude that, with the rules provided earlier, the equation is covariant in terms of orthogonal transformations in space (rotational transformations); and the rules for transforming the quantities in the equation to ensure that it remains covariant also become clear.

The co-variance of the equation of continuity, $\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u_{\nu})}{\partial x_{\nu}} = 0, \qquad\text{(17)}$ requires, from the foregoing, no particular discussion.

The co-variance of the continuity equation, $\frac{\partial\rho}{\partial t} + \frac{\partial(\rho u_{\nu})}{\partial x_{\nu}} = 0, \qquad\text{(17)}$ doesn't need any special discussion based on what we've covered.

We shall also test for co-variance the equations which express the dependence of the stress components upon the properties of [Pg 21] the matter, and set up these equations for the case of a compressible viscous fluid with the aid of the conditions of co-variance. If we neglect the viscosity, the pressure, $p$ , will be a scalar, and will depend only upon the density and the temperature of the fluid. The contribution to the stress tensor is then evidently $p \delta_{\mu\nu}$ in which $\delta_{\mu\nu}$ is the special symmetrical tensor. This term will also be present in the case of a viscous fluid. But in this case there will also be pressure terms, which depend upon the space derivatives of the $u_{\nu}$ . We shall assume that this dependence is a linear one. Since these terms must be symmetrical tensors, the only ones which enter will be $\alpha\left(\frac{\partial u_{\mu}}{\partial x_{\nu}} + \frac{\partial u_{\nu}}{\partial x_{\mu}}\right) + \beta\delta_{\mu\nu} \frac{\partial u_{\alpha}}{\partial x_{\alpha}}$ (for $\dfrac{\partial u_{\alpha}}{\partial x_{\alpha}}$ is a scalar). For physical reasons (no slipping) it is assumed that for symmetrical dilatations in all directions, i.e. when $\frac{\partial u_{1}}{\partial x_{1}} = \frac{\partial u_{2}}{\partial x_{2}} = \frac{\partial u_{3}}{\partial x_{3}};\quad \frac{\partial u_{1}}{\partial x_{2}}, \text{ etc.,} = 0,$ there are no frictional forces present, from which it follows that $\beta$ = $-\dfrac{2}{3}\alpha$ . If only $\dfrac{\partial u_{1}}{\partial x_{3}}$ is different from zero, let $p_{31} = -\alpha \dfrac{\partial u_{1}}{\partial x_{3}}$ , by which $\alpha$ is determined. We then obtain for the complete stress tensor, $p_{\mu\nu} = p \delta_{\mu\nu} - \alpha \biggl[ \biggl(\frac{\partial u_{\mu}}{\partial x_{\nu}} + \frac{\partial u_{\nu}}{\partial x_{\mu}}\biggr) - \frac{2}{3} \biggl(\frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}}\biggr)\delta_{\mu\nu} \biggr]. \qquad\text{(18)}$ [Pg 22]

We will also examine the co-variance of the equations that show how the stress components depend on the properties of the matter, and we will develop these equations for a compressible viscous fluid using the conditions of co-variance. If we ignore viscosity, the pressure, $p$ , will be a scalar, depending only on the density and temperature of the fluid. The contribution to the stress tensor is then clearly $p \delta_{\mu\nu}$ , where $\delta_{\mu\nu}$ is the special symmetrical tensor. This term will also exist for a viscous fluid. However, in this case, there will be pressure terms that depend on the spatial derivatives of $u_{\nu}$ . We will assume that this dependence is linear. Since these terms must be symmetrical tensors, the only ones included will be $\alpha\left(\frac{\partial u_{\mu}}{\partial x_{\nu}} + \frac{\partial u_{\nu}}{\partial x_{\mu}}\right) + \beta\delta_{\mu\nu} \frac{\partial u_{\alpha}}{\partial x_{\alpha}}$ (since $\dfrac{\partial u_{\alpha}}{\partial x_{\alpha}}$ is a scalar). For physical reasons (no slipping), it is assumed that for symmetrical expansions in all directions, i.e., when $\frac{\partial u_{1}}{\partial x_{1}} = \frac{\partial u_{2}}{\partial x_{2}} = \frac{\partial u_{3}}{\partial x_{3}};\quad \frac{\partial u_{1}}{\partial x_{2}}, \text{ etc.,} = 0,$ there are no frictional forces, which implies that $\beta$ = $-\dfrac{2}{3}\alpha$ . If only $\dfrac{\partial u_{1}}{\partial x_{3}}$ is different from zero, let $p_{31} = -\alpha \dfrac{\partial u_{1}}{\partial x_{3}}$ , which will determine $\alpha$ . We then obtain for the complete stress tensor, $p_{\mu\nu} = p \delta_{\mu\nu} - \alpha \biggl[ \biggl(\frac{\partial u_{\mu}}{\partial x_{\nu}} + \frac{\partial u_{\nu}}{\partial x_{\mu}}\biggr) - \frac{2}{3} \biggl(\frac{\partial u_{1}}{\partial x_{1}} + \frac{\partial u_{2}}{\partial x_{2}} + \frac{\partial u_{3}}{\partial x_{3}}\biggr)\delta_{\mu\nu} \biggr]. \qquad\text{(18)}$ [Pg 22]

The heuristic value of the theory of invariants, which arises from the isotropy of space (equivalence of all directions), becomes evident from this example.

The practical value of the theory of invariants, which comes from the isotropy of space (the equivalence of all directions), is clear from this example.

We consider, finally, Maxwell's equations in the form which are the foundation of the electron theory of Lorentz.

We finally look at Maxwell's equations in their form, which are the basis of Lorentz's electron theory.

$\begin{align*} \left. \begin{aligned} &\begin{alignedat}{3} \frac{\partial h_{3}}{\partial x_{2}} &- \frac{\partial h_{2}}{\partial x_{3}} &&= \frac{1}{c}\, \frac{\partial e_{1}}{\partial t} &&+ \frac{1}{c}\, i_{1}, \qquad \\ \frac{\partial h_{1}}{\partial x_{3}} &- \frac{\partial h_{3}}{\partial x_{1}} &&= \frac{1}{c}\, \frac{\partial e_{2}}{\partial t} &&+ \frac{1}{c}\, i_{2}, \\ \frac{\partial h_{2}}{\partial x_{1}} &- \frac{\partial h_{1}}{\partial x_{2}} &&= \frac{1}{c}\, \frac{\partial e_{3}}{\partial t} &&+ \frac{1}{c}\, i_{3}, \end{alignedat} \\ & \frac{\partial e_{1}}{\partial x_{1}} + \frac{\partial e_{2}}{\partial x_{2}} + \frac{\partial e_{3}}{\partial x_{3}} = \rho; \end{aligned} \right\} \qquad\text{(19)}\\ \left. \begin{aligned} &\begin{alignedat}{2} \frac{\partial e_{3}}{\partial x_{2}} &- \frac{\partial e_{2}}{\partial x_{3}} &&= -\frac{1}{c}\, \frac{\partial h_{1}}{\partial t}, \qquad \\ \frac{\partial e_{1}}{\partial x_{3}} &- \frac{\partial e_{3}}{\partial x_{1}} &&= -\frac{1}{c}\, \frac{\partial h_{2}}{\partial t}, \\ \frac{\partial e_{2}}{\partial x_{1}} &- \frac{\partial e_{1}}{\partial x_{2}} &&= -\frac{1}{c}\, \frac{\partial h_{3}}{\partial t}, \end{alignedat} \\ & \frac{\partial h_{1}}{\partial x_{1}} + \frac{\partial h_{2}}{\partial x_{2}} + \frac{\partial h_{3}}{\partial x_{3}} = 0. \end{aligned} \right\} \qquad \text{(20)} \end{align*}$

$\mathbf i$ is a vector, because the current density is defined as the density of electricity multiplied by the vector velocity of the electricity. According to the first three equations it is evident that $\mathbf e$ is also to be regarded as a vector. Then $\mathbf h$ cannot be regarded as a vector.[6] The equations may, however, easily be [Pg 23] interpreted if $\mathbf h$ is regarded as a skew-symmetrical tensor of the second rank. In this sense, we write $h_{23}$ , $h_{31}$ , $h_{12}$ in place of $h_{1}$ , $h_{2}$ , $h_{3}$ respectively. Paying attention to the skew-symmetry of $h_{\mu\nu}$ , the first three equations of (19) and (20) may be written in the form $\begin{gather*} \frac{\partial h_{\mu\nu}}{\partial x_{\nu}} = \frac{1}{c}\, \frac{\partial e_{\mu}}{\partial t} + \frac{1}{c} i_{\mu}, \qquad & &\text{(19a)} \\ \frac{\partial e_{\mu}}{\partial x_{\nu}} - \frac{\partial e_{\nu}}{\partial x_{\mu}} = +\frac{1}{c}\, \frac{\partial h_{\mu\nu}}{\partial t}. \qquad & &\text{(20a)} \end{gather*}$ In contrast to $\mathbf e$ , $\mathbf h$ appears as a quantity which has the same type of symmetry as an angular velocity. The divergence equations then take the form $\begin{gather*} \frac{\partial e_{\nu}}{\partial x_{\nu}} = \rho\, \qquad & &\text{(19b)} \\ \frac{\partial h_{\mu\nu}}{\partial x_{\rho}} + \frac{\partial h_{\nu\rho}}{\partial x_{\mu}} + \frac{\partial h_{\rho\mu}}{\partial x_{\nu}} = 0. \qquad & &\text{(20b)} \end{gather*}$ The last equation is a skew-symmetrical tensor equation of the third rank (the skew-symmetry of the left-hand side with respect to every pair of indices may easily be proved, if attention is paid to the skew-symmetry of $h_{\mu\nu}$ ). This notation is more natural than the usual one, because, in contrast to the latter, it is applicable to Cartesian left-handed systems as well as to right-handed systems without change of sign. [Pg 24]

$\mathbf i$ is a vector because current density is defined as the electricity density multiplied by the vector velocity of the electricity. Based on the first three equations, it’s clear that $\mathbf e$ should also be considered a vector. Therefore, $\mathbf h$ cannot be treated as a vector.[6] The equations can, however, be easily interpreted if $\mathbf h$ is treated as a skew-symmetric tensor of the second rank. In this context, we write $h_{23}$ , $h_{31}$ , $h_{12}$ instead of $h_{1}$ , $h_{2}$ , $h_{3}$ respectively. Taking into account the skew-symmetry of $h_{\mu\nu}$ , the first three equations of (19) and (20) can be expressed as $\begin{gather*} \frac{\partial h_{\mu\nu}}{\partial x_{\nu}} = \frac{1}{c}\, \frac{\partial e_{\mu}}{\partial t} + \frac{1}{c} i_{\mu}, \qquad & &\text{(19a)} \\ \frac{\partial e_{\mu}}{\partial x_{\nu}} - \frac{\partial e_{\nu}}{\partial x_{\mu}} = +\frac{1}{c}\, \frac{\partial h_{\mu\nu}}{\partial t}. \qquad & &\text{(20a)} \end{gather*}$ Unlike $\mathbf e$ , $\mathbf h$ appears to have a type of symmetry similar to angular velocity. The divergence equations then take the form $\begin{gather*} \frac{\partial e_{\nu}}{\partial x_{\nu}} = \rho\, \qquad & &\text{(19b)} \\ \frac{\partial h_{\mu\nu}}{\partial x_{\rho}} + \frac{\partial h_{\nu\rho}}{\partial x_{\mu}} + \frac{\partial h_{\rho\mu}}{\partial x_{\nu}} = 0. \qquad & &\text{(20b)} \end{gather*}$ The last equation is a skew-symmetric tensor equation of the third rank (the skew-symmetry of the left-hand side with respect to every pair of indices can be easily demonstrated, considering the skew-symmetry of $h_{\mu\nu}$ ). This notation is more intuitive than the typical one because, unlike the usual approach, it applies to both left-handed and right-handed Cartesian systems without changing the sign. [Pg 24]

[6]These considerations will make the reader familiar with tensor operations without the special difficulties of the four-dimensional treatment; corresponding considerations in the theory of special relativity (Minkowski's interpretation of the field) will then offer fewer difficulties.

[6]These points will help the reader understand tensor operations without the complex challenges of four-dimensional analysis; similar points in the theory of special relativity (Minkowski's interpretation of the field) will then be easier to grasp.

LECTURE II

THE THEORY OF SPECIAL RELATIVITY

THE previous considerations concerning the configuration of rigid bodies have been founded, irrespective of the assumption as to the validity of the Euclidean geometry, upon the hypothesis that all directions in space, or all configurations of Cartesian systems of co-ordinates, are physically equivalent. We may express this as the "principle of relativity with respect to direction," and it has been shown how equations (laws of nature) may be found, in accord with this principle, by the aid of the calculus of tensors. We now inquire whether there is a relativity with respect to the state of motion of the space of reference; in other words, whether there are spaces of reference in motion relatively to each other which are physically equivalent. From the standpoint of mechanics it appears that equivalent spaces of reference do exist. For experiments upon the earth tell us nothing of the fact that we are moving about the sun with a velocity of approximately 30 kilometres a second. On the other hand, this physical equivalence does not seem to hold for spaces of reference in arbitrary motion; for mechanical effects do not seem to be subject to the same laws in a jolting railway train as in one moving with uniform velocity; the rotation of the earth must be considered in writing down the equations of motion relatively to the earth. It appears, therefore, as if there were Cartesian systems of co-ordinates, the so-called inertial systems, with reference to which the laws of mechanics (more generally the laws of physics) are expressed in the simplest form. We may infer the validity of the following theorem: If $K$ is an inertial system, then every [Pg 25] other system $K$ ' which moves uniformly and without rotation relatively to $K$ , is also an inertial system; the laws of nature are in concordance for all inertial systems. This statement we shall call the "principle of special relativity." We shall draw certain conclusions from this principle of "relativity of translation" just as we have already done for relativity of direction.

THE previous discussions about the arrangement of rigid bodies have been based, regardless of the assumption about the validity of Euclidean geometry, on the idea that all directions in space, or all setups of Cartesian coordinate systems, are physically equal. We can refer to this as the "principle of relativity concerning direction," and it has been demonstrated how equations (laws of nature) can be derived, in line with this principle, using tensor calculus. Now, we ask whether there is relativity when it comes to the state of motion of the reference frame; in other words, are there reference frames in motion relative to each other that are physically equivalent? From a mechanical perspective, it appears that equivalent reference frames do exist. Experiments conducted on Earth provide no indication that we are orbiting the sun at about 30 kilometers per second. However, this physical equivalence doesn’t seem to apply to reference frames in arbitrary motion; for instance, mechanical effects behave differently in a bumpy train compared to one moving at a constant speed; the Earth's rotation must be considered when formulating the equations of motion relative to the Earth. Thus, it appears there are Cartesian coordinate systems, referred to as inertial systems, with respect to which the laws of mechanics (and more broadly, the laws of physics) are expressed most simply. We can conclude the following theorem: If $K$ is an inertial system, then every other system $K$ ' that moves uniformly and without rotation in relation to $K$ is also an inertial system; the laws of nature are consistent across all inertial systems. We will refer to this assertion as the "principle of special relativity." We will draw certain conclusions from this principle of "relativity of translation," just as we have already done for the relativity of direction.

In order to be able to do this, we must first solve the following problem. If we are given the Cartesian co-ordinates, $x_{\nu}$ , and the time, $t$ , of an event relatively to one inertial system, $K$ , how can we calculate the co-ordinates, $x'_{\nu}$ , and the time, $t$ ', of the same event relatively to an inertial system $K$ ' which moves with uniform translation relatively to $K$ ? In the pre-relativity physics this problem was solved by making unconsciously two hypotheses:—

To do this, we first need to solve the following problem. If we have the Cartesian coordinates, $x_{\nu}$ , and the time, $t$ , of an event in one inertial system, $K$ , how do we calculate the coordinates, $x'_{\nu}$ , and the time, $t$ ', of the same event in another inertial system $K$ , which is moving uniformly in relation to $K$ ? In pre-relativity physics, this problem was solved by unconsciously making two assumptions:—

1. The time is absolute; the time of an event, $t$ ', relatively to $K$ ' is the same as the time relatively to $K$ . If instantaneous signals could be sent to a distance, and if one knew that the state of motion of a clock had no influence on its rate, then this assumption would be physically established. For then clocks, similar to one another, and regulated alike, could be distributed over the systems $K$ and $K$ ', at rest relatively to them, and their indications would be independent of the state of motion of the systems; the time of an event would then be given by the clock in its immediate neighbourhood.

1. Time is absolute; the timing of an event, $t$ ', in relation to $K$ ' is the same as the timing in relation to $K$ . If instant signals could be sent over a distance, and if we knew that the motion of a clock didn’t affect its rate, then this idea would be proven. This way, clocks that are similar and set the same could be placed in the systems $K$ and $K$ ', resting in relation to them, and their readings would not depend on the motion of the systems; the timing of an event would then be indicated by the clock nearby.

2. Length is absolute; if an interval, at rest relatively to $K$ , has a length $s$ , then it has the same length $s$ relatively to a system $K$ ' which is in motion relatively to $K$ .

2. Length is absolute; if an interval, at rest relative to $K$ , has a length $s$ , then it has the same length $s$ relative to a system $K$ ' that is moving relative to $K$ .

If the axes of $K$ and $K$ ' are parallel to each other, a simple calculation based on these two assumptions, gives the equations [Pg 26] of transformation $\left. \begin{aligned} {x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\text{,} \\ t' &= t - b\text{.} \end{aligned} \right\} \qquad \text{(21)}$

If the axes of $K$ and $K$ are parallel, a straightforward calculation based on these two assumptions results in the transformation equations [Pg 26] $\left. \begin{aligned} {x'}_{\nu} &= x_{\nu} - a_{\nu} - b_{\nu}t\text{,} \\ t' &= t - b\text{.} \end{aligned} \right\} \qquad \text{(21)}$

This transformation is known as the "Galilean Transformation." Differentiating twice by the time, we get $\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}.$ Further, it follows that for two simultaneous events, ${{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}.$ The invariance of the distance between the two points results from squaring and adding. From this easily follows the co-variance of Newton's equations of motion with respect to the Galilean transformation (21). Hence it follows that classical mechanics is in accord with the principle of special relativity if the two hypotheses respecting scales and clocks are made.

This change is called the "Galilean Transformation." If we differentiate twice with respect to time, we find $\frac{d^{2} {x'}_{\nu}}{dt^{2}} = \frac{d^{2} x_{\nu}}{dt^{2}}.$ Additionally, it follows that for two simultaneous events, ${{x'}_{\nu}}^{(1)} - {{x'}_{\nu}}^{(2)} = {x_{\nu}}^{(1)} - {x_{\nu}}^{(2)}.$ The invariance of the distance between the two points comes from squaring and adding. This naturally leads to the covariance of Newton's equations of motion concerning the Galilean transformation (21). Therefore, it can be concluded that classical mechanics aligns with the principle of special relativity if the two assumptions about scales and clocks are accepted.

But this attempt to found relativity of translation upon the Galilean transformation fails when applied to electromagnetic phenomena. The Maxwell-Lorentz electromagnetic equations are not co-variant with respect to the Galilean transformation. In particular, we note, by (21), that a ray of light which referred to $K$ has a velocity $c$ , has a different velocity referred to $K$ ', depending upon its direction. The space of reference of $K$ is therefore distinguished, with respect to its physical properties, from all spaces of reference which are in motion relatively to it (quiescent æther). But all experiments have shown that electromagnetic and optical phenomena, relatively to the earth as the [Pg 27] body of reference, are not influenced by the translational velocity of the earth. The most important of these experiments are those of Michelson and Morley, which I shall assume are known. The validity of the principle of special relativity can therefore hardly be doubted.

But this effort to base the relativity of translation on the Galilean transformation doesn't hold up when we look at electromagnetic phenomena. The Maxwell-Lorentz electromagnetic equations don’t remain consistent with the Galilean transformation. Specifically, we note, by (21), that a ray of light referenced to $K$ travels at a speed of $c$ , but has a different speed when referenced to $K$ depending on its direction. The reference frame of $K$ is therefore distinct, regarding its physical properties, from all reference frames that are in motion relative to it (stationary ether). However, all experiments have shown that electromagnetic and optical phenomena, in relation to the Earth as the reference body, are not affected by the Earth's translational velocity. The most significant of these experiments are those conducted by Michelson and Morley, which I assume are familiar to you. Thus, the validity of the principle of special relativity is difficult to dispute.

On the other hand, the Maxwell-Lorentz equations have proved their validity in the treatment of optical problems in moving bodies. No other theory has satisfactorily explained the facts of aberration, the propagation of light in moving bodies (Fizeau), and phenomena observed in double stars (De Sitter). The consequence of the Maxwell-Lorentz equations that in a vacuum light is propagated with the velocity $c$ , at least with respect to a definite inertial system $K$ , must therefore be regarded as proved. According to the principle of special relativity, we must also assume the truth of this principle for every other inertial system.

On the other hand, the Maxwell-Lorentz equations have demonstrated their accuracy in addressing optical issues in moving objects. No other theory has successfully explained the phenomena of aberration, the propagation of light in moving bodies (Fizeau), and the observations made in double stars (De Sitter). The implication of the Maxwell-Lorentz equations is that in a vacuum, light travels at the velocity $c$ , at least relative to a specific inertial system $K$ , and this must be considered proven. According to the principle of special relativity, we also have to accept this principle as true for every other inertial system.

Before we draw any conclusions from these two principles we must first review the physical significance of the concepts "time" and "velocity." It follows from what has gone before, that co-ordinates with respect to an inertial system are physically defined by means of measurements and constructions with the aid of rigid bodies. In order to measure time, we have supposed a clock, $U$ , present somewhere, at rest relatively to $K$ . But we cannot fix the time, by means of this clock, of an event whose distance from the clock is not negligible; for there are no "instantaneous signals" that we can use in order to compare the time of the event with that of the clock. In order to complete the definition of time we may employ the principle of the constancy of the velocity of light in a vacuum. Let us suppose that we place similar clocks at points of the system $K$ , at rest relatively [Pg 28] to it, and regulated according to the following scheme. A ray of light is sent out from one of the clocks, $U_{m}$ , at the instant when it indicates the time $t_{m}$ , and travels through a vacuum a distance $r_{mn}$ , to the clock $U_{n}$ ; at the instant when this ray meets the clock $U_{n}$ the latter is set to indicate the time $t_{n}$ = $t_{m}$ + $\dfrac{r_{mn}}{c}$ .[7] The principle of the constancy of the velocity of light then states that this adjustment of the clocks will not lead to contradictions. With clocks so adjusted, we can assign the time to events which take place near any one of them. It is essential to note that this definition of time relates only to the inertial system $K$ , since we have used a system of clocks at rest relatively to $K$ . The assumption which was made in the pre-relativity physics of the absolute character of time (i.e. the independence of time of the choice of the inertial system) does not follow at all from this definition.

Before we make any conclusions based on these two principles, we first need to look at the physical meaning of "time" and "velocity." It is clear from what we've discussed before that coordinates in an inertial system are physically defined through measurements and constructions using rigid bodies. To measure time, we assume there is a clock, $U$ located somewhere that is at rest relative to $K$ . However, we can't determine the time of an event that is not close to this clock because there are no "instantaneous signals" we can use to compare the event's time with that of the clock. To fully define time, we can use the principle that the speed of light in a vacuum is constant. Let's say we place identical clocks at various points in system $K$ that are at rest compared to it, and we set them according to the following method. A light beam is sent out from one of the clocks, $U_{m}$ , at the moment it shows the time $t_{m}$ , and travels a distance $r_{mn}$ through a vacuum to the clock $U_{n}$ . At the moment this light beam reaches the clock $U_{n}$ , this clock is set to show the time $t_{n}$ = $t_{m}$ + $\dfrac{r_{mn}}{c}$ .[7] The principle of the constancy of the speed of light states that this adjustment of the clocks won't cause any contradictions. With the clocks adjusted this way, we can assign times to events that happen near any of them. It's important to note that this definition of time applies only to the inertial system $K$ since we've used clocks that are at rest with respect to $K$ . The assumption made in pre-relativity physics about the absolute nature of time (i.e., that time is independent of the choice of the inertial system) does not follow from this definition at all.

[7]Strictly speaking, it would be more correct to define simultaneity first, somewhat as follows: two events taking place at the points $A$ and $B$ of the system $K$ are simultaneous if they appear at the same instant when observed from the middle point, $M$ , of the interval $AB$ . Time is then defined as the ensemble of the indications of similar clocks, at rest relatively to $K$ , which register the same simultaneously.

[7]To be precise, it would be better to first define simultaneity as follows: two events occurring at points $A$ and $B$ within the system $K$ are simultaneous if they seem to happen at the same moment when viewed from the midpoint, $M$ , of the interval $K$ , which register the same simultaneity.

The theory of relativity is often criticized for giving, without justification, a central theoretical role to the propagation of light, in that it founds the concept of time upon the law of propagation of light. The situation, however, is somewhat as follows. In order to give physical significance to the concept of time, processes of some kind are required which enable relations to be established between different places. It is immaterial what kind of processes one chooses for such a definition of time. It is advantageous, however, for the theory, to choose only those [Pg 29] processes concerning which we know something certain. This holds for the propagation of light in vacuo in a higher degree than for any other process which could be considered, thanks to the investigations of Maxwell and H. A. Lorentz.

The theory of relativity often faces criticism for assigning a central theoretical role to the propagation of light without providing proper justification, as it bases the concept of time on the law of light propagation. However, the situation is somewhat like this: To give physical meaning to the idea of time, some processes are needed to establish relationships between different locations. It doesn’t really matter which processes are chosen for this definition of time. However, it's beneficial for the theory to select only those processes that we know something definite about. This is more applicable to the propagation of light in vacuo than to any other process that could be considered, thanks to the research done by Maxwell and H. A. Lorentz. [Pg 29]

From all of these considerations, space and time data have a physically real, and not a mere fictitious, significance; in particular this holds for all the relations in which co-ordinates and time enter, e.g. the relations (21). There is, therefore, sense in asking whether those equations are true or not, as well as in asking what the true equations of transformation are by which we pass from one inertial system $K$ to another, $K$ ', moving relatively to it. It may be shown that this is uniquely settled by means of the principle of the constancy of the velocity of light and the principle of special relativity.

From all of these considerations, space and time data have a real physical significance, not just a fictional one; this is especially true for all the relationships involving coordinates and time, such as the relationships (21). Therefore, it makes sense to question whether those equations are accurate and to determine what the correct transformation equations are that allow us to move from one inertial system $K$ to another, $K$ while moving relatively to it. It can be shown that this is uniquely determined by the principle of the constancy of the speed of light and the principle of special relativity.

To this end we think of space and time physically defined with respect to two inertial systems, $K$ and $K$ ', in the way that has been shown. Further, let a ray of light pass from one point $P_{1}$ to another point $P_{2}$ of $K$ through a vacuum. If $r$ is the measured distance between the two points, then the propagation of light must satisfy the equation $r = c·\Delta t.$

To this end, we define space and time in relation to two inertial systems, $K$ and $K$ ', as has been demonstrated. Furthermore, let a beam of light travel from one point $P_{1}$ to another point $P_{2}$ of $K$ through a vacuum. If $r$ is the distance measured between the two points, then the light's propagation must satisfy the equation $r = c·\Delta t.$

If we square this equation, and express $r^{2}$ by the differences of the co-ordinates, $\Delta x_{\nu}$ , in place of this equation we can write $\sum (\Delta x_{\nu})^{2} - c^{2} \Delta t^{2} = 0. \qquad\text{(22)}$ This equation formulates the principle of the constancy of the velocity of light relatively to $K$ . It must hold whatever may be the motion of the source which emits the ray of light. [Pg 30]

If we square this equation and express $r^{2}$ in terms of the differences in the coordinates, $\Delta x_{\nu}$ , instead of this equation we can write $\sum (\Delta x_{\nu})^{2} - c^{2} \Delta t^{2} = 0. \qquad\text{(22)}$ This equation expresses the principle that the speed of light is constant relative to $K$ . It must apply regardless of the motion of the source emitting the light ray. [Pg 30]

The same propagation of light may also be considered relatively to $K$ ', in which case also the principle of the constancy of the velocity of light must be satisfied. Therefore, with respect to $K$ ', we have the equation $\sum ({\Delta x'}_{\nu})^{2} - c^{2} \Delta t'^{2} = 0. \qquad\text{(22a)}$

The same way light spreads can also be viewed in relation to $K$ ', meaning the principle of the constant speed of light must also hold true. So, in relation to $K$ ', we have the equation $\sum ({\Delta x'}_{\nu})^{2} - c^{2} \Delta t'^{2} = 0. \qquad\text{(22a)}$

Equations (22a) and (22) must be mutually consistent with each other with respect to the transformation which transforms from $K$ to $K$ '. A transformation which effects this we shall call a "Lorentz transformation."

Equations (22a) and (22) need to be consistent with each other regarding the transformation that changes from $K$ to $K$ '. We'll refer to this type of transformation as a "Lorentz transformation."

Before considering these transformations in detail we shall make a few general remarks about space and time. In the pre-relativity physics space and time were separate entities. Specifications of time were independent of the choice of the space of reference. The Newtonian mechanics was relative with respect to the space of reference, so that, e.g. the statement that two non-simultaneous events happened at the same place had no objective meaning (that is, independent of the space of reference). But this relativity had no role in building up the theory. One spoke of points of space, as of instants of time, as if they were absolute realities. It was not observed that the true element of the space-time specification was the event, specified by the four numbers $x_{1}$ , $x_{2}$ , $x_{3}$ , $t$ . The conception of something happening was always that of a four-dimensional continuum; but the recognition of this was obscured by the absolute character of the pre-relativity time. Upon giving up the hypothesis of the absolute character of time, particularly that of simultaneity, the four-dimensionality of the time-space concept was immediately recognized. It is neither the point in space, nor the instant in [Pg 31] time, at which something happens that has physical reality, but only the event itself. There is no absolute (independent of the space of reference) relation in space, and no absolute relation in time between two events, but there is an absolute (independent of the space of reference) relation in space and time, as will appear in the sequel. The circumstance that there is no objective rational division of the four-dimensional continuum into a three-dimensional space and a one-dimensional time continuum indicates that the laws of nature will assume a form which is logically most satisfactory when expressed as laws in the four-dimensional space-time continuum. Upon this depends the great advance in method which the theory of relativity owes to Minkowski. Considered from this standpoint, we must regard $x_{1}$ , $x_{2}$ , $x_{3}$ , $t$ as the four co-ordinates of an event in the four-dimensional continuum. We have far less success in picturing to ourselves relations in this four-dimensional continuum than in the three-dimensional Euclidean continuum; but it must be emphasized that even in the Euclidean three-dimensional geometry its concepts and relations are only of an abstract nature in our minds, and are not at all identical with the images we form visually and through our sense of touch. The non-divisibility of the four-dimensional continuum of events does not at all, however, involve the equivalence of the space co-ordinates with the time co-ordinate. On the contrary, we must remember that the time co-ordinate is defined physically wholly differently from the space co-ordinates. The relations (22) and (22a) which when equated define the Lorentz transformation show, further, a difference in the role of the time co-ordinate from that of the space co-ordinates; for the term $\Delta t^{2}$ has the opposite sign to the space terms, $\Delta x_{1}^{2}$ , $\Delta x_{2}^{2}$ , $\Delta x_{3}^{2}$ . [Pg 32]

Before diving into these transformations in detail, let's make a few general comments about space and time. In pre-relativity physics, space and time were seen as separate entities. Descriptions of time were independent of the chosen reference space. Newtonian mechanics depended on the reference space, meaning that, for instance, stating that two non-simultaneous events occurred at the same place had no objective meaning (i.e., regardless of the reference space). However, this relativity didn't play a role in shaping the theory. People treated points in space and instants in time as if they were absolute realities. It wasn't recognized that the fundamental aspect of space-time specification was the event, represented by the four numbers $x_{1}$ , $x_{2}$ , $x_{3}$ , $t$ . The concept of an event was always understood as part of a four-dimensional continuum, but this understanding was clouded by the absolute nature of pre-relativity time. Once the idea of absolute time—especially that of simultaneity—was abandoned, the four-dimensional nature of space-time was immediately acknowledged. It's not the point in space or the instant in time where something occurs that has physical reality; it's the event itself. There are no absolute (independent of the reference space) relations in space and no absolute relations in time between two events, but there is an absolute (independent of the reference space) relation in space and time, as will be shown later. The fact that we can't objectively divide the four-dimensional continuum into a three-dimensional space and a one-dimensional time continuum suggests that the laws of nature will be most logically expressed as laws within the four-dimensional space-time continuum. This is a significant methodological advancement that relativity owes to Minkowski. From this perspective, we should consider $x_{1}$ , $x_{2}$ , $x_{3}$ , $t$ as the four coordinates of an event in the four-dimensional continuum. We find it much harder to visualize relationships in this four-dimensional continuum than in the three-dimensional Euclidean continuum; however, it's important to stress that even in the Euclidean three-dimensional geometry, its concepts and relationships exist only as abstract ideas in our minds, not identical to the images we create visually and through our sense of touch. The indivisibility of the four-dimensional continuum of events does not imply that the space coordinates are equivalent to the time coordinate. On the contrary, we must remember that the time coordinate is defined physically in a completely different way from the space coordinates. The equations (22) and (22a), which define the Lorentz transformation when equated, further illustrate a difference in the role of the time coordinate compared to the space coordinates; for the term $\Delta t^{2}$ has the opposite sign to the space terms $\Delta x_{1}^{2}$ , $\Delta x_{2}^{2}$ , $\Delta x_{3}^{2}$ . [Pg 32]

Before we analyse further the conditions which define the Lorentz transformation, we shall introduce the light-time, $l = ct$ , in place of the time, $t$ , in order that the constant $c$ shall not enter explicitly into the formulas to be developed later. Then the Lorentz transformation is defined in such a way that, first, it makes the equation ${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 \qquad\text{(22b)}$ a co-variant equation, that is, an equation which is satisfied with respect to every inertial system if it is satisfied in the inertial system to which we refer the two given events (emission and reception of the ray of light). Finally, with Minkowski, we introduce in place of the real time co-ordinate $l = ct$ , the imaginary time co-ordinate $x_{4} = il = ict\quad (\sqrt{-1} = i).$ Then the equation defining the propagation of light, which must be co-variant with respect to the Lorentz transformation, becomes $\sum_{(4)} {\Delta x_{\nu}}^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} = 0. \qquad\text{(22c)}$ This condition is always satisfied [8] if we satisfy the more general condition that $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} \qquad\text{(23)}$ [Pg 33] shall be an invariant with respect to the transformation. This condition is satisfied only by linear transformations, that is, transformations of the type ${x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} \qquad\text{(24)}$ in which the summation over the $\alpha$ is to be extended from $\alpha$ = 1 to $\alpha$ = 4. A glance at equations (23) and (24) shows that the Lorentz transformation so defined is identical with the translational and rotational transformations of the Euclidean geometry, if we disregard the number of dimensions and the relations of reality. We can also conclude that the coefficients $b_{\mu\alpha}$ must satisfy the conditions $b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}. \qquad\text{(25)}$ Since the ratios of the $x_{\nu}$ are real, it follows that all the $a_{\mu}$ and the $b_{\mu\alpha}$ are real, except $a_{4}$ , $b_{41}$ , $b_{42}$ , $b_{43}$ , $b_{14}$ , $b_{24}$ and $b_{34}$ , which are purely imaginary.

Before we dive deeper into the conditions that define the Lorentz transformation, let’s introduce the light-time, $l = ct$ , instead of the time, $t$ , so that the constant $c$ does not explicitly appear in the formulas we will develop later. Then the Lorentz transformation is defined in such a way that, firstly, it makes the equation ${\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - \Delta l^{2} = 0 \qquad\text{(22b)}$ a co-variant equation, meaning it holds true for every inertial system if it is satisfied in the inertial system from which we refer the two given events (the emission and reception of the light ray). Finally, with Minkowski, we substitute the real time coordinate $l = ct$ , with the imaginary time coordinate $x_{4} = il = ict\quad (\sqrt{-1} = i).$ The equation defining the propagation of light, which must also be co-variant with respect to the Lorentz transformation, becomes $\sum_{(4)} {\Delta x_{\nu}}^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} = 0. \qquad\text{(22c)}$ This condition is always satisfied [8] if we meet the broader condition that $s^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} + {\Delta x_{4}}^{2} \qquad\text{(23)}$ [Pg 33] is an invariant with respect to the transformation. This condition is only satisfied by linear transformations, which are transformations of the form ${x'}_{\mu} = a_{\mu} + b_{\mu\alpha} x_{\alpha} \qquad\text{(24)}$ where the summation over the $\alpha$ extends from $\alpha$ = 1 to $\alpha$ = 4. A look at equations (23) and (24) shows that the Lorentz transformation, as defined, is the same as the translational and rotational transformations of Euclidean geometry, except for the number of dimensions and the nature of reality. We can also conclude that the coefficients $b_{\mu\alpha}$ must meet the conditions $b_{\mu\alpha}b_{\nu\alpha} = \delta_{\mu\nu} = b_{\alpha\mu}b_{\alpha\nu}. \qquad\text{(25)}$ Since the ratios of the $x_{\nu}$ are real, it follows that all the $a_{\mu}$ and the $b_{\mu\alpha}$ are real, except for $a_{4}$ , $b_{41}$ , $b_{42}$ , $b_{43}$ , $b_{14}$ , $b_{24}$ and $b_{34}$ , which are purely imaginary.

[8]That this specialization lies in the nature of the case will be evident later.

[8]It will become clear later that this specialization is inherent to the situation.

Special Lorentz Transformation. We obtain the simplest transformations of the type of (24) and (25) if only two of the co-ordinates are to be transformed, and if all the $a_{\mu}$ , which determine the new origin, vanish. We obtain then for the indices 1 and 2, on account of the three independent conditions which the relations (25) furnish, $\left. \begin{alignedat}{2} {x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi, \\ {x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi, \\ {x'}_{3} &= x_{3}, && \\ {x'}_{4} &= x_{4}. && \end{alignedat} \right\} \qquad\text{(26)}$ [Pg 34]

Special Lorentz Transformation. We can derive the simplest transformations similar to (24) and (25) if we're only changing two of the coordinates and if all the $a_{\mu}$ that identify the new origin are zero. Therefore, for indices 1 and 2, based on the three independent conditions provided by the relations (25), we arrive at: $\left. \begin{alignedat}{2} {x'}_{1} &= x_{1} \cos\phi &&- x_{2} \sin\phi, \\ {x'}_{2} &= x_{1} \sin\phi &&+ x_{2} \cos\phi, \\ {x'}_{3} &= x_{3}, && \\ {x'}_{4} &= x_{4}. && \end{alignedat} \right\} \qquad\text{(26)}$ [Pg 34]

This is a simple rotation in space of the (space) co-ordinate system about $x_{3}$ -axis. We see that the rotational transformation in space (without the time transformation) which we studied before is contained in the Lorentz transformation as a special case. For the indices 1 and 4 we obtain, in an analogous manner, $\left. \begin{alignedat}{2} {x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi, \\ {x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi, \\ {x'}_{2} &= x_{2}, && \\ {x'}_{3} &= x_{3}. && \end{alignedat} \right\} \qquad\text{(26a)}$

This is a straightforward rotation in space of the (space) coordinate system around the $x_{3}$ -axis. We can see that the rotational transformation in space (without considering time transformation) that we studied earlier is included in the Lorentz transformation as a specific case. For the indices 1 and 4, we get, in a similar way, $\left. \begin{alignedat}{2} {x'}_{1} &= x_{1} \cos\psi &&- x_{4} \sin\psi, \\ {x'}_{4} &= x_{1} \sin\psi &&+ x_{4} \cos\psi, \\ {x'}_{2} &= x_{2}, && \\ {x'}_{3} &= x_{3}. && \end{alignedat} \right\} \qquad\text{(26a)}$

On account of the relations of reality $\psi$ must be taken as imaginary. To interpret these equations physically, we introduce the real light-time $l$ and the velocity $v$ of $K$ ' relatively to $K$ , instead of the imaginary angle $\psi$ . We have, first, $\begin{alignat*}{4} {x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi, \\ l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. \end{alignat*}$ Since for the origin of $K$ ' i.e., for $x_{1}$ = 0, we must have $x_{1} = vl$ , it follows from the first of these equations that $v = i\tan\psi, \qquad\text{(27)}$ and also $\begin{aligned} \left. \begin{aligned} \sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}, \\ \cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}, \end{aligned} \right\} \qquad\text{(28)} \end{aligned}$ [Pg 35] so that we obtain $\left. \begin{aligned} {x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}, \\ l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}, \\ {x'}_{2} &= x_{2}, \\ {x'}_{3} &= x_{3}. \end{aligned} \right\} \qquad\text{(29)}$

On the basis of the relationships in reality, $\psi$ must be considered imaginary. To give these equations a physical interpretation, we introduce the real light-time $l$ and the velocity $v$ of $K$ ' relative to $K$ instead of the imaginary angle $\psi$ . Firstly, we have, $\begin{alignat*}{4} {x'}_{1} &= &&x_{1} \cos\psi &{}-{}& i&&l \sin \psi, \\ l' &= -i&&x_{1} \sin\psi &{}+{}& &&l \cos\psi. \end{alignat*}$ Since for the origin of $K$ , that is, when $x_{1}$ = 0, we must have $x_{1} = vl$ , it follows from the first of these equations that $v = i\tan\psi, \qquad\text{(27)}$ and also $\begin{aligned} \left. \begin{aligned} \sin\psi &= \frac{-iv}{\sqrt{1 - v^{2}}}, \\ \cos\psi &= \frac{1}{\sqrt{1 - v^{2}}}, \end{aligned} \right\} \qquad\text{(28)} \end{aligned}$ [Pg 35] so that we obtain $\left. \begin{aligned} {x'}_{1} &= \frac{x_{1} - vl}{\sqrt{1 - v^{2}}}, \\ l' &= \frac{l - vx_{1}}{\sqrt{1 - v^{2}}}, \\ {x'}_{2} &= x_{2}, \\ {x'}_{3} &= x_{3}. \end{aligned} \right\} \qquad\text{(29)}$

These equations form the well-known special Lorentz transformation, which in the general theory represents a rotation, through an imaginary angle, of the four-dimensional system of co-ordinates. If we introduce the ordinary time $t$ , in place of the light-time $l$ , then in (29) we must replace $l$ by $ct$ and $v$ by $\dfrac{v}{c}$ .

These equations represent the well-known special Lorentz transformation, which in the general theory signifies a rotation through an imaginary angle of the four-dimensional coordinate system. If we substitute ordinary time $t$ for light-time $l$ , then in (29) we need to replace $l$ with $ct$ and $v$ with $\dfrac{v}{c}$ .

We must now fill in a gap. From the principle of the constancy of the velocity of light it follows that the equation $\sum {\Delta x_{\nu}}^{2} = 0$ has a significance which is independent of the choice of the inertial system; but the invariance of the quantity $\Delta x_{\nu}^{2}$ does not at all follow from this. This quantity might be transformed with a factor. This depends upon the fact that the right-hand side of (29) might be multiplied by a factor $\lambda$ , independent of $v$ . But the principle of relativity does not permit this factor to be different from 1, as we shall now show. Let us assume that we have a rigid circular cylinder moving in the direction of its axis. If its radius, measured at rest with a unit measuring rod is equal to $R_{0}$ , its radius $R$ in motion, might be different from $R_{0}$ , since the theory of relativity does not make the assumption that the shape of bodies with respect to a space of reference is independent of their motion relatively to this space of reference. But [Pg 36] all directions in space must be equivalent to each other. $R$ may therefore depend upon the magnitude $q$ of the velocity, but not upon its direction; $R$ must therefore be an even function of $q$ . If the cylinder is at rest relatively to $K$ ' the equation of its lateral surface is $x'^{2} + y'^{2} = {R_{0}}^{2}.$ If we write the last two equations of (29) more generally $\begin{align*} {x'}_{2} = \lambda x_{2}, \\ {x'}_{3} = \lambda x_{3}, \end{align*}$ then the lateral surface of the cylinder referred to $K$ satisfies the equation $x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}.$ The factor $\lambda$ therefore measures the lateral contraction of the cylinder, and can thus, from the above, be only an even function of $v$ .

We now need to address a gap. From the principle of the constancy of the speed of light, it follows that the equation $\sum {\Delta x_{\nu}}^{2} = 0$ has a meaning that doesn’t depend on which inertial system you choose; however, the invariance of the quantity $\Delta x_{\nu}^{2}$ does not necessarily follow from this. This quantity could be transformed with a factor. This hinges on the fact that the right side of (29) could be multiplied by a factor $\lambda$ , which is independent of $v$ . However, the principle of relativity does not allow this factor to be anything other than 1, as we will now demonstrate. Let’s suppose we have a rigid circular cylinder moving along its axis. If its radius, measured while at rest with a unit measuring rod, equals $R_{0}$ , its radius $R$ when in motion, could be different from $R_{0}$ , since relativity doesn’t assume that the shape of bodies is independent of their motion relative to the frame of reference. But [Pg 36] all directions in space must be equivalent. $R$ may therefore depend on the magnitude $q$ of the velocity, but not its direction; $R$ must thus be an even function of $K$ , the equation for its lateral surface is $x'^{2} + y'^{2} = {R_{0}}^{2}.$ If we express the last two equations of (29) more generally $\begin{align*} {x'}_{2} = \lambda x_{2}, \\ {x'}_{3} = \lambda x_{3}, \end{align*}$ then the lateral surface of the cylinder relative to $K$ meets the equation $x^{2} + y^{2} = \frac{{R_{0}}^{2}}{\lambda^{2}}.$ The factor $\lambda$ therefore represents the lateral contraction of the cylinder and can only be an even function of $v$ .

If we introduce a third system of co-ordinates, $K$ ", which moves relatively to $K$ ' with velocity $v$ in the direction of the negative $x$ -axis of $K$ , we obtain, by applying (29) twice, $\begin{align*} {x''}_{1} &= \lambda(v) \lambda(-v) x_{1}, \\ {x''}_{2} &= \lambda(v) \lambda(-v) x_{2}, \\ {x''}_{3} &= \lambda(v) \lambda(-v) x_{3}, \\ l'' &= \lambda(v) \lambda(-v) l. \end{align*}$ Now, since $\lambda(v)$ must be equal to $\lambda(-v)$ and since we assume that we use the same measuring rods in all the systems, it follows that the transformation of $K$ " to $K$ must be the identical [Pg 37] transformation (since the possibility $\lambda = -1$ does not need to be considered). It is essential for these considerations to assume that the behaviour of the measuring rods does not depend upon the history of their previous motion.

If we add a third coordinate system, $K$ ", that moves relative to $K$ ' at a velocity of $v$ in the negative $x$ -axis of $K$ , applying (29) twice gives us: $\begin{align*} {x''}_{1} &= \lambda(v) \lambda(-v) x_{1}, \\ {x''}_{2} &= \lambda(v) \lambda(-v) x_{2}, \\ {x''}_{3} &= \lambda(v) \lambda(-v) x_{3}, \\ l'' &= \lambda(v) \lambda(-v) l. \end{align*}$ Now, since $\lambda(v)$ must equal $\lambda(-v)$ and we assume we use the same measuring rods across all systems, it follows that the transformation from $K$ " to $K$ must be the same transformation (since the possibility $\lambda = -1$ doesn't need to be considered). It's crucial for these considerations to assume that the behavior of the measuring rods doesn't depend on their previous motion history.

Moving Measuring Rods and Clocks. At the definite $K$ -time, $l = 0$ , the position of the points given by the integers $x'_{1} = n$ , is with respect to $K$ , given by $x_{1} = n {\sqrt{1 - v^{2}}}$ ; this follows from the first of equations (29) and expresses the Lorentz contraction. A clock at rest at the origin $x_{1} = 0$ of $K$ , whose beats are characterized by $l = n$ , will, when observed from $K$ ', have beats characterized by $l' = \frac{n}{\sqrt{1 - v^{2}}};$ this follows from the second of equations (29) and shows that the clock goes slower than if it were at rest relatively to $K$ '. These two consequences, which hold, mutatis mutandis, for every system of reference, form the physical content, free from convention, of the Lorentz transformation.

Moving Measuring Rods and Clocks. At a specific $K$ -time, $l = 0$ , the position of the points represented by the integers $x'_{1} = n$ is related to $K$ and given by $x_{1} = n {\sqrt{1 - v^{2}}}$ ; this is derived from the first of equations (29) and illustrates the Lorentz contraction. A clock at rest at the origin $x_{1} = 0$ of $K$ , which has beats represented by $l = n$ , will, when viewed from $K$ ', have beats represented by $l' = \frac{n}{\sqrt{1 - v^{2}}};$ this comes from the second of equations (29) and indicates that the clock ticks slower than it would if it were at rest relative to $K$ . These two outcomes, which apply, mutatis mutandis, to every reference system, represent the physical essence, free from convention, of the Lorentz transformation.

Addition Theorem for Velocities. If we combine two special Lorentz transformations with the relative velocities $v_{1}$ and $v_{2}$ , then the velocity of the single Lorentz transformation which takes the place of the two separate ones is, according to (27), given by $v_{12} = i \tan (\psi_{1} + \psi_{2}) = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. \qquad\text{(30)}$ [Pg 38]

Addition Theorem for Velocities. When we combine two specific Lorentz transformations with the relative velocities $v_{1}$ and $v_{2}$ , the velocity of the single Lorentz transformation that replaces the two separate ones is given by (27) as follows: $v_{12} = i \tan (\psi_{1} + \psi_{2}) = i \frac{\tan\psi_{1} + \tan\psi_{2}}{1 - \tan\psi_{1} \tan\psi_{2}} = \frac{v_{1} + v_{2}}{1 + v_{1}v_{2}}. \qquad\text{(30)}$ [Pg 38]

General Statements about the Lorentz Transformation and its Theory of Invariants. The whole theory of invariants of the special theory of relativity depends upon the invariant $s^{2}$ (23). Formally, it has the same rôle in the four-dimensional space-time continuum as the invariant $\Delta x_{1}^{2}$ + $\Delta x_{2}^{2}$ + $\Delta x_{3}^{2}$ in the Euclidean geometry and in the pre-relativity physics. The latter quantity is not an invariant with respect to all the Lorentz transformations; the quantity $s^{2}$ of equation (23) assumes the rôle of this invariant. With respect to an arbitrary inertial system, $s^{2}$ may be determined by measurements; with a given unit of measure it is a completely determinate quantity, associated with an arbitrary pair of events.

General Statements about the Lorentz Transformation and its Theory of Invariants. The entire theory of invariants in the special theory of relativity relies on the invariant $s^{2}$ (23). Formally, it serves the same purpose in the four-dimensional space-time continuum as the invariant $\Delta x_{1}^{2}$ + $\Delta x_{2}^{2}$ + $\Delta x_{3}^{2}$ in Euclidean geometry and pre-relativity physics. The latter quantity is not invariant under all Lorentz transformations; the quantity $s^{2}$ from equation (23) takes on the role of this invariant. In relation to any inertial system, $s^{2}$ can be determined through measurements; with a specific unit of measurement, it is a fully determined quantity linked to any pair of events.

The invariant $s^{2}$ differs, disregarding the number of dimensions, from the corresponding invariant of the Euclidean geometry in the following points. In the Euclidean geometry $s^{2}$ is necessarily positive; it vanishes only when the two points concerned come together. On the other hand, from the vanishing of $s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2}$ it cannot be concluded that the two space-time points fall together; the vanishing of this quantity $s^{2}$ , is the invariant condition that the two space-time points can be connected by a light signal in vacuo. If $P$ a point (event) represented in the four-dimensional space of the $x_{1}$ , $x_{2}$ , $x_{3}$ , $l$ then all the "points" which can be connected to $P$ by means of a light signal lie upon the cone $s^{2}$ = 0 (compare Fig. 1, in which the dimension $x_{3}$ is suppressed). The "upper" half of the cone may contain the "points" to which light signals can be sent from $P$ ; then the "lower" half [Pg 39] of the cone will contain the "points" from which light signals can be sent to $P$ . The points $P'$ enclosed by the conical surface furnish, with $P$ , a negative $s^{2}$ ; $PP'$ as well as $P'P$ is then, according to Minkowski, of the nature of a time. Such intervals represent elements of possible paths of motion, the velocity being less than that of light.[9] In this case the $l$ -axis may be drawn [Pg 40] in the direction of $PP'$ by suitably choosing the state of motion of the inertial system. If $P'$ lies outside of the "light-cone" then $PP'$ is of the nature of a space; in this case, by properly choosing the inertial system, $\Delta l$ can be made to vanish.

The invariant $s^{2}$ is different from the corresponding invariant in Euclidean geometry in a few key ways, regardless of the number of dimensions. In Euclidean geometry, $s^{2}$ is always positive and only equals zero when the two points being considered are the same. However, if $s^{2} = \sum_{(4)} {\Delta x_{\nu}}^{2} = {\Delta x_{1}}^{2} + {\Delta x_{2}}^{2} + {\Delta x_{3}}^{2} - {\Delta t}^{2}$ equals zero, it doesn't mean the two space-time points coincide; instead, this zero value $s^{2}$ is the invariant condition that allows a light signal to connect the two points in vacuo. If $P$ represents a point (event) in the four-dimensional space of the coordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , and $l$ then all the "points" that can be reached from $P$ via a light signal lie on the cone defined by $s^{2}$ = 0 (see Fig. 1, which omits the $x_{3}$ dimension). The "upper" half of the cone might include the "points" to which light signals can be sent from $P$ . Meanwhile, the "lower" half [Pg 39] of the cone will include the "points" from which light signals can be sent to $P$ . The points $P'$ inside the conical surface, together with $P$ , result in a negative $s^{2}$ ; $PP'$ as well as $P'P$ are, according to Minkowski, time-like intervals. These intervals represent potential paths of motion, with speeds less than that of light.[9] In this scenario, the $l$ -axis can be oriented along $PP'$ by appropriately selecting the state of motion of the inertial system. If $P'$ falls outside the "light-cone," then $PP'$ is space-like; in this case, by properly selecting the inertial system, $\Delta l$ can be made to equal zero.

FIG. 1.

[9]That material velocities exceeding that of light are not possible, follows from the appearance of the radical $\sqrt{1 - v^{2}}$ in the special Lorentz transformation (29).

[9]The fact that anything moving faster than the speed of light is impossible comes from the presence of the radical $\sqrt{1 - v^{2}}$ in the special Lorentz transformation (29).

By the introduction of the imaginary time variable, $x_{4} = il$ , Minkowski has made the theory of invariants for the four-dimensional continuum of physical phenomena fully analogous to the theory of invariants for the three-dimensional continuum of Euclidean space. The theory of four-dimensional tensors of special relativity differs from the theory of tensors in three-dimensional space, therefore, only in the number of dimensions and the relations of reality.

By introducing the imaginary time variable, $x_{4} = il$ , Minkowski has made the theory of invariants for the four-dimensional continuum of physical phenomena completely similar to the theory of invariants for the three-dimensional continuum of Euclidean space. The theory of four-dimensional tensors in special relativity only differs from the theory of tensors in three-dimensional space in terms of the number of dimensions and the relationships of reality.

A physical entity which is specified by four quantities, $A_{\nu}$ , in an arbitrary inertial system of the $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ , is called a 4-vector, with the components $A_{\nu}$ , if the $A_{\nu}$ correspond in their relations of reality and the properties of transformation to the $\Delta x_{\nu}$ ; it may be of the nature of a space or of a time. The sixteen quantities $A_{\mu\nu}$ , then form the components of a tensor of the second rank, if they transform according to the scheme ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}.$ It follows from this that the $A_{\mu\nu}$ behave, with respect to their properties of transformation and their properties of reality, as the products of components, $U_{\mu}V_{\nu}$ of two 4-vectors, ( $U$ ) and ( $V$ ). All the components are real except those which contain the index 4 once, those being purely imaginary. Tensors [Pg 41] of the third and higher ranks may be defined in an analogous way. The operations of addition, subtraction, multiplication, contraction and differentiation for these tensors are wholly analogous to the corresponding operations for tensors in three-dimensional space.

A physical entity defined by four quantities, $A_{\nu}$ , exists in any inertial system represented by the coordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ . This is referred to as a 4-vector, with components $A_{\nu}$ , if the $A_{\nu}$ correspond in their reality and transformation properties to the $\Delta x_{\nu}$ . This can represent space or time. The sixteen quantities $A_{\mu\nu}$ , then, are the components of a second-rank tensor if they transform according to the equation ${A'}_{\mu\nu} = b_{\mu\alpha} b_{\nu\beta} A_{\alpha\beta}.$ This means that the $A_{\mu\nu}$ behave with their transformation and reality properties like the products of components, $U_{\mu}V_{\nu}$ of two 4-vectors, ( $U$ ) and ( $V$ ). All components are real, except those that contain the index 4 once, which are purely imaginary. Tensors [Pg 41] of third rank and higher can be defined similarly. The operations of addition, subtraction, multiplication, contraction, and differentiation for these tensors are entirely similar to the corresponding operations for tensors in three-dimensional space.

Before we apply the tensor theory to the four-dimensional space-time continuum, we shall examine more particularly the skew-symmetrical tensors. The tensor of the second rank has, in general, 16 = 4·4 components. In the case of skew-symmetry the components with two equal indices vanish, and the components with unequal indices are equal and opposite in pairs. There exist, therefore, only six independent components, as is the case in the electromagnetic field. In fact, it will be shown when we consider Maxwell's equations that these may be looked upon as tensor equations, provided we regard the electromagnetic field as a skew-symmetrical tensor. Further, it is clear that the skew-symmetrical tensor of the third rank (skew-symmetrical in all pairs of indices) has only four independent components, since there are only four combinations of three different indices.

Before we apply tensor theory to the four-dimensional space-time continuum, let's take a closer look at skew-symmetrical tensors. Generally, a second-rank tensor has 16 components (4·4). In the case of skew-symmetry, the components with two equal indices drop out, and the components with unequal indices are equal in magnitude but opposite in sign when paired. As a result, there are only six independent components, just like in the electromagnetic field. In fact, when we examine Maxwell's equations, we will see that these can be considered tensor equations if we treat the electromagnetic field as a skew-symmetrical tensor. Additionally, it's evident that a third-rank skew-symmetrical tensor (skew-symmetrical in all index pairs) has only four independent components, as there are only four combinations of three different indices.

We now turn to Maxwell's equations (19a), (19b), (20a), (20b), and introduce the notation:[10] $\begin{align*} &\left. \begin{aligned} &\phi_{23} && \phi_{31} && \phi_{12} && \phi_{14} && \phi_{24} && \phi_{34} \\ &\mathbf h_{23} && \mathbf h_{31} && \mathbf h_{12} & -&i\mathbf e_{x} & -&i\mathbf e_{y} & -&i\mathbf e_{z} \end{aligned} \right\} &&\text{(30a)}\\ &\qquad\qquad\left. \begin{aligned} &J_{1} && J_{2} && J_{3} && J_{4} \\ &\frac{1}{c}{\mathbf i_{x}} && \frac{1}{c}{\mathbf i_{y}} && \frac{1}{c}{\mathbf i_{z}} && i\rho \end{aligned} \right\} &&\text{(31)} \end{align*}$ with the convention that $\phi_{\mu\nu}$ shall be equal to $-\phi_{\mu\nu}$ . Then [Pg 42] Maxwell's equations may be combined into the forms $\begin{gather*} \frac{\partial \phi_{\mu\nu}}{\partial x_{\nu}} = J_{\mu}, \qquad & &\text{(32)}\\ \frac{\partial \phi_{\mu\nu}}{\partial x_{\sigma}} + \frac{\partial \phi_{\nu\sigma}}{\partial x_{\mu}} + \frac{\partial \phi_{\sigma\mu}}{\partial x_{\nu}} = 0, \qquad & &\text{(33)} \end{gather*}$ as one can easily verify by substituting from (30a) and (31). Equations (32) and (33) have a tensor character, and are therefore co-variant with respect to Lorentz transformations, if the $\phi_{\mu\nu}$ and the $J_{\nu}$ have a tensor character, which we assume. Consequently, the laws for transforming these quantities from one to another allowable (inertial) system of co-ordinates are uniquely determined. The progress in method which electrodynamics owes to the theory of special relativity lies principally in this, that the number of independent hypotheses is diminished. If we consider, for example, equations (19a) only from the standpoint of relativity of direction, as we have done above, we see that they have three logically independent terms. The way in which the electric intensity enters these equations appears to be wholly independent of the way in which the magnetic intensity enters them; it would not be surprising if instead of $\dfrac{\partial e_{\mu}}{\partial l}$ , we had, say, $\dfrac{\partial^{2} e_{\mu}}{\partial l^{2}}$ or if this term were absent. On the other hand, only two independent terms appear in equation (32). The electromagnetic field appears as a formal unit; the way in which the electric field enters this equation is determined by the way in which the magnetic field enters it. Besides the electromagnetic field, only the electric current density appears as an independent entity. This advance in method arises from the fact that the [Pg 43] electric and magnetic fields draw their separate existences from the relativity of motion. A field which appears to be purely an electric field, judged from one system, has also magnetic field components when judged from another inertial system. When applied to an electromagnetic field, the general law of transformation furnishes, for the special case of the special Lorentz transformation, the equations $\left. \begin{alignedat}{2} {\mathbf e'}_{x} &= \mathbf e_{x}\qquad & {\mathbf h'}_{x} &= \mathbf h_{x}, \\ {\mathbf e'}_{y} &= \frac{\mathbf e_{y} - v\mathbf h_{z}}{\sqrt{1 - v^{2}}}\qquad & {\mathbf h'}_{y} &= \frac{\mathbf h_{y} + v\mathbf e_{z}}{\sqrt{1 - v^{2}}}, \\ {\mathbf e'}_{z} &= \frac{\mathbf e_{z} + v\mathbf h_{y}}{\sqrt{1 - v^{2}}}\qquad & {\mathbf h'}_{z} &= \frac{\mathbf h_{z} - v\mathbf e_{y}}{\sqrt{1 - v^{2}}}. \end{alignedat} \right\} \qquad \text{(34)}$

We now move on to Maxwell's equations (19a), (19b), (20a), (20b), and introduce the notation:[10] $\begin{align*} &\left. \begin{aligned} &\phi_{23} && \phi_{31} && \phi_{12} && \phi_{14} && \phi_{24} && \phi_{34} \\ &\mathbf h_{23} && \mathbf h_{31} && \mathbf h_{12} & -i\mathbf e_{x} & -i\mathbf e_{y} & -i\mathbf e_{z} \end{aligned} \right\} &&\text{(30a)}\\ &\qquad\qquad\left. \begin{aligned} &J_{1} && J_{2} && J_{3} && J_{4} \\ &\frac{1}{c}{\mathbf i_{x}} && \frac{1}{c}{\mathbf i_{y}} && \frac{1}{c}{\mathbf i_{z}} && i\rho \end{aligned} \right\} &&\text{(31)} \end{align*}$ with the convention that $\phi_{\mu\nu}$ is equal to $-\phi_{\mu\nu}$ . Then [Pg 42] Maxwell's equations can be combined into the forms $\begin{gather*} \frac{\partial \phi_{\mu\nu}}{\partial x_{\nu}} = J_{\mu}, \qquad & &\text{(32)}\\ \frac{\partial \phi_{\mu\nu}}{\partial x_{\sigma}} + \frac{\partial \phi_{\nu\sigma}}{\partial x_{\mu}} + \frac{\partial \phi_{\sigma\mu}}{\partial x_{\nu}} = 0, \qquad & &\text{(33)} \end{gather*}$ as can be easily verified by substituting from (30a) and (31). Equations (32) and (33) have a tensor nature, and are therefore co-variant with respect to Lorentz transformations, if the $\phi_{\mu\nu}$ and the $J_{\nu}$ have a tensor character, which we assume. Consequently, the rules for transforming these quantities from one permissible (inertial) system of coordinates are uniquely determined. The advancement in method that electrodynamics owes to the theory of special relativity mainly lies in the fact that the number of independent hypotheses has decreased. If we look, for example, at equations (19a) only from the perspective of the relativity of direction, as we have done above, we see that they contain three logically independent terms. The manner in which the electric intensity is involved in these equations seems to be completely independent of how the magnetic intensity is involved; it wouldn't be surprising if instead of $\dfrac{\partial e_{\mu}}{\partial l}$ , we had, for instance, $\dfrac{\partial^{2} e_{\mu}}{\partial l^{2}}$ or if this term were absent. On the other hand, only two independent terms appear in equation (32). The electromagnetic field seems to act as a unified entity; the way the electric field is included in this equation is dictated by the way the magnetic field is included. Besides the electromagnetic field, only the electric current density appears as a separate entity. This progression in method results from the fact that the [Pg 43] electric and magnetic fields derive their separate existences from the relativity of motion. A field that looks purely electric from one system has also magnetic field components when viewed from another inertial system. When applied to an electromagnetic field, the general law of transformation provides, for the specific case of the Lorentz transformation, the equations $\left. \begin{alignedat}{2} {\mathbf e'}_{x} &= \mathbf e_{x}\qquad & {\mathbf h'}_{x} &= \mathbf h_{x}, \\ {\mathbf e'}_{y} &= \frac{\mathbf e_{y} - v\mathbf h_{z}}{\sqrt{1 - v^{2}}}\qquad & {\mathbf h'}_{y} &= \frac{\mathbf h_{y} + v\mathbf e_{z}}{\sqrt{1 - v^{2}}}, \\ {\mathbf e'}_{z} &= \frac{\mathbf e_{z} + v\mathbf h_{y}}{\sqrt{1 - v^{2}}}\qquad & {\mathbf h'}_{z} &= \frac{\mathbf h_{z} - v\mathbf e_{y}}{\sqrt{1 - v^{2}}}. \end{alignedat} \right\} \qquad \text{(34)}$

[10]In order to avoid confusion from now on we shall use the three-dimensional space indices, $x$ , $y$ , $z$ instead of 1, 2, 3, and we shall reserve the numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.

[10]To avoid confusion moving forward, we will use the three-dimensional space indices, $x$ , $y$ , $z$ instead of 1, 2, 3, and we will use the numeral indices 1, 2, 3, 4 for the four-dimensional space-time continuum.

If there exists with respect to $K$ only a magnetic field, $\mathbf h$ , but no electric field, $\mathbf e$ , then with respect to $K$ ' there exists an electric field $\mathbf e$ as well, which would act upon an electric particle at rest relatively to $K$ '. An observer at rest relatively to $K$ would designate this force as the Biot-Savart force, or the Lorentz electromotive force. It therefore appears as if this electromotive force had become fused with the electric field intensity into a single entity.

If there is only a magnetic field, $K$ , and no electric field, $\mathbf h$ , then there is also an electric field $\mathbf e$ with respect to $K$ , which would affect an electric particle at rest relative to $K$ . An observer at rest relative to $K$ would identify this force as the Biot-Savart force or the Lorentz electromotive force. It thus seems that this electromotive force has merged with the intensity of the electric field into one unified concept.

In order to view this relation formally, let us consider the expression for the force acting upon unit volume of electricity, $\mathbf k = \rho\mathbf e + [\mathbf i, \mathbf h], \qquad \text{(35)}$ in which $\mathbf i$ is the vector velocity of electricity, with the velocity of light as the unit. If we introduce $J_{\mu}$ and $\phi_{\mu\nu}$ according to (30a) and (31), we obtain for the first component the expression $\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}.$ [Pg 44] Observing that $\phi_{11}$ vanishes on account of the skew-symmetry of the tensor ( $\phi$ ), the components of $\mathbf k$ are given by the first three components of the four-dimensional vector $K_{\mu} = \phi_{\mu\nu} J_{\nu}, \qquad \text{(36)}$ and the fourth component is given by $K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} = i(\mathbf e_{x} \mathbf i_{x} + \mathbf e_{y} \mathbf i_{y} + \mathbf e_{z} \mathbf i_{z}) = i \lambda. \qquad \text{(37)}$ There is, therefore, a four-dimensional vector of force per unit volume, whose first three components, $K_{1}$ , $K_{2}$ , $K_{3}$ , are the ponderomotive force components per unit volume, and whose fourth component is the rate of working of the field per unit volume, multiplied by $\sqrt{-1}$ .

To formally analyze this relationship, let's consider the expression for the force acting on a unit volume of electricity, $\mathbf k = \rho\mathbf e + [\mathbf i, \mathbf h], \qquad \text{(35)}$ where $\mathbf i$ is the vector velocity of electricity, measured in terms of the speed of light. By introducing $J_{\mu}$ and $\phi_{\mu\nu}$ according to (30a) and (31), we find that the first component is expressed as $\phi_{12} J_{2} + \phi_{13} J_{3} + \phi_{14} J_{4}.$ [Pg 44] Noting that $\phi_{11}$ equals zero due to the skew-symmetry of the tensor ( $\phi$ ), the components of $\mathbf k$ are represented by the first three components of the four-dimensional vector $K_{\mu} = \phi_{\mu\nu} J_{\nu}, \qquad \text{(36)}$ and the fourth component is given by $K_{4} = \phi_{41} J_{1} + \phi_{42} J_{2} + \phi_{43} J_{3} = i(\mathbf e_{x} \mathbf i_{x} + \mathbf e_{y} \mathbf i_{y} + \mathbf e_{z} \mathbf i_{z}) = i \lambda. \qquad \text{(37)}$ Thus, there is a four-dimensional vector of force per unit volume, where the first three components, $K_{1}$ , $K_{2}$ , $K_{3}$ , represent the ponderomotive force components per unit volume, while the fourth component indicates the rate of work done by the field per unit volume, multiplied by $\sqrt{-1}$ .

A comparison of (36) and (35) shows that the theory of relativity formally unites the ponderomotive force of the electric field, $\rho$ $\mathbf e$ , and the Biot-Savart or Lorentz force [ $\mathbf i$ , $\mathbf h$ ].

A comparison of (36) and (35) shows that the theory of relativity formally links the ponderomotive force of the electric field, $\rho$ $\mathbf e$ , and the Biot-Savart or Lorentz force [ $\mathbf i$ , $\mathbf h$ ].

Mass and Energy. An important conclusion can be drawn from the existence and significance of the 4-vector $K_{\mu}$ . Let us imagine a body upon which the electromagnetic field acts for a time. In the symbolic figure (Fig. 2) $Ox_{1}$ designates the $x_{1}$ -axis, and is at the same time a substitute for the three space axes $Ox_{1}$ , $Ox_{2}$ , $Ox_{1}$ ; $Ol$ designates the real time axis. In this diagram a body of finite extent is represented, at a definite time $l$ , by the interval $AB$ the whole space-time existence of the body is represented by a strip whose boundary is everywhere inclined less than 45° to the $l$ -axis. Between the time sections, $l$ = $l_{1}$ and $l$ = $l_{2}$ , but not extending to them, a portion of the strip is shaded. This represents the portion of the space-time manifold [Pg 45] in which the electromagnetic field acts upon the body, or upon the electric charges contained in it, the action upon them being transmitted to the body. We shall now consider the changes which take place in the momentum and energy of the body as a result of this action.

Mass and Energy. An important conclusion can be drawn from the significance of the 4-vector $K_{\mu}$ . Imagine a body that is subjected to an electromagnetic field for a certain period of time. In the symbolic figure (Fig. 2), $Ox_{1}$ represents the $x_{1}$ -axis, and simultaneously serves as a substitute for the three spatial axes $Ox_{1}$ , $Ox_{2}$ , $Ox_{1}$ ; $Ol$ represents the actual time axis. In this diagram, a body of finite size is shown at a specific time $l$ , represented by the interval $AB$ . The entire space-time existence of the body is depicted as a strip whose boundary is angled at less than 45° to the $l$ -axis. Between the time sections, $l$ = $l_{1}$ and $l$ = $l_{2}$ , but not extending to them, a portion of the strip is shaded. This shading indicates the area of the space-time manifold [Pg 45] where the electromagnetic field interacts with the body or the electric charges within it, the effects on the charges being transmitted to the body. Next, we will examine the changes in the momentum and energy of the body due to this interaction.

FIG. 2.

We shall assume that the principles of momentum and energy are valid for the body. The change in momentum, $\Delta I_{x}$ , $\Delta I_{y}$ , $\Delta I_{z}$ , and the change in energy, $\Delta E$ are then given [Pg 46] by the expressions $\begin{alignat*}{2} \begin{aligned} \Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{x}\, dx\, dy\, dz} &&= \frac{1}{i}\int{K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{y}\, dx\, dy\, dz} &&= \frac{1}{i}\int{K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{z}\, dx\, dy\, dz\,} &&= \frac{1}{i}\int{K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta E &= \int_{l_{1}}^{l_{2}}dl \int{\lambda\, dx\, dy\, dz} &&= \frac{1}{i}\int{ \frac{1}{i}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}. \end{aligned} \end{alignat*}$ Since the four-dimensional element of volume is an invariant, and ( $K_{1}$ , $K_{2}$ , $K_{3}$ , $K_{4}$ ) forms a 4-vector, the four-dimensional integral extended over the shaded portion transforms as a 4-vector, as does also the integral between the limits $l_{1}$ and $l_{2}$ , because the portion of the region which is not shaded contributes nothing to the integral. It follows, therefore, that $\Delta I_{x}$ , $\Delta I_{y}$ , $\Delta I_{z}$ , $i \Delta E$ form a 4-vector. Since the quantities themselves transform in the same way as their increments, it follows that the aggregate of the four quantities $I_{x},\ I_{y},\ I_{z},\ iE$ has itself the properties of a vector; these quantities are referred to an instantaneous condition of the body (e.g. at the time $l$ = $l_{1}$ ).

We will assume that the principles of momentum and energy apply to the body. The change in momentum, $\Delta I_{x}$ , $\Delta I_{y}$ , $\Delta I_{z}$ , and the change in energy, $\Delta E$ are given by the expressions [Pg 46] $\begin{alignat*}{2} \begin{aligned} \Delta I_{x} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{x}\, dx\, dy\, dz} &&= \frac{1}{i}\int{K_{1}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta I_{y} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{y}\, dx\, dy\, dz} &&= \frac{1}{i}\int{K_{2}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta I_{z} &= \int_{l_{1}}^{l_{2}}dl \int{\mathbf k_{z}\, dx\, dy\, dz\,} &&= \frac{1}{i}\int{K_{3}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}, \\ \Delta E &= \int_{l_{1}}^{l_{2}}dl \int{\lambda\, dx\, dy\, dz} &&= \frac{1}{i}\int{ \frac{1}{i}K_{4}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}}. \end{aligned} \end{alignat*}$ Since the four-dimensional volume element is an invariant, and ( $K_{1}$ , $K_{2}$ , $K_{3}$ , $K_{4}$ ) forms a 4-vector, the four-dimensional integral over the shaded area transforms as a 4-vector, just like the integral between the limits $l_{1}$ and $l_{2}$ , because the unshaded part of the region contributes nothing to the integral. Therefore, it follows that $\Delta I_{x}$ , $\Delta I_{y}$ , $\Delta I_{z}$ , $i \Delta E$ form a 4-vector. Since the quantities themselves transform in the same way as their changes, it follows that the collection of the four quantities $I_{x},\ I_{y},\ I_{z},\ iE$ also has the properties of a vector; these quantities are related to an instantaneous state of the body (for example, at the time $l$ = $l_{1}$

This 4-vector may also be expressed in terms of the mass $m$ , and the velocity of the body, considered as a material particle. To form this expression, we note first, that $-ds^{2} = d\tau^{2} = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} = dl^{2}(1 - q^{2}) \qquad \text{(38)}$ [Pg 47] is an invariant which refers to an infinitely short portion of the four-dimensional line which represents the motion of the material particle. The physical significance of the invariant $d \tau$ may easily be given. If the time axis is chosen in such a way that it has the direction of the line differential which we are considering, or, in other words, if we reduce the material particle to rest, we shall then have $d \tau = dl$ ; this will therefore be measured by the light-seconds clock which is at the same place, and at rest relatively to the material particle. We therefore call $\tau$ the proper time of the material particle. As opposed to $dl$ , $d \tau$ is therefore an invariant, and is practically equivalent to $dl$ for motions whose velocity is small compared to that of light. Hence we see that $u_{\sigma} = \frac{dx_{\sigma}}{d\tau} \qquad \text{(39)}$ has, just as the $dx_{\nu}$ , the character of a vector; we shall designate ( $u_{\sigma}$ ) as the four-dimensional vector (in brief, 4-vector) of velocity. Its components satisfy, by (38), the condition $\sum {u_{\sigma}}^{2} = -1. \qquad \text{(40)}$ We see that this 4-vector, whose components in the ordinary notation are $\frac{\mathbf q_{x}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf q_{y}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf q_{z}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf i}{\sqrt{1 - q^{2}}} \qquad \text{(41)}$ is the only 4-vector which can be formed from the velocity components of the material particle which are defined in three dimensions by $\mathbf q_{x} = \frac{dx}{dl},\quad \mathbf q_{y} = \frac{dy}{dl},\quad \mathbf q_{z} = \frac{dz}{dl}.$ [Pg 48] We therefore see that $\left(m \frac{dx_{\mu}}{d\tau}\right) \qquad \text{(42)}$ must be that 4-vector which is to be equated to the 4-vector of momentum and energy whose existence we have proved above. By equating the components, we obtain, in three-dimensional notation, $\left. \begin{aligned} I_{x} = \frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}, \\ I_{y} = \frac{m\mathbf q_{y}}{\sqrt{1 - q^{2}}}, \\ I_{z} = \frac{m\mathbf q_{z}}{\sqrt{1 - q^{2}}}, \\ E = \frac{m}{\sqrt{1 - q^{2}}}. \end{aligned} \right\} \qquad \text{(43)}$

This 4-vector can also be described using the mass $m$ and the velocity of the body, treated as a material particle. To create this expression, we first note that $-ds^{2} = d\tau^{2} = - ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) - {dx_{4}}^{2} = dl^{2}(1 - q^{2}) \qquad \text{(38)}$ [Pg 47] is an invariant that relates to an infinitely small segment of the four-dimensional line representing the motion of the material particle. The physical meaning of the invariant $d \tau$ can be easily explained. If we choose the time axis in such a way that it aligns with the direction of the differential line we are considering, or in other words, if we bring the material particle to rest, we then have $d \tau = dl$ ; this will thus be measured by the light-seconds clock located at the same place and at rest relative to the material particle. Therefore, we refer to $\tau$ as the proper time of the material particle. Unlike $dl$ , $d \tau$ is indeed an invariant, and it is practically equivalent to $dl$ for motions where the velocity is small compared to that of light. Therefore, we find that $u_{\sigma} = \frac{dx_{\sigma}}{d\tau} \qquad \text{(39)}$ has, like the $dx_{\nu}$ , the nature of a vector; we will refer to ( $u_{\sigma}$ ) as the four-dimensional vector (or 4-vector) of velocity. Its components comply with the condition given by (38): $\sum {u_{\sigma}}^{2} = -1. \qquad \text{(40)}$ We see that this 4-vector, whose components in ordinary notation are $\frac{\mathbf q_{x}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf q_{y}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf q_{z}}{\sqrt{1 - q^{2}}},\quad \frac{\mathbf i}{\sqrt{1 - q^{2}}} \qquad \text{(41)}$ is the only 4-vector that can be formed from the velocity components of the material particle, which are defined in three dimensions by $\mathbf q_{x} = \frac{dx}{dl},\quad \mathbf q_{y} = \frac{dy}{dl},\quad \mathbf q_{z} = \frac{dz}{dl}.$ [Pg 48] We therefore conclude that $\left(m \frac{dx_{\mu}}{d\tau}\right) \qquad \text{(42)}$ must be that 4-vector which is equivalent to the 4-vector of momentum and energy whose existence we demonstrated earlier. By equating the components, we get, in three-dimensional notation: $\left. \begin{aligned} I_{x} = \frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}, \\ I_{y} = \frac{m\mathbf q_{y}}{\sqrt{1 - q^{2}}}, \\ I_{z} = \frac{m\mathbf q_{z}}{\sqrt{1 - q^{2}}}, \\ E = \frac{m}{\sqrt{1 - q^{2}}}. \end{aligned} \right\} \qquad \text{(43)}$

We recognize, in fact, that these components of momentum agree with those of classical mechanics for velocities which are small compared to that of light. For large velocities the momentum increases more rapidly than linearly with the velocity, so as to become infinite on approaching the velocity of light.

We acknowledge that these aspects of momentum align with those of classical mechanics when the velocities are small compared to the speed of light. However, at high velocities, momentum increases faster than linearly with speed, ultimately approaching infinity as one gets closer to the speed of light.

If we apply the last of equations (43) to a material particle at rest ( $q$ = 0), we see that the energy, $E_{0}$ of a, body at rest is equal to its mass. Had we chosen the second as our unit of time, we would have obtained $E_{0} = mc^{2}. \qquad \text{(44)}$ Mass and energy are therefore essentially alike; they are only different expressions for the same thing. The mass of a body [Pg 49] is not a constant; it varies with changes in its energy.[11] We see from the last of equations (43) that $E$ becomes infinite when $q$ approaches 1, the velocity of light. If we develop $E$ in powers of $q^{2}$ , we obtain, $E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots. \qquad \text{(45)}$ The second term of this expansion corresponds to the kinetic energy of the material particle in classical mechanics.

If we use the last equation from (43) for a material particle that is at rest ( $q$ = 0), we find that the energy, $E_{0}$ of a body at rest is equal to its mass. If we had chosen the second as our unit of time, we would get $E_{0} = mc^{2}. \qquad \text{(44)}$ Mass and energy are fundamentally the same; they're just different expressions of the same concept. The mass of a body [Pg 49] is not constant; it changes with variations in its energy.[11] From the last equation (43), we see that $E$ becomes infinite when $q$ approaches 1, which is the speed of light. If we expand $E$ in terms of $q^{2}$ , we get, $E = m + \frac{m}{2}q^{2} + \frac{3}{8}mq^{4} +\dots. \qquad \text{(45)}$ The second term of this expansion represents the kinetic energy of the material particle in classical mechanics.

[11]The emission of energy in radioactive processes is evidently connected with the fact that the atomic weights are not integers. Attempts have been made to draw conclusions from this concerning the structure and stability of the atomic nuclei.

[11]The release of energy in radioactive processes is clearly related to the fact that atomic weights are not whole numbers. There have been efforts to make inferences about the structure and stability of atomic nuclei based on this.

Equations of Motion of Material Particles. From (43) we obtain, by differentiating by the time $l$ , and using the principle of momentum, in the notation of three-dimensional vectors, $\mathbf{K} = \frac{d}{dl}\left(\frac{m\mathbf q}{\sqrt{1 - q^{2}}}\right). \qquad \text{(46)}$

Equations of Motion of Material Particles. From (43) we get, by differentiating with respect to time $l$ , and applying the principle of momentum, in the notation of three-dimensional vectors, $\mathbf{K} = \frac{d}{dl}\left(\frac{m\mathbf q}{\sqrt{1 - q^{2}}}\right). \qquad \text{(46)}$

This equation, which was previously employed by H. A. Lorentz for the motion of electrons, has been proved to be true, with great accuracy, by experiments with $\beta$ -rays.

This equation, which H. A. Lorentz used earlier for electron motion, has been shown to be accurate through experiments with $\beta$ -rays.

Energy Tensor of the Electromagnetic Field. Before the development of the theory of relativity it was known that the principles of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important conception, [Pg 50] that of the energy tensor, which is important for the further development of the theory of relativity.

Energy Tensor of the Electromagnetic Field. Before the theory of relativity was developed, it was understood that the principles of energy and momentum could be expressed in a differential form for the electromagnetic field. The four-dimensional formulation of these principles leads to an important concept, [Pg 50] that of the energy tensor, which is crucial for the further advancement of the theory of relativity.

If in the expression for the 4-vector of force per unit volume, $K_{\mu} = \phi_{\mu\nu} J_{\nu},$ using the field equations (32), we express $J_{\nu}$ in terms of the field intensities, $\phi_{\mu\nu}$ , we obtain, after some transformations and repeated application of the field equations (32) and (33), the expression $K_{\mu} = -\frac{\partial T_{\mu\nu}}{\partial x_{\nu}}, \qquad \text{(47)}$ where we have written [12] $T_{\mu\nu} = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + \phi_{\mu\alpha} \phi_{\nu\alpha}. \qquad \text{(48)}$

If we take the equation for the 4-vector of force per unit volume, $K_{\mu} = \phi_{\mu\nu} J_{\nu},$ and use the field equations (32) to express $J_{\nu}$ in terms of the field intensities, $\phi_{\mu\nu}$ , we arrive at the expression $K_{\mu} = -\frac{\partial T_{\mu\nu}}{\partial x_{\nu}}, \qquad \text{(47)}$ after making some transformations and applying the field equations (32) and (33) multiple times, where we have written [12] $T_{\mu\nu} = -\tfrac{1}{4}{\phi_{\alpha\beta}}^{2} \delta_{\mu\nu} + \phi_{\mu\alpha} \phi_{\nu\alpha}. \qquad \text{(48)}$

[12]To be summed for the indices $\alpha$ and $\beta$ .

The physical meaning of equation (47) becomes evident if in place of this equation we write, using a new notation, $\left. \begin{alignedat}{4} \mathbf k_{x} &= -\frac{\partial p_{xx}}{\partial x} &&- \frac{\partial p_{xy}}{\partial y} &&- \frac{\partial p_{xz}}{\partial z} &&- \frac{\partial (i b_{x})}{\partial (il)}, \\ \mathbf k_{y} &= -\frac{\partial p_{yx}}{\partial x} &&- \frac{\partial p_{yy}}{\partial y} &&- \frac{\partial p_{yz}}{\partial z} &&- \frac{\partial (i b_{y})}{\partial (il)}, \\ \mathbf k_{z} &= -\frac{\partial p_{zx}}{\partial x} &&- \frac{\partial p_{zy}}{\partial y} &&- \frac{\partial p_{zz}}{\partial z} &&- \frac{\partial (i b_{z})}{\partial (il)}, \\ i\lambda &= -\frac{\partial (i\mathbf s_{x})}{\partial x} &&- \frac{\partial (i\mathbf s_{y})}{\partial y} &&- \frac{\partial (i\mathbf s_{z})}{\partial z} &&- \frac{\partial (-\eta)}{\partial (i l)}; \end{alignedat} \right\} \qquad \text{(47a)}$ [Pg 51] or, on eliminating the imaginary, $\left. \begin{alignedat}{4} k_{x} &= -\frac{\partial p_{xx}}{\partial x} &&- \frac{\partial p_{xy}}{\partial y} &&- \frac{\partial p_{xz}}{\partial z} &&- \frac{\partial b_{x}}{\partial l}, \\ k_{y} &= -\frac{\partial p_{yx}}{\partial x} &&- \frac{\partial p_{yy}}{\partial y} &&- \frac{\partial p_{yz}}{\partial z} &&- \frac{\partial b_{y}}{\partial l}, \\ k_{z} &= -\frac{\partial p_{zx}}{\partial x} &&- \frac{\partial p_{zy}}{\partial y} &&- \frac{\partial p_{zz}}{\partial z} &&- \frac{\partial b_{z}}{\partial l}, \\ \lambda &= -\frac{\partial s_{x}}{\partial x} &&- \frac{\partial s_{y}}{\partial y} &&- \frac{\partial s_{z}}{\partial z} &&- \frac{\partial \eta}{\partial l}; \end{alignedat} \right\} \qquad \text{(47b)}$

The physical meaning of equation (47) becomes clear if we replace this equation with a new notation, $\left. \begin{alignedat}{4} \mathbf k_{x} &= -\frac{\partial p_{xx}}{\partial x} &&- \frac{\partial p_{xy}}{\partial y} &&- \frac{\partial p_{xz}}{\partial z} &&- \frac{\partial (i b_{x})}{\partial (il)}, \\ \mathbf k_{y} &= -\frac{\partial p_{yx}}{\partial x} &&- \frac{\partial p_{yy}}{\partial y} &&- \frac{\partial p_{yz}}{\partial z} &&- \frac{\partial (i b_{y})}{\partial (il)}, \\ \mathbf k_{z} &= -\frac{\partial p_{zx}}{\partial x} &&- \frac{\partial p_{zy}}{\partial y} &&- \frac{\partial p_{zz}}{\partial z} &&- \frac{\partial (i b_{z})}{\partial (il)}, \\ i\lambda &= -\frac{\partial (i\mathbf s_{x})}{\partial x} &&- \frac{\partial (i\mathbf s_{y})}{\partial y} &&- \frac{\partial (i\mathbf s_{z})}{\partial z} &&- \frac{\partial (-\eta)}{\partial (i l)}; \end{alignedat} \right\} \qquad \text{(47a)}$ [Pg 51] or, if we eliminate the imaginary part, $\left. \begin{alignedat}{4} k_{x} &= -\frac{\partial p_{xx}}{\partial x} &&- \frac{\partial p_{xy}}{\partial y} &&- \frac{\partial p_{xz}}{\partial z} &&- \frac{\partial b_{x}}{\partial l}, \\ k_{y} &= -\frac{\partial p_{yx}}{\partial x} &&- \frac{\partial p_{yy}}{\partial y} &&- \frac{\partial p_{yz}}{\partial z} &&- \frac{\partial b_{y}}{\partial l}, \\ k_{z} &= -\frac{\partial p_{zx}}{\partial x} &&- \frac{\partial p_{zy}}{\partial y} &&- \frac{\partial p_{zz}}{\partial z} &&- \frac{\partial b_{z}}{\partial l}, \\ \lambda &= -\frac{\partial s_{x}}{\partial x} &&- \frac{\partial s_{y}}{\partial y} &&- \frac{\partial s_{z}}{\partial z} &&- \frac{\partial \eta}{\partial l}; \end{alignedat} \right\} \qquad \text{(47b)}$

When expressed in the latter form, we see that the first three equations state the principle of momentum; $p_{xx}$ ,..., $p_{zx}$ are the Maxwell stresses in the electromagnetic field, and ( $b_{x}$ , $b_{y}$ , $b_{z}$ ) is the vector momentum per unit volume of the field. The last of equations (47b) expresses the energy principle; $\mathbf s$ is the vector flow of energy, and $\eta$ the energy per unit volume of the field. In fact, we get from (48) by introducing the well-known expressions for the components of the field intensity from electrodynamics, $\left. \begin{aligned} &\begin{alignedat}{4} p_{xx} = &{} - \mathbf h_{x} \mathbf h_{x} &&+ \tfrac{1}{2}({\mathbf h_{x}}^{2} &&+ {\mathbf h_{y}}^{2} &&+ {\mathbf h_{z}}^{2}) \\ &{} - \mathbf e_{x} \mathbf e_{y} &&+ \tfrac{1}{2}({\mathbf e_{x}}^{2} &&+ {\mathbf e_{y}}^{2} &&+ {\mathbf e_{z}}^{2}), \end{alignedat} \\ &\qquad\qquad\qquad\qquad\qquad \begin{alignedat}{3} p_{xy} = &{} - \mathbf h_{x} \mathbf h_{y}\quad && p_{xz} = && {} - \mathbf h_{x} \mathbf h_{z} \\ &{} - \mathbf e_{x} \mathbf e_{y},\quad && && {} - \mathbf e_{x} \mathbf e_{z}, \end{alignedat} \\ &\qquad\qquad\qquad\qquad\qquad\quad\vdots \\ &b_{x} = \mathbf s_{x} = \mathbf e_{y} \mathbf h_{z} - \mathbf e_{z}\mathbf h_{y}, \\ &b_{y} = \mathbf s_{y} = \mathbf e_{z} \mathbf h_{x} - \mathbf e_{x}\mathbf h_{z}, \\ &b_{z} = \mathbf s_{z} = \mathbf e_{x} \mathbf h_{y} - \mathbf e_{y}\mathbf h_{x}, \\ &\eta = +\tfrac{1}{2}({\mathbf e_{x}}^{2} + {\mathbf e_{y}}^{2} + {\mathbf e_{z}}^{2} + {\mathbf h_{x}}^{2} + {\mathbf h_{y}}^{2} + {\mathbf h_{z}}^{2}). \end{aligned} \right\} \qquad \text{(48a)}$ [Pg 52] We conclude from (48) that the energy tensor of the electromagnetic field is symmetrical; with this is connected the fact that the momentum per unit volume and the how of energy are equal to each other (relation between energy and inertia).

When stated in this way, the first three equations outline the principle of momentum; $p_{xx}$ ,..., $p_{zx}$ represent the Maxwell stresses in the electromagnetic field, and ( $b_{x}$ , $b_{y}$ , $b_{z}$ ) is the vector momentum per unit volume of the field. The last of the equations (47b) expresses the energy principle; $\mathbf s$ is the vector flow of energy, and $\eta$ is the energy per unit volume of the field. In fact, we obtain from (48) by integrating the well-known expressions for the components of the field intensity from electrodynamics, $\left. \begin{aligned} &\begin{alignedat}{4} p_{xx} = &{} - \mathbf h_{x} \mathbf h_{x} &&+ \tfrac{1}{2}({\mathbf h_{x}}^{2} &&+ {\mathbf h_{y}}^{2} &&+ {\mathbf h_{z}}^{2}) \\ &{} - \mathbf e_{x} \mathbf e_{y} &&+ \tfrac{1}{2}({\mathbf e_{x}}^{2} &&+ {\mathbf e_{y}}^{2} &&+ {\mathbf e_{z}}^{2}), \end{alignedat} \\ &\qquad\qquad\qquad\qquad\qquad \begin{alignedat}{3} p_{xy} = &{} - \mathbf h_{x} \mathbf h_{y}\quad && p_{xz} = && {} - \mathbf h_{x} \mathbf h_{z} \\ &{} - \mathbf e_{x} \mathbf e_{y},\quad && && {} - \mathbf e_{x} \mathbf e_{z}, \end{alignedat} \\ &\qquad\qquad\qquad\qquad\qquad\quad\vdots \\ &b_{x} = \mathbf s_{x} = \mathbf e_{y} \mathbf h_{z} - \mathbf e_{z}\mathbf h_{y}, \\ &b_{y} = \mathbf s_{y} = \mathbf e_{z} \mathbf h_{x} - \mathbf e_{x}\mathbf h_{z}, \\ &b_{z} = \mathbf s_{z} = \mathbf e_{x} \mathbf h_{y} - \mathbf e_{y}\mathbf h_{x}, \\ &\eta = +\tfrac{1}{2}({\mathbf e_{x}}^{2} + {\mathbf e_{y}}^{2} + {\mathbf e_{z}}^{2} + {\mathbf h_{x}}^{2} + {\mathbf h_{y}}^{2} + {\mathbf h_{z}}^{2}). \end{aligned} \right\} \qquad \text{(48a)}$ [Pg 52] We conclude from (48) that the energy tensor of the electromagnetic field is symmetrical; this is related to the fact that the momentum per unit volume and the flow of energy are equal to one another (the relation between energy and inertia).

We therefore conclude from these considerations that the energy per unit volume has the character of a tensor. This has been proved directly only for an electromagnetic field, although we may claim universal validity for it. Maxwell's equations determine the electromagnetic field when the distribution of electric charges and currents is known. But we do not know the laws which govern the currents and charges. We do know, indeed, that electricity consists of elementary particles (electrons, positive nuclei), but from a theoretical point of view we cannot comprehend this. We do not know the energy factors which determine the distribution of electricity in particles of definite size and charge, and all attempts to complete the theory in this direction have failed. If then we can build upon Maxwell's equations in general, the energy tensor of the electromagnetic field is known only outside the charged particles.[13] In these regions, outside of charged particles, the only regions in which we can believe that we have the complete expression for the energy tensor, we have, by (47), $\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = 0. \qquad \text{(47c)}$

We can conclude from these considerations that the energy per unit volume behaves like a tensor. This has only been directly proven for an electromagnetic field, although we can claim it applies universally. Maxwell's equations define the electromagnetic field when we know the distribution of electric charges and currents. However, we don't understand the laws that govern these currents and charges. While we know electricity is made up of elementary particles (electrons and positive nuclei), we can't fully understand this from a theoretical standpoint. We don't know the energy factors that determine how electricity is distributed among particles of specific sizes and charges, and all attempts to expand the theory in this area have not succeeded. Therefore, while we can generally rely on Maxwell's equations, the energy tensor of the electromagnetic field is only known outside the charged particles.[13] In these areas, which are outside of charged particles and are the only regions where we can be confident we have the complete expression for the energy tensor, we find by (47), $\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = 0. \qquad \text{(47c)}$

[13]It has been attempted to remedy this lack of knowledge by considering the charged particles as proper singularities. But in my opinion this means giving up a real understanding of the structure of matter. It seems to me much better to give in to our present inability rather than to be satisfied by a solution that is only apparent.

[13]People have tried to fix this knowledge gap by treating charged particles as distinct singularities. However, I believe this approach sacrifices a true understanding of the makeup of matter. It seems better to acknowledge our current limitations than to settle for a solution that only appears to work.

[Pg 53]

General Expressions for the Conservation Principles. We can hardly avoid making the assumption that in all other cases, also, the space distribution of energy is given by a symmetrical tensor, $T_{\mu\nu}$ , and that this complete energy tensor everywhere satisfies the relation (47c). At any rate we shall see that by means of this assumption we obtain the correct expression for the integral energy principle.

General Expressions for the Conservation Principles. It’s almost impossible to avoid assuming that in all other cases, the energy distribution in space is represented by a symmetrical tensor, $T_{\mu\nu}$ , and that this complete energy tensor consistently meets the relation (47c) everywhere. In any case, we will see that with this assumption, we derive the correct expression for the integral energy principle.

Let us consider a spatially bounded, closed system, which, four-dimensionally, we may represent as a strip, outside of which the $T_{\mu\nu}$ vanish. Integrate equation (47c) over a space section. Since the integrals of $\dfrac{\partial T_{\mu1}}{\partial x_{1}}$ , $\dfrac{\partial T_{\mu2}}{\partial x_{2}}$ and $\dfrac{\partial T_{\mu3}}{\partial x_{3}}$ vanish because the $T_{\mu\nu}$ vanish at the limits of integration, we obtain $\frac{\partial}{\partial l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} = 0. \qquad \text{(49)}$ Inside the parentheses are the expressions for the momentum of the whole system, multiplied by $i$ , together with the negative energy of the system, so that (49) expresses the conservation principles in their integral form. That this gives the right conception of energy and the conservation principles will be seen from the following considerations.

Let’s look at a closed, bounded system in space, which we can represent as a strip in four dimensions, beyond which the $T_{\mu\nu}$ disappear. We can integrate equation (47c) over a spatial section. Since the integrals of $\dfrac{\partial T_{\mu1}}{\partial x_{1}}$ , $\dfrac{\partial T_{\mu2}}{\partial x_{2}}$ and $\dfrac{\partial T_{\mu3}}{\partial x_{3}}$ vanish because the $T_{\mu\nu}$ go to zero at the limits of integration, we get $\frac{\partial}{\partial l}\left\{\int T_{\mu4}\, dx_{1}\, dx_{2}\, dx_{3} \right\} = 0. \qquad \text{(49)}$ Inside the parentheses are the momentum expressions of the entire system, multiplied by $i$ , along with the system's negative energy, so (49) represents the conservation principles in their integral form. We will see from the following considerations that this provides the correct understanding of energy and the conservation principles.

PHENOMENOLOGICAL REPRESENTATION OF THE ENERGY TENSOR OF MATTER.

Hydrodynamical Equations. We know that matter is built up of electrically charged particles, but we do not know the laws which govern the constitution of these particles. In treating mechanical problems, we are therefore obliged to make use of an [Pg 54] inexact description of matter, which corresponds to that of classical mechanics. The density $\sigma$ , of a material substance and the hydrodynamical pressures are the fundamental concepts upon which such a description is based.

Hydrodynamical Equations. We know that matter is made up of electrically charged particles, but we don’t fully understand the rules that define these particles. When dealing with mechanical problems, we have to rely on an [Pg 54] imperfect description of matter that aligns with classical mechanics. The density $\sigma$ of a material substance and the hydrodynamic pressures are the core concepts on which this description is based.

FIG. 3.

Let $\sigma_{0}$ be the density of matter at a place, estimated with reference to a system of co-ordinates moving with the matter. Then $\sigma_{0}$ , the density at rest, is an invariant. If we think of the matter in arbitrary motion and neglect the pressures (particles of dust in vacuo, neglecting the size of the particles and the temperature), then the energy tensor will depend only upon the [Pg 55] velocity components, $u_{\nu}$ and $\sigma_{0}$ . We secure the tensor character of $T_{\mu\nu}$ by putting $T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}, \qquad \text{(50)}$ in which the $u_{\mu}$ , in the three-dimensional representation, are given by (41). In fact, it follows from (50) that for $q = 0$ , $T_{44} = -\sigma_{0}$ (equal to the negative energy per unit volume), as it should, according to the theorem of the equivalence of mass and energy, and according to the physical interpretation of the energy tensor given above. If an external force (four-dimensional vector, $K_{\mu}$ ) acts upon the matter, by the principles of momentum and energy the equation $K_{\mu} = \frac{\partial T_{\mu\nu}}{\partial x_{\nu}}$ must hold. We shall now show that this equation leads to the same law of motion of a material particle as that already obtained. Let us imagine the matter to be of infinitely small extent in space, that is, a four-dimensional thread; then by integration over the whole thread with respect to the space co-ordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , we obtain $\begin{align*} \int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} &= \int \frac{\partial T_{14}}{\partial x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ &= -i \frac{d}{dl}\left\{ \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} \right\}. \end{align*}$

Let $\sigma_{0}$ be the matter density at a location, measured with respect to a coordinate system moving with the matter. Then $\sigma_{0}$ , the density in a stationary frame, is an invariant. If we consider matter in arbitrary motion and ignore the pressures (particles of dust in vacuo, disregarding the size of the particles and temperature), then the energy tensor will depend only on the velocity components, $u_{\nu}$ and $\sigma_{0}$ . We establish the tensor nature of $T_{\mu\nu}$ by writing $T_{\mu\nu} = \sigma_{0} u_{\mu} u_{\nu}, \qquad \text{(50)}$ where the $u_{\mu}$ , in the three-dimensional representation, are given by (41). In fact, it follows from (50) that for $q = 0$ , $T_{44} = -\sigma_{0}$ (equal to the negative energy per unit volume), which aligns with the theorem of mass-energy equivalence, and with the physical interpretation of the energy tensor given earlier. If an external force (four-dimensional vector, $K_{\mu}$ ) acts on the matter, by the principles of momentum and energy, the equation $K_{\mu} = \frac{\partial T_{\mu\nu}}{\partial x_{\nu}}$ must hold. We will now demonstrate that this equation leads to the same law of motion for a material particle as previously obtained. Let's assume the matter is infinitely small in space, essentially a four-dimensional thread; then by integrating over the entire thread with respect to the spatial coordinates $x_{1}$ , $x_{2}$ , $x_{3}$ , we obtain $\begin{align*} \int K_{1}\, dx_{1}\, dx_{2}\, dx_{3} &= \int \frac{\partial T_{14}}{\partial x_{4}}\, dx_{1}\, dx_{2}\, dx_{3} \\ &= -i \frac{d}{dl}\left\{ \int \sigma_{0}\frac{dx_{1}}{d\tau}\, \frac{dx_{4}}{d\tau}\, dx_{1}\, dx_{2}\, dx_{3} \right\}. \end{align*}$

Now $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ is an invariant, as is, therefore, also $\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ . We shall calculate this integral, first with respect to the inertial system which we have chosen, and second, with respect to a system relatively to which the matter has the velocity zero. The integration is to be extended over a filament [Pg 56] of the thread for which $\sigma_{0}$ may be regarded as constant over the whole section. If the space volumes of the filament referred to the two systems are $dV$ and $dV_{0}$ respectively, then we have $\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau$ and therefore also $\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} = \int dm\, i\, \frac{d\tau}{dx_{4}}.$

Now $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ is an invariant, and so is $\int \sigma_{0}\, dx_{1}\, dx_{2}\, dx_{3}\, dx_{4}$ . We'll calculate this integral, first with respect to the inertial system we've chosen, and second, with respect to a system where the matter has zero velocity. The integration will be done over a filament [Pg 56] of the thread for which $\sigma_{0}$ can be considered constant across the whole section. If the space volumes of the filament in relation to the two systems are $dV$ and $dV_{0}$ respectively, then we have $\int \sigma_{0}\, dV\, dl = \int \sigma_{0}\, dV_{0}\, d\tau$ and thus also $\int \sigma_{0}\, dV = \int \sigma_{0}\, dV_{0}\, \frac{d\tau}{dl} = \int dm\, i\, \frac{d\tau}{dx_{4}}.$

If we substitute the right-hand side for the left-hand side in the former integral, and put $\dfrac{dx_{1}}{d\tau}$ outside the sign of integration, we obtain, $\mathbf K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) = \frac{d}{dl}\left(\frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}\right).$ We see, therefore, that the generalized conception of the energy tensor is in agreement with our former result.

If we replace the right side with the left side in the previous integral and move $\dfrac{dx_{1}}{d\tau}$ outside the integral sign, we get, $\mathbf K_{x} = \frac{d}{dl}\left(m \frac{dx_{1}}{d\tau}\right) = \frac{d}{dl}\left(\frac{m\mathbf q_{x}}{\sqrt{1 - q^{2}}}\right).$ So, we can see that the generalized concept of the energy tensor aligns with our previous result.

The Eulerian Equations for Perfect Fluids. In order to get nearer to the behaviour of real matter we must add to the energy tensor a term which corresponds to the pressures. The simplest case is that of a perfect fluid in which the pressure is determined by a scalar $p$ . Since the tangential stresses $p_{xy}$ , etc., vanish in this case, the contribution to the energy tensor must be of the form $p\delta_{\nu\mu}$ . We must therefore put $T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}. \qquad \text{(51)}$ [Pg 57] At rest, the density of the matter, or the energy per unit volume, is in this case, not $\sigma$ but $\sigma - p$ . For $-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} = \sigma - p.$ In the absence of any force, we have $\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = \sigma u_{\nu} \frac{\partial u_{\mu}}{\partial x_{\nu}} + u_{\mu} \frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{\partial p}{\partial x_{\mu}} = 0.$ If we multiply this equation by $u_{\mu} \left(= \dfrac{dx_{\mu}}{d\tau}\right)$ and sum for the $\mu$ 's we obtain, using (40), $-\frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{dp}{d\tau} = 0, \qquad \text{(52)}$ where we have put $\dfrac{\partial p}{\partial x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}$ . This is the equation of continuity, which differs from that of classical mechanics by the term $\dfrac{dp}{d\tau}$ , which, practically, is vanishingly small. Observing (52), the conservation principles take the form $\sigma \frac{du_{\mu}}{d\tau} + u_{\mu} \frac{dp}{d\tau} + \frac{\partial p}{\partial x_{\mu}} = 0. \qquad \text{(53)}$ The equations for the first three indices evidently correspond to the Eulerian equations. That the equations (52) and (53) correspond, to a first approximation, to the hydrodynamical equations of classical mechanics, is a further confirmation of the generalized energy principle. The density of matter and of energy has the character of a symmetrical tensor. [Pg 58]

The Eulerian Equations for Perfect Fluids. To better mimic the behavior of real matter, we need to add a term to the energy tensor that reflects the pressures. The simplest case is a perfect fluid where the pressure is determined by a scalar $p$ . In this scenario, the tangential stresses $p_{xy}$ and similar terms disappear, so the contribution to the energy tensor must be in the form $p\delta_{\nu\mu}$ . Therefore, we have to set $T_{\mu\nu} = \sigma u_{\mu} u_{\nu} + p\delta_{\mu\nu}. \qquad \text{(51)}$ [Pg 57] At rest, the density of the matter, or the energy per unit volume, is in this case not $\sigma$ but $\sigma - p$ . For $-T_{44} = -\sigma \frac{dx_{4}}{d\tau}\, \frac{dx_{4}}{d\tau} - p\delta_{44} = \sigma - p.$ In the absence of any force, we get $\frac{\partial T_{\mu\nu}}{\partial x_{\nu}} = \sigma u_{\nu} \frac{\partial u_{\mu}}{\partial x_{\nu}} + u_{\mu} \frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{\partial p}{\partial x_{\mu}} = 0.$ If we multiply this equation by $u_{\mu} \left(= \dfrac{dx_{\mu}}{d\tau}\right)$ and sum over the $\mu$ indices, we find, using (40), $-\frac{\partial (\sigma u_{\nu})}{\partial x_{\nu}} + \frac{dp}{d\tau} = 0, \qquad \text{(52)}$ where we have set $\dfrac{\partial p}{\partial x_{\mu}}\, \dfrac{dx_{\mu}}{d\tau} = \dfrac{dp}{d\tau}$ . This is the continuity equation, which differs from the one in classical mechanics by the term $\dfrac{dp}{d\tau}$ , which, in practice, is extremely small. Observing (52), the conservation principles take the form $\sigma \frac{du_{\mu}}{d\tau} + u_{\mu} \frac{dp}{d\tau} + \frac{\partial p}{\partial x_{\mu}} = 0. \qquad \text{(53)}$ The equations for the first three indices clearly match the Eulerian equations. That equations (52) and (53) correspond, at a first approximation, to the hydrodynamic equations of classical mechanics further confirms the generalized energy principle. The density of matter and energy behaves as a symmetrical tensor. [Pg 58]

LECTURE III

THE GENERAL THEORY OF RELATIVITY

ALL of the previous considerations have been based upon the assumption that all inertial systems are equivalent for the description of physical phenomena, but that they are preferred, for the formulation of the laws of nature, to spaces of reference in a different state of motion. We can think of no cause for this preference for definite states of motion to all others, according to our previous considerations, either in the perceptible bodies or in the concept of motion; on the contrary, it must be regarded as an independent property of the space-time continuum. The principle of inertia, in particular, seems to compel us to ascribe physically objective properties to the space-time continuum. Just as it was necessary from the Newtonian standpoint to make both the statements, tempus est absolutum, spatium est absolutum, so from the standpoint of the special theory of relativity we must say, continuum spatii et temporis est absolutum. In this latter statement absolutum means not only "physically real," but also "independent in its physical properties, having a physical effect, but not itself influenced by physical conditions."

ALL of the previous considerations have been based on the assumption that all inertial systems are equivalent for describing physical phenomena. However, they are preferred for formulating the laws of nature to reference spaces in a different state of motion. We can't identify any reason for this preference for certain states of motion over others, based on what we've discussed before, whether in the physical bodies or in the concept of motion; on the contrary, it should be seen as an independent property of the space-time continuum. The principle of inertia, in particular, seems to require us to attribute physically objective properties to the space-time continuum. Just as it was necessary from the Newtonian perspective to assert both statements, tempus est absolutum and spatium est absolutum, we must say from the perspective of the special theory of relativity that continuum spatii et temporis est absolutum. In this latter statement, absolutum means not only "physically real," but also "independent in its physical properties, having a physical effect, but not influenced by physical conditions."

As long as the principle of inertia is regarded as the keystone of physics, this standpoint is certainly the only one which is justified. But there are two serious criticisms of the ordinary conception. In the first place, it is contrary to the mode of thinking in science to conceive of a thing (the space-time continuum) which acts itself, but which cannot be acted upon. This is the reason why E. Mach was led to make the attempt to eliminate space as an active cause in the system of mechanics. According [Pg 59] to him, a material particle does not move in unaccelerated motion relatively to space, but relatively to the centre of all the other masses in the universe; in this way the series of causes of mechanical phenomena was closed, in contrast to the mechanics of Newton and Galileo. In order to develop this idea within the limits of the modern theory of action through a medium, the properties of the space-time continuum which determine inertia must be regarded as field properties of space, analogous to the electromagnetic field. The concepts of classical mechanics afford no way of expressing this. For this reason Mach's attempt at a solution failed for the time being. We shall come back to this point of view later. In the second place, classical mechanics indicates a limitation which directly demands an extension of the principle of relativity to spaces of reference which are not in uniform motion relatively to each other. The ratio of the masses of two bodies is defined in mechanics in two ways which differ from each other fundamentally; in the first place, as the reciprocal ratio of the accelerations which the same motional force imparts to them (inert mass), and in the second place, as the ratio of the forces which act upon them in the same gravitational field (gravitational mass). The equality of these two masses, so differently defined, is a fact which is confirmed by experiments of very high accuracy (experiments of Eötvös), and classical mechanics offers no explanation for this equality. It is, however, clear that science is fully justified in assigning such a numerical equality only after this numerical equality is reduced to an equality of the real nature of the two concepts.

As long as the principle of inertia is considered the foundation of physics, this perspective is certainly the only one that makes sense. However, there are two significant criticisms of the conventional view. First, it goes against the scientific way of thinking to imagine something (the space-time continuum) that acts on its own but cannot be influenced by anything. This is why E. Mach tried to remove space as an active cause in mechanics. According to him, a material particle doesn’t move in uniform motion relative to space, but rather in relation to the center of all other masses in the universe. This way, the series of causes for mechanical phenomena was closed off, unlike in Newton and Galileo's mechanics. To develop this idea within the framework of modern theories of action through a medium, the properties of the space-time continuum that determine inertia must be viewed as field properties of space, similar to the electromagnetic field. Classical mechanics does not provide a way to express this. For this reason, Mach's attempt at a solution was unsuccessful for the time being. We will revisit this perspective later. Secondly, classical mechanics shows a limitation that directly calls for an extension of the principle of relativity to reference frames that are not in uniform motion relative to each other. The ratio of the masses of two bodies is defined in mechanics in two fundamentally different ways: first, as the reciprocal ratio of the accelerations that the same force applies to them (inert mass), and second, as the ratio of the forces acting on them in the same gravitational field (gravitational mass). The equality of these two masses, so differently defined, is a fact confirmed by highly accurate experiments (Eötvös experiments), and classical mechanics does not explain this equality. However, it is clear that science is fully justified in establishing such a numerical equality only after reducing that equality to an equality in the actual nature of the two concepts.

That this object may actually be attained by an extension of the principle of relativity, follows from the following consideration. A little reflection will show that the theorem of the [Pg 60] equality of the inert and the gravitational mass is equivalent to the theorem that the acceleration imparted to a body by a gravitational field is independent of the nature of the body. For Newton's equation of motion in a gravitational field, written out in full, is $\begin{aligned} (\text{Inert mass})·(\text{Acceleration}) = \,(\text{Intensity of the} \\ \qquad \text{gravitational field}) &· (\text{Gravitational mass}). \end{aligned}$ It is only when there is numerical equality between the inert and gravitational mass that the acceleration is independent of the nature of the body. Let now $K$ be an inertial system. Masses which are sufficiently far from each other and from other bodies are then, with respect to $K$ , free from acceleration. We shall also refer these masses to a system of co-ordinates $K$ ' uniformly accelerated with respect to $K$ . Relatively to $K$ ' all the masses have equal and parallel accelerations; with respect to $K$ ' they behave just as if a gravitational field were present and $K$ ' were unaccelerated. Overlooking for the present the question as to the "cause" of such a gravitational field, which will occupy us later, there is nothing to prevent our conceiving this gravitational field as real, that is, the conception that $K$ ' is "at rest" and a gravitational field is present we may consider as equivalent to the conception that only $K$ is an "allowable" system of co-ordinates and no gravitational field is present. The assumption of the complete physical equivalence of the systems of co-ordinates, $K$ and $K$ ', we call the "principle of equivalence;" this principle is evidently intimately connected with the theorem of the equality between the inert and the gravitational mass, and signifies an extension of the principle of relativity to co-ordinate systems which are in [Pg 61] non-uniform motion relatively to each other. In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. For according to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to $K$ ) or under the combined action of inertia and gravitation (with respect to $K$ '). The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered in development must be considered as small in comparison.

That this goal can actually be achieved by expanding the principle of relativity can be understood through the following reasoning. A bit of thought will reveal that the theorem stating the equality of inertial and gravitational mass is the same as the theorem asserting that the acceleration experienced by an object in a gravitational field does not depend on what that object is. Newton's full equation of motion in a gravitational field is: $\begin{aligned} (\text{Inert mass})·(\text{Acceleration}) = \,(\text{Intensity of the} \\ \qquad \text{gravitational field}) &· (\text{Gravitational mass}). \end{aligned}$ Acceleration is only independent of the type of object when the inertial and gravitational masses are numerically equal. Now let $K$ be an inertial system. Masses that are far enough apart from each other and from other objects are, in relation to $K$ , free from acceleration. We'll also consider these masses in a coordinate system $K$ , which is uniformly accelerating in relation to $K$ . In relation to $K$ , all the masses share equal and parallel accelerations; they behave as if a gravitational field were present and $K$ were unaccelerated. Ignoring for now the question of what causes such a gravitational field—a topic we'll address later—there's nothing stopping us from thinking of this gravitational field as real. That is, the idea that $K$ is "at rest" and a gravitational field exists can be considered equivalent to the idea that only $K$ is an "allowed" coordinate system, with no gravitational field present. The assumption of complete physical equivalence between the coordinate systems, $K$ and $K$ , is what we call the "principle of equivalence." This principle is clearly closely linked to the theorem of equality between inertial and gravitational mass and signifies an extension of the principle of relativity to coordinate systems that are in non-uniform motion relative to each other. In fact, through this concept, we achieve an understanding of the unity of inertia and gravitation. According to our perspective, the same masses can be viewed as either reacting to inertia alone (in relation to $K$ ) or responding to a combination of inertia and gravitation (in relation to $K$ ). The ability to explain the numerical equivalence of inertia and gravitation through their fundamental unity gives the general theory of relativity, in my opinion, such an advantage over classical mechanics that all the challenges faced in its development are relatively minor.

What justifies us in dispensing with the preference for inertial systems over all other co-ordinate systems, a preference that seems so securely established by experiment based upon the principle of inertia? The weakness of the principle of inertia lies in this, that it involves an argument in a circle: a mass moves without acceleration if it is sufficiently far from other bodies; we know that it is sufficiently far from other bodies only by the fact that it moves without acceleration. Are there, in general, any inertial systems for very extended portions of the space-time continuum, or, indeed, for the whole universe? We may look upon the principle of inertia as established, to a high degree of approximation, for the space of our planetary system, provided that we neglect the perturbations due to the sun and planets. Stated more exactly, there are finite regions, where, with respect to a suitably chosen space of reference, material particles move freely without acceleration, and in which the laws of the special theory of relativity, which have been developed above, hold with remarkable accuracy. Such regions we shall call "Galilean regions." We shall proceed from the consideration of such regions [Pg 62] as a special case of known properties.

What justifies us in prioritizing inertial systems over all other coordinate systems, a preference that seems firmly established by experiments based on the principle of inertia? The weakness of the principle of inertia lies in the fact that it involves circular reasoning: a mass moves without acceleration if it is far enough away from other bodies; we recognize that it is far enough away from other bodies only because it moves without acceleration. Are there, in general, any inertial systems for very large regions of the space-time continuum, or even for the entire universe? We can consider the principle of inertia to be established, with a high degree of approximation, for the space within our planetary system, as long as we ignore the disturbances caused by the sun and planets. More specifically, there are finite regions where, with respect to a suitably chosen reference frame, material particles move freely without acceleration, and in which the laws of the special theory of relativity, previously developed, hold with remarkable accuracy. We will refer to these regions as "Galilean regions." We will start by examining such regions as a special case of known properties. [Pg 62]

The principle of equivalence demands that in dealing with Galilean regions we may equally well make use of non-inertial systems, that is, such co-ordinate systems as, relatively to inertial systems, are not free from acceleration and rotation. If, further, we are going to do away completely with the difficult question as to the objective reason for the preference of certain systems of co-ordinates, then we must allow the use of arbitrarily moving systems of co-ordinates. As soon as we make this attempt seriously we come into conflict with that physical interpretation of space and time to which we were led by the special theory of relativity. For let $K$ ' be a system of co-ordinates whose $z$ '-axis coincides with the $z$ -axis of $K$ , and which rotates about the latter axis with constant angular velocity. Are the configurations of rigid bodies, at rest relatively to $K$ ', in accordance with the laws of Euclidean geometry? Since $K$ ' is not an inertial system, we do not know directly the laws of configuration of rigid bodies with respect to $K$ ', nor the laws of nature, in general. But we do know these laws with respect to the inertial system $K$ , and we can therefore estimate them with respect to $K$ '. Imagine a circle drawn about the origin in the $x'-y'$ plane of $K$ ' and a diameter of this circle. Imagine, further, that we have given a large number of rigid rods, all equal to each other. We suppose these laid in series along the periphery and the diameter of the circle, at rest relatively to $K$ '. If $U$ is the number of these rods along the periphery, $D$ the number along the diameter, then, if $K$ ' does not rotate relatively to $K$ , we shall have $\frac{U}{D} = \pi.$ [Pg 63] But if $K$ ' rotates we get a different result. Suppose that at a definite time $t$ of $K$ we determine the ends of all the rods. With respect to $K$ all the rods upon the periphery experience the Lorentz contraction, but the rods upon the diameter do not experience this contraction (along their lengths!).[14] It therefore follows that $\frac{U}{D} > \pi.$

The principle of equivalence states that when working in Galilean regions, we can also use non-inertial systems—meaning coordinate systems that are not free from acceleration and rotation compared to inertial systems. If we are going to completely eliminate the complex issue of why we prefer certain coordinate systems, then we must accept the use of arbitrarily moving coordinate systems. Once we take this approach seriously, we clash with the physical interpretation of space and time that we arrived at through the special theory of relativity. Let $K$ be a coordinate system whose $z$ -axis aligns with the $z$ -axis of $K$ , and that rotates around this axis at a constant angular velocity. Are the configurations of rigid bodies, which are at rest relative to $K$ ', consistent with the laws of Euclidean geometry? Since $K$ ' is not an inertial system, we don't have direct knowledge of the laws of configuration for rigid bodies concerning $K$ ', nor do we have a clear understanding of the laws of nature in general. However, we do understand these laws concerning the inertial system $K$ , and we can estimate them concerning $K$ '. Picture a circle centered at the origin in the $x'-y'$ plane of $K$ , along with a diameter of this circle. Now, imagine we have a large number of rigid rods, all identical. We place these rods in a line along the edge and the diameter of the circle, at rest relative to $K$ '. If $U$ is the number of rods along the perimeter, and $D$ is the number along the diameter, then if $K$ ' does not rotate relative to $K$ , we would have $\frac{U}{D} = \pi.$ [Pg 63] However, if $K$ rotates, we get a different result. Let's say that at a specific time $t$ of $K$ we identify the ends of all the rods. In relation to $K$ all the rods on the perimeter experience Lorentz contraction, but those on the diameter do not have this contraction (along their lengths!).[14] This leads to the conclusion that $\frac{U}{D} > \pi.$

It therefore follows that the laws of configuration of rigid bodies with respect to $K$ ' do not agree with the laws of configuration of rigid bodies that are in accordance with Euclidean geometry. If, further, we place two similar clocks (rotating with $K$ '), one upon the periphery, and the other at the centre of the circle, then, judged from $K$ , the clock on the periphery will go slower than the clock at the centre. The same thing must take place, judged from $K$ ', if we define time with respect to $K$ ' in a not wholly unnatural way, that is, in such a way that the laws with respect to $K$ ' depend explicitly upon the time. Space and time, therefore, cannot be defined with respect to $K$ ' as they were in the special theory of relativity with respect to inertial systems. But, according to the principle of equivalence, $K$ ' is also to be considered as a system at rest, with respect to which there is a gravitational field (field of centrifugal force, and force of Coriolis). We therefore arrive at the result: the gravitational field influences and even determines the metrical laws of the space-time continuum. If the laws of configuration of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field the geometry is not Euclidean.

It follows that the ways rigid bodies are arranged with respect to $K$ do not match the ways rigid bodies are arranged according to Euclidean geometry. Furthermore, if we place two similar clocks (rotating with $K$ ), one at the edge and the other at the center of the circle, then, viewed from $K$ , the clock on the edge will run slower than the clock at the center. The same phenomenon will occur, viewed from $K$ , if we define time with respect to $K$ in a somewhat natural way, meaning that the laws concerning $K$ depend clearly on time. Therefore, space and time cannot be defined regarding $K$ as they were in the special theory of relativity regarding inertial systems. However, according to the principle of equivalence, $K$ should also be seen as a system at rest, in relation to which there is a gravitational field (a field of centrifugal force and Coriolis force). Thus, we conclude that the gravitational field affects and even determines the geometric laws of the space-time continuum. If the arrangements of ideal rigid bodies are to be expressed geometrically, then in the presence of a gravitational field, the geometry is not Euclidean.

[14]These considerations assume that the behaviour of rods and clocks depends only upon velocities, and not upon accelerations, or, at least, that the influence of acceleration does not counteract that of velocity.

[14]These considerations assume that the behavior of rods and clocks depends only on speeds, and not on accelerations, or, at least, that the effect of acceleration doesn't negate the effect of speed.

[Pg 64]

The case that we have been considering is analogous to that which is presented in the two-dimensional treatment of surfaces. It is impossible in the latter case also, to introduce co-ordinates on a surface (e.g. the surface of an ellipsoid) which have a simple metrical significance, while on a plane the Cartesian co-ordinates, $x_{1}$ , $x_{2}$ , signify directly lengths measured by a unit measuring rod. Gauss overcame this difficulty, in his theory of surfaces, by introducing curvilinear co-ordinates which, apart from satisfying conditions of continuity, were wholly arbitrary, and afterwards these co-ordinates were related to the metrical properties of the surface. In an analogous way we shall introduce in the general theory of relativity arbitrary co-ordinates, $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ , which shall number uniquely the space-time points, so that neighbouring events are associated with neighbouring values of the co-ordinates; otherwise, the choice of co-ordinates is arbitrary. We shall be true to the principle of relativity in its broadest sense if we give such a form to the laws that they are valid in every such four-dimensional system of co-ordinates, that is, if the equations expressing the laws are co-variant with respect to arbitrary transformations.

The case we’ve been looking at is similar to what's seen in the two-dimensional analysis of surfaces. Just like in that scenario, it's impossible to create coordinates on a surface (like the surface of an ellipsoid) that have a clear metric meaning. However, on a plane, Cartesian coordinates, $x_{1}$ , $x_{2}$ , directly represent lengths measured with a standard measuring stick. Gauss tackled this problem in his theory of surfaces by using curvilinear coordinates that, aside from meeting continuity requirements, were entirely arbitrary. He then connected these coordinates to the metric properties of the surface. Similarly, in the general theory of relativity, we will introduce arbitrary coordinates, $x_{1}$ , $x_{2}$ , $x_{3}$ , $x_{4}$ , to uniquely identify the points in space-time, ensuring that nearby events correspond to nearby coordinate values; otherwise, the choice of coordinates is arbitrary. We will uphold the principle of relativity in its broadest interpretation by structuring the laws so that they hold true in any such four-dimensional coordinate system, meaning that the equations describing the laws are covariant with respect to arbitrary transformations.

The most important point of contact between Gauss's theory of surfaces and the general theory of relativity lies in the metrical properties upon which the concepts of both theories, in the main, are based. In the case of the theory of surfaces, Gauss's argument is as follows. Plane geometry may be based upon the concept of the distance $ds$ , between two indefinitely near points. The concept of this distance is physically significant because the distance can be measured directly by means of a rigid measuring rod. By a suitable choice of Cartesian co-ordinates this [Pg 65] distance may be expressed by the formula $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}$ . We may base upon this quantity the concepts of the straight line as the geodesic ( $\delta\! \int\!\! ds = 0$ ), the interval, the circle, and the angle, upon which the Euclidean plane geometry is built. A geometry may be developed upon another continuously curved surface, if we observe that an infinitesimally small portion of the surface may be regarded as plane, to within relatively infinitesimal quantities. There are Cartesian co-ordinates, $X_{1}$ , $X_{2}$ , upon such a small portion of the surface, and the distance between two points, measured by a measuring rod, is given by $ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}.$ If we introduce arbitrary curvilinear co-ordinates, $x_{1}$ , $x_{2}$ , on the surface, then $dX_{1}$ , $dX_{2}$ , may be expressed linearly in terms of $dx_{1}$ , $dx_{2}$ . Then everywhere upon the surface we have $ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + g_{22}\, {dx_{2}}^{2},$ where $g_{11}$ , $g_{12}$ , $g_{22}$ are determined by the nature of the surface and the choice of co-ordinates; if these quantities are known, then it is also known how networks of rigid rods may be laid upon the surface. In other words, the geometry of surfaces may be based upon this expression for $ds^{2}$ exactly as plane geometry is based upon the corresponding expression.

The main connection between Gauss's surface theory and general relativity lies in the metrical properties that both theories fundamentally rely on. In Gauss's surface theory, he asserts that plane geometry can be derived from the idea of distance $ds$ between two very close points. This distance is significant in a physical sense because it can be measured directly with a rigid measuring rod. By appropriately choosing Cartesian coordinates, this distance can be expressed with the formula $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2}$ . From this quantity, we can define concepts such as the straight line as the geodesic ( $\delta\! \int\!\! ds = 0$ ), the interval, the circle, and the angle, which together form the basis of Euclidean plane geometry. A different geometry can emerge on another continuously curved surface if we note that a tiny section of the surface can be approximated as flat to a degree that is negligible. There are Cartesian coordinates, $X_{1}$ , $X_{2}$ , for such a small section of the surface, and the distance between two points, measured with a measuring rod, is calculated as $ds^{2} = {dX_{1}}^{2} + {dX_{2}}^{2}.$ If we use arbitrary curvilinear coordinates, $x_{1}$ , $x_{2}$ , on the surface, then $dX_{1}$ , $dX_{2}$ can be expressed linearly in terms of $dx_{1}$ , $dx_{2}$ . Thus, everywhere on the surface, we have $ds^{2} = g_{11}\, {dx_{1}}^{2} + 2g_{12}\, dx_{1}\, dx_{2} + g_{22}\, {dx_{2}}^{2},$ where $g_{11}$ , $g_{12}$ , $g_{22}$ depend on the properties of the surface and the coordinate choice; once these quantities are established, the layout of rigid rods on the surface can also be understood. In other words, the geometry of surfaces can be founded on this expression for $ds^{2}$ just like plane geometry is grounded on the corresponding expression.

There are analogous relations in the four-dimensional space-time continuum of physics. In the immediate neighbourhood of an observer, falling freely in a gravitational field, there exists no gravitational field. We can therefore always regard an infinitesimally small region of the space-time continuum as Galilean. For such an infinitely small region there will be an inertial system (with the space co-ordinates, $X_{1}$ , $X_{2}$ , $X_{3}$ , and the time [Pg 66] co-ordinate $X_{4}$ ) relatively to which we are to regard the laws of the special theory of relativity as valid. The quantity which is directly measurable by our unit measuring rods and clocks, ${dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2},$ or its negative, $ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}, \qquad \text{(54)}$ is therefore a uniquely determinate invariant for two neighbouring events (points in the four-dimensional continuum), provided that we use measuring rods that are equal to each other when brought together and superimposed, and clocks whose rates are the same when they are brought together. In this the physical assumption is essential that the relative lengths of two measuring rods and the relative rates of two clocks are independent, in principle, of their previous history. But this assumption is certainly warranted by experience; if it did not hold there could be no sharp spectral lines; for the single atoms of the same element certainly do not have the same history, and it would be absurd to suppose any relative difference in the structure of the single atoms due to their previous history if the mass and frequencies of the single atoms of the same element were always the same.

There are similar relationships in the four-dimensional space-time continuum of physics. Right next to an observer falling freely in a gravitational field, there is no gravitational field. Therefore, we can always consider an infinitely small part of the space-time continuum as Galilean. In such an infinitely small area, there will be an inertial system (with the spatial coordinates, $X_{1}$ , $X_{2}$ , $X_{3}$ , and the time coordinate $X_{4}$ ) within which we can apply the laws of the special theory of relativity. The quantity that can be directly measured with our unit measuring rods and clocks, ${dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2} - {dX_{4}}^{2},$ or its negative, $ds^{2} = -{dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + {dX_{4}}^{2}, \qquad \text{(54)}$ is thus a uniquely defined invariant for two neighboring events (points in the four-dimensional continuum), as long as we use measuring rods that are equal when they are aligned and superimposed, and clocks that run at the same rate when brought together. The fundamental physical assumption here is that the relative lengths of two measuring rods and the relative rates of two clocks are, in principle, independent of their past. This assumption is definitely supported by experience; if it weren’t true, there wouldn’t be any sharp spectral lines, because individual atoms of the same element certainly do not have the same history, and it would be unreasonable to suggest any relative differences in the structure of individual atoms based on their previous history if the mass and frequencies of the individual atoms of the same element were always identical.

Space-time regions of finite extent are, in general, not Galilean, so that a gravitational field cannot be done away with by any choice of co-ordinates in a finite region. There is, therefore, no choice of co-ordinates for which the metrical relations of the special theory of relativity hold in a finite region. But the invariant $ds$ always exists for two neighbouring points (events) of the continuum. This invariant $ds$ may be [Pg 67] expressed in arbitrary co-ordinates. If one observes that the local $dX_{\nu}$ may be expressed linearly in terms of the co-ordinate differentials $dx_{\nu}$ , $ds^{2}$ may be expressed in the form $ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. \qquad \text{(55)}$

Space-time regions of finite size are generally not Galilean, meaning that you can't eliminate a gravitational field by just choosing different coordinates in a limited area. Therefore, there isn't any set of coordinates where the metric relations of the special theory of relativity are valid in a finite region. However, the invariant $ds$ always exists for two neighboring points (events) in the continuum. This invariant $ds$ can be expressed in any coordinates. If you notice that the local $dX_{\nu}$ can be expressed linearly in terms of the coordinate differentials $dx_{\nu}$ , then $ds^{2}$ can be expressed as $ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. \qquad \text{(55)}$

The functions $g_{\mu\nu}$ describe, with respect to the arbitrarily chosen system of co-ordinates, the metrical relations of the space-time continuum and also the gravitational field. As in the special theory of relativity, we have to discriminate between time-like and space-like line elements in the four-dimensional continuum; owing to the change of sign introduced, time-like line elements have a real, space-like line elements an imaginary $ds$ . The time-like $ds$ can be measured directly by a suitably chosen clock.

The functions $g_{\mu\nu}$ describe, regarding an arbitrarily chosen coordinate system, the geometric relationships of the space-time continuum and the gravitational field. Similar to the special theory of relativity, we need to distinguish between time-like and space-like line elements in the four-dimensional continuum; because of the change in sign, time-like line elements are real, while space-like line elements are imaginary $ds$ . The time-like $ds$ can be measured directly using a properly chosen clock.

According to what has been said, it is evident that the formulation of the general theory of relativity assumes a generalization of the theory of invariants and the theory of tensors; the question is raised as to the form of the equations which are co-variant with respect to arbitrary point transformations. The generalized calculus of tensors was developed by mathematicians long before the theory of relativity. Riemann first extended Gauss's train of thought to continua of any number of dimensions; with prophetic vision he saw the physical meaning of this generalization of Euclid's geometry. Then followed the development of the theory in the form of the calculus of tensors, particularly by Ricci and Levi-Civita. This is the place for a brief presentation of the most important mathematical concepts and operations of this calculus of tensors.

According to what has been stated, it's clear that the formulation of the general theory of relativity involves a broader understanding of the theory of invariants and the theory of tensors. The question arises regarding the form of the equations that are covariant with respect to any arbitrary point transformations. The generalized calculus of tensors was developed by mathematicians long before the theory of relativity. Riemann was the first to extend Gauss’s ideas to continua of any number of dimensions; with foresight, he recognized the physical implications of this generalization of Euclidean geometry. This was followed by the development of the theory in the form of tensor calculus, especially by Ricci and Levi-Civita. Here, we will briefly present the most important mathematical concepts and operations of this tensor calculus.

We designate four quantities, which are defined as functions of the $x_{\nu}$ with respect to every system of co-ordinates, as components, [Pg 68] $A^{\nu}$ , of a contra-variant vector, if they transform in a change of co-ordinates as the co-ordinate differentials $dx_{\nu}$ . We therefore have ${{A'}^{\mu}} = \frac{{\partial x'}_{\mu}}{\partial x_{\nu}} A^{\nu}. \qquad \text{(56)}$ Besides these contra-variant vectors, there are also co-variant vectors. If $B_{\nu}$ are the components of a co-variant vector, these vectors are transformed according to the rule ${B'}_{\mu} = \frac{\partial x_{\nu}}{{\partial {x'}}_{\mu}} B_{\nu}. \qquad \text{(57)}$ The definition of a co-variant vector is chosen in such a way that a co-variant vector and a contra-variant vector together form a scalar according to the scheme, $\phi = B_{\nu} A^{\nu}\quad \text{(summed over the } \nu).$ Accordingly, ${B'}_{\mu} {{A'}^{\mu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} B_{\alpha} A^{\beta} = B_{\alpha} A^{\alpha}.$ In particular, the derivatives $\dfrac{\partial \phi}{\partial x_{\alpha}}$ of a scalar $\phi$ , are components of a co-variant vector, which, with the co-ordinate differentials, form the scalar $\dfrac{\partial \phi}{\partial x_{\alpha}}\, dx_{\alpha}$ ; we see from this example how natural is the definition of the co-variant vectors.

We define four quantities as functions of the $x_{\nu}$ for each coordinate system, which we call components, [Pg 68] $A^{\nu}$ , of a contravariant vector, if they change during a coordinate transformation like the coordinate differentials $dx_{\nu}$ . Therefore, we have ${{A'}^{\mu}} = \frac{{\partial x'}_{\mu}}{\partial x_{\nu}} A^{\nu}. \qquad \text{(56)}$ In addition to these contravariant vectors, there are also covariant vectors. If $B_{\nu}$ are the components of a covariant vector, these vectors transform according to the rule ${B'}_{\mu} = \frac{\partial x_{\nu}}{{\partial {x'}}_{\mu}} B_{\nu}. \qquad \text{(57)}$ The definition of a covariant vector is designed so that a covariant vector and a contravariant vector together form a scalar according to the formula, $\phi = B_{\nu} A^{\nu}\quad \text{(summed over the } \nu).$ Thus, ${B'}_{\mu} {{A'}^{\mu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} B_{\alpha} A^{\beta} = B_{\alpha} A^{\alpha}.$ In particular, the derivatives $\dfrac{\partial \phi}{\partial x_{\alpha}}$ of a scalar $\phi$ , are components of a covariant vector, which, together with the coordinate differentials, form the scalar $\dfrac{\partial \phi}{\partial x_{\alpha}}\, dx_{\alpha}$ . This example shows how natural the definition of covariant vectors is.

There are here, also, tensors of any rank, which may have co-variant or contra-variant character with respect to each index; as with vectors, the character is designated by the position [Pg 69] of the index. For example, ${A_{\mu}^{\nu}}$ denotes a tensor of the second rank, which is co-variant with respect to the index $\mu$ , and contra-variant with respect to the index $\nu$ . The tensor character indicates that the equation of transformation is ${{A'}_{\mu}^{\nu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\nu}}{\partial x_{\beta}} A_{\alpha}^{\beta}. \qquad \text{(58)}$

There are also tensors of any rank here, which can have either co-variant or contra-variant characteristics depending on each index; similar to vectors, the characteristic is indicated by the position of the index. For example, ${A_{\mu}^{\nu}}$ represents a second-rank tensor that is co-variant with respect to the index $\mu$ , and contra-variant with respect to the index $\nu$ . The tensor characteristic shows that the transformation equation is ${{A'}_{\mu}^{\nu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\nu}}{\partial x_{\beta}} A_{\alpha}^{\beta}. \qquad \text{(58)}$

Tensors may be formed by the addition and subtraction of tensors of equal rank and like character, as in the theory of invariants of orthogonal linear substitutions, for example, $A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}. \qquad \text{(59)}$ The proof of the tensor character of $C_{\mu}^{\nu}$ depends upon (58).

Tensors can be created by adding and subtracting tensors of the same rank and similar type, just like in the theory of invariants of orthogonal linear transformations, for example, $A_{\mu}^{\nu} + B_{\mu}^{\nu} = C_{\mu}^{\nu}. \qquad \text{(59)}$ The proof that $C_{\mu}^{\nu}$ has tensor characteristics relies on (58).

Tensors may be formed by multiplication, keeping the character of the indices, just as in the theory of invariants of linear orthogonal transformations, for example, $A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}. \qquad \text{(60)}$ The proof follows directly from the rule of transformation.

Tensors can be created through multiplication, maintaining the nature of the indices, similar to the theory of invariants of linear orthogonal transformations. For example, $A_{\mu}^{\nu} B_{\sigma\tau} = C_{\mu\sigma\tau}^{\nu}. \qquad \text{(60)}$ The proof comes directly from the transformation rule.

Tensors may be formed by contraction with respect to two indices of different character, for example, $A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}. \qquad \text{(61)}$ The tensor character of $A_{\mu\sigma\tau}^{\mu}$ determines the tensor character of $B_{\sigma\tau}$ . Proof— ${{A'}_{\mu\sigma\tau}^{\mu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} {A_{\alpha st}^{\beta}} = \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} A_{\alpha st}^{\alpha}.$ [Pg 70]

Tensors can be created by contracting two indices of different types. For example, $A_{\mu\sigma\tau}^{\mu} = B_{\sigma\tau}. \qquad \text{(61)}$ The tensor nature of $A_{\mu\sigma\tau}^{\mu}$ determines the tensor nature of $B_{\sigma\tau}$ . Proof— ${{A'}_{\mu\sigma\tau}^{\mu}} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}} \frac{{\partial x'}_{\mu}}{\partial x_{\beta}} \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} {A_{\alpha st}^{\beta}} = \frac{\partial x_{s}}{{\partial x'}_{\sigma}} \frac{\partial x_{t}}{{\partial x'}_{\tau}} A_{\alpha st}^{\alpha}.$ [Pg 70]

The properties of symmetry and skew-symmetry of a tensor with respect to two indices of like character have the same significance as in the theory of invariants.

The properties of symmetry and skew-symmetry of a tensor concerning two similar indices have the same importance as in the theory of invariants.

With this, everything essential has been said with regard to the algebraic properties of tensors.

With this, everything important has been said about the algebraic properties of tensors.

The Fundamental Tensor. It follows from the invariance of $ds^{2}$ for an arbitrary choice of the $dx_{\nu}$ , in connexion with the condition of symmetry consistent with (55), that the $g_{\mu\nu}$ are components of a symmetrical co-variant tensor (Fundamental Tensor). Let us form the determinant, $g$ , of the $g_{\mu\nu}$ , and also the minors, divided by $g$ , corresponding to the single $g_{\mu\nu}$ . These minors, divided by $g$ , will be denoted by $g^{\mu\nu}$ , and their co-variant character is not yet known. Then we have $g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} = \begin{cases} 1 & \text{if $\alpha = \beta,$} \\ 0 & \text{if $\alpha \neq \beta.$} \end{cases} \qquad \text{(62)}$

The Fundamental Tensor. It follows from the invariance of $ds^{2}$ for any choice of the $dx_{\nu}$ , in connection with the symmetry condition consistent with (55), that the $g_{\mu\nu}$ are components of a symmetric co-variant tensor (Fundamental Tensor). Let's compute the determinant, $g$ , of the $g_{\mu\nu}$ , as well as the minors, divided by $g$ that correspond to the single $g_{\mu\nu}$ . These minors, divided by $g$ , will be labeled as $g^{\mu\nu}$ , and their co-variant nature is not yet determined. Then we have $g_{\mu\alpha} g^{\mu\beta} = \delta_{\alpha}^{\beta} = \begin{cases} 1 & \text{if $\alpha = \beta,$} \\ 0 & \text{if $\alpha \neq \beta.$} \end{cases} \qquad \text{(62)}$

If we form the infinitely small quantities (co-variant vectors) $d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}, \qquad \text{(63)}$ multiply by $g^{\mu\nu}$ and sum over the $\mu$ , we obtain, by the use of (62), $dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}. \qquad \text{(64)}$ Since the ratios of the $d\xi_{\mu}$ , are arbitrary, and the $dx_{\beta}$ as well as the $dx_{\mu}$ are components of vectors, it follows that the $g^{\mu\nu}$ are the components of a contra-variant tensor [15] (contra-variant fundamental tensor). The tensor character of $\delta_{\alpha}^{\beta}$ (mixed fundamental [Pg 71] tensor) accordingly follows, by (62). By means of the fundamental tensor, instead of tensors with co-variant index character, we can introduce tensors with contra-variant index character, and conversely. For example, $\begin{align*} A^{\mu} &= g^{\mu\alpha} A_{\alpha}, \\ A_{\mu} &= g_{\mu\alpha} A^{\alpha}, \\ T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. \end{align*}$

If we take the infinitely small quantities (co-variant vectors) $d\xi_{\mu} = g_{\mu\alpha}\, dx_{\alpha}, \qquad \text{(63)}$ multiply by $g^{\mu\nu}$ and sum over the $\mu$ , we find, using (62), $dx_{\beta} = g^{\beta\mu}\, d\xi_{\mu}. \qquad \text{(64)}$ Since the ratios of the $d\xi_{\mu}$ are arbitrary, and the $dx_{\beta}$ as well as the $dx_{\mu}$ are components of vectors, it follows that the $g^{\mu\nu}$ are the components of a contra-variant tensor [15] (contra-variant fundamental tensor). The tensor nature of $\delta_{\alpha}^{\beta}$ (mixed fundamental tensor) therefore follows from (62). Using the fundamental tensor, instead of tensors with co-variant index characteristics, we can introduce tensors with contra-variant index characteristics, and vice versa. For example, $\begin{align*} A^{\mu} &= g^{\mu\alpha} A_{\alpha}, \\ A_{\mu} &= g_{\mu\alpha} A^{\alpha}, \\ T_{\mu}^{\sigma} &= g^{\sigma\nu} T_{\mu\nu}. \end{align*}$

[15]If we multiply (64) by $\dfrac{{\partial x'}_{\alpha}}{\partial x_{\beta}}$ , sum over the $\beta$ , and replace the $d\xi^{\mu}$ by a transformation to the accented system, we obtain ${dx'}_{\alpha} = \frac{{\partial x'}_{\sigma}}{\partial x_{\mu}}\, \frac{{\partial x'}_{\alpha}}{\partial x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}.$ The statement made above follows from this, since, by (64), we must also have ${dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}$ and both equations must hold for every choice of ${d\xi'}_{\sigma}$ .

[15]If we multiply (64) by $\dfrac{{\partial x'}_{\alpha}}{\partial x_{\beta}}$ , sum over the $\beta$ , and replace the $d\xi^{\mu}$ with a transformation to the accented system, we get ${dx'}_{\alpha} = \frac{{\partial x'}_{\sigma}}{\partial x_{\mu}}\, \frac{{\partial x'}_{\alpha}}{\partial x_{\beta}}\, g^{\mu\beta}\, {d\xi'}_{\sigma}.$ The statement made above follows from this, since, according to (64), we must also have ${dx'}_{\alpha} = g^{\sigma\alpha'}\, {d\xi'}_{\alpha}$ and both equations must hold for every choice of ${d\xi'}_{\sigma}$ .

Volume Invariants. The volume element $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx$ is not an invariant. For by Jacobi's theorem, $dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx. \qquad \text{(65)}$ But we can complement $dx$ so that it becomes an invariant. If we form the determinant of the quantities ${g'}_{\mu\nu} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}}\, \frac{\partial x_{\beta}}{{\partial x'}_{\nu}}\, g_{\alpha\beta},$ [Pg 72] we obtain, by a double application of the theorem of multiplication of determinants, $g' = |{g'}_{\mu\nu}| = \left|\frac{\partial x_{\nu}}{{\partial x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| = \left|\frac{{\partial x'}_{\mu}}{\partial x_{\nu}}\right|^{-2} g. \qquad \text{(66)}$ We therefore get the invariant, $\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx.$

Volume Invariants. The volume element $\int dx_{1}\, dx_{2}\, dx_{3}\, dx_{4} = dx$ is not invariant. According to Jacobi's theorem, $dx' = \left| \frac{{dx'}_{\mu}}{dx_{\nu}}\right| dx. \qquad \text{(65)}$ However, we can supplement $dx$ to make it invariant. If we compute the determinant of the quantities ${g'}_{\mu\nu} = \frac{\partial x_{\alpha}}{{\partial x'}_{\mu}}\, \frac{\partial x_{\beta}}{{\partial x'}_{\nu}}\, g_{\alpha\beta},$ [Pg 72] we obtain, through a double application of the determinant multiplication theorem, $g' = |{g'}_{\mu\nu}| = \left|\frac{\partial x_{\nu}}{{\partial x'}_{\mu}}\right|^{2}·|g_{\mu\nu}| = \left|\frac{{\partial x'}_{\mu}}{\partial x_{\nu}}\right|^{-2} g. \qquad \text{(66)}$ Thus, we obtain the invariant, $\sqrt{g'}\, dx' = \sqrt{g\vphantom{g'}}\, dx.$

Formation of Tensors by Differentiation. Although the algebraic operations of tensor formation have proved to be as simple as in the special case of invariance with respect to linear orthogonal transformations, nevertheless in the general case, the invariant differential operations are, unfortunately, considerably more complicated. The reason for this is as follows. If $A^{\mu}$ is a contra-variant vector, the coefficients of its transformation, $\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}$ , are independent of position only if the transformation is a linear one. For then the vector components, $A^{\mu} + \dfrac{\partial A^{\mu}}{\partial x_{\alpha}}\, dx_{\alpha}$ , at a neighbouring point transform in the same way as the $A^{\mu}$ , from which follows the vector character of the vector differentials, and the tensor character of $\dfrac{\partial A^{\mu}}{\partial x_{\alpha}}$ . But if the $\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}$ are variable this is no longer true.

Formation of Tensors by Differentiation. While the algebraic operations for creating tensors are straightforward in cases involving invariance under linear orthogonal transformations, in more general situations, the invariant differential operations unfortunately become much more complex. The reason for this is as follows. If $A^{\mu}$ is a contravariant vector, the transformation coefficients $\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}$ are only position-independent if the transformation is linear. In this case, the vector components $A^{\mu} + \dfrac{\partial A^{\mu}}{\partial x_{\alpha}}\, dx_{\alpha}$ at a neighboring point transform in the same way as the $A^{\mu}$ , which establishes the vector nature of the vector differentials, and the tensor nature of $\dfrac{{\partial x'}_{\mu}}{\partial x_{\nu}}$ are variable, this is no longer the case.

That there are, nevertheless, in the general case, invariant differential operations for tensors, is recognized most satisfactorily in the following way, introduced by Levi-Civita and Weyl. Let ( $A^{\mu}$ ) be a contra-variant vector whose components are given with respect to the co-ordinate system of the $x_{\nu}$ . Let $P_{1}$ and $P_{2}$ [Pg 73] be two infinitesimally near points of the continuum. For the infinitesimal region surrounding the point $P_{1}$ , there is, according to our way of considering the matter, a co-ordinate system of the $X_{\nu}$ (with imaginary $X_{4}$ -co-ordinate) for which the continuum is Euclidean. Let $A_{(1)}^{\mu}$ be the co-ordinates of the vector at the point $P_{1}$ . Imagine a vector drawn at the point $P_{2}$ , using the local system of the $X_{\nu}$ , with the same co-ordinates (parallel vector through $P_{2}$ ), then this parallel vector is uniquely determined by the vector at $P_{1}$ and the displacement. We designate this operation, whose uniqueness will appear in the sequel, the parallel displacement of the vector $A_{\mu}$ from $P_{1}$ to the infinitesimally near point $P_{2}$ . If we form the vector difference of the vector ( $A^{\mu}$ ) at the point $P_{2}$ and the vector obtained by parallel displacement from $P_{1}$ to $P_{2}$ , we get a vector which may be regarded as the differential of the vector ( $A^{\mu}$ ) for the given displacement $dx_{\nu}$ .

That there are, nevertheless, in general, consistent differential operations for tensors, is recognized most clearly in the following manner, introduced by Levi-Civita and Weyl. Let $A^{\mu}$ be a contravariant vector whose components are defined in relation to the coordinate system of the $x_{\nu}$ . Let $P_{1}$ and $P_{2}$ be two points that are infinitesimally close in the continuum. For the infinitesimal area around the point $P_{1}$ , there is, based on our approach, a coordinate system of the $X_{\nu}$ (with an imaginary $X_{4}$ -coordinate) where the continuum is Euclidean. Let $A_{(1)}^{\mu}$ be the coordinates of the vector at the point $P_{1}$ . Imagine a vector drawn at the point $P_{2}$ , using the local system of the $X_{\nu}$ , with the same coordinates (a parallel vector through $P_{2}$ ), then this parallel vector is uniquely determined by the vector at $P_{1}$ and the displacement. We call this operation, whose uniqueness will become clear later, the parallel displacement of the vector $A_{\mu}$ from $P_{1}$ to the infinitesimally close point $P_{2}$ . If we take the vector difference between the vector ( $A^{\mu}$ ) at the point $P_{2}$ and the vector obtained by parallel displacement from $P_{1}$ to $P_{2}$ , we get a vector that can be viewed as the differential of the vector ( $A^{\mu}$ ) for the given displacement $dx_{\nu}$ .

This vector displacement can naturally also be considered with respect to the co-ordinate system of the $x_{\nu}$ . If $A^{\nu}$ are the co-ordinates of the vector at $P_{1}$ , $A^{\nu} + \delta A^{\nu}$ the co-ordinates of the vector displaced to $P_{2}$ along the interval ( $dx_{\nu}$ ), then the $\delta A^{\nu}$ do not vanish in this case. We know of these quantities, which do not have a vector character, that they must depend linearly and homogeneously upon the $dx_{\nu}$ and the $A^{\nu}$ . We therefore put $\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}. \qquad \text{(67)}$

This vector displacement can also be viewed in relation to the coordinate system of the $x_{\nu}$ . If $A^{\nu}$ represents the coordinates of the vector at $P_{1}$ , then $A^{\nu} + \delta A^{\nu}$ are the coordinates of the vector moved to $P_{2}$ along the interval ( $dx_{\nu}$ ). In this case, the $\delta A^{\nu}$ does not disappear. We understand that these quantities, which don’t have a vector character, must depend linearly and homogeneously on the $dx_{\nu}$ and the $A^{\nu}$ . Therefore, we express it as $\delta A^{\nu} = -\Gamma_{\alpha\beta}^{\nu} A^{\alpha}\, dx_{\beta}. \qquad \text{(67)}$

In addition, we can state that the $\Gamma_{\alpha\beta}^{\nu}$ must be symmetrical with respect to the indices $\alpha$ and $\beta$ . For we can assume from a representation by the aid of a Euclidean system of local co-ordinates that the same parallelogram will be described by the displacement of an element $d^{(1)}x_{\nu}$ along a second element $d^{(2)}x_{\nu}$ [Pg 74] as by a displacement of $d^{(2)}x_{\nu}$ along $d^{(1)}x_{\nu}$ . We must therefore have $\begin{aligned} d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ \quad &= d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). \end{aligned}$ The statement made above follows from this, after interchanging the indices of summation, $\alpha$ and $\beta$ , on the right-hand side.

In addition, we can say that the $\Gamma_{\alpha\beta}^{\nu}$ must be symmetrical with respect to the indices $\alpha$ and $d^{(1)}x_{\nu}$ along a second element $d^{(2)}x_{\nu}$ just as it would by moving $d^{(2)}x_{\nu}$ along $d^{(1)}x_{\nu}$ $\begin{aligned} d^{(2)}x_{\nu} + (d^{(1)}x_{\nu} - \Gamma_{\alpha\beta}^{\nu}\, d^{(1)}x_{\alpha}\, d^{(2)}x_{\beta}) \\ \quad &= d^{(1)}x_{\nu} + (d^{(2)}x_{\nu} - \Gamma_{\alpha\beta}^{\nu}\, d^{(2)}x_{\alpha}\, d^{(1)}x_{\beta}). \end{aligned}$ The statement above follows from this, after swapping the indices of summation, $\alpha$ and $\beta$ on the right-hand side.

Since the quantities $g_{\mu\nu}$ determine all the metrical properties of the continuum, they must also determine the $\Gamma_{\alpha\beta}^{\nu}$ . If we consider the invariant of the vector $A^{\nu}$ that is, the square of its magnitude, $g_{\mu\nu} A^{\mu} A^{\nu},$ which is an invariant, this cannot change in a parallel displacement. We therefore have $0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) = \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu}$ or, by (67), $\left(\frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} = 0.$

Since the quantities $g_{\mu\nu}$ determine all the metric properties of the continuum, they must also determine the $\Gamma_{\alpha\beta}^{\nu}$ . If we consider the invariant of the vector $A^{\nu}$ — that is, the square of its magnitude, $g_{\mu\nu} A^{\mu} A^{\nu},$ which is an invariant, this cannot change during parallel displacement. We therefore have $0 = \delta(g_{\mu\nu} A^{\mu} A^{\nu}) = \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} A^{\mu} A^{\nu}\, dx_{\alpha} + g_{\mu\nu} A^{\mu} \delta A^{\nu} + g_{\mu\nu} A^{\nu} \delta A^{\mu}$ or, by (67), $\left(\frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} - g_{\mu\beta} \Gamma_{\nu\alpha}^{\beta} - g_{\nu\beta} \Gamma_{\mu\alpha}^{\beta}\right) A^{\mu} A^{\nu}\, dx_{\alpha} = 0.$

Owing to the symmetry of the expression in the brackets with respect to the indices $\mu$ and $\nu$ , this equation can be valid for an arbitrary choice of the vectors ( $A^{\mu}$ ) and $dx_{\nu}$ only when the expression in the brackets vanishes for all combinations of the indices. By a cyclic interchange of the indices $\mu$ , $\nu$ , $\alpha$ , we obtain thus altogether three equations, from which we obtain, on taking into account the symmetrical property of the $\Gamma_{\mu\nu}^{\alpha}$ , $\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}, \qquad \text{(68)}$ [Pg 75] in which, following Christoffel, the abbreviation has been used, $\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] = \tfrac{1}{2}\left( \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} \right). \qquad \text{(69)}$

Due to the symmetry of the expression in the brackets regarding the indices $\mu$ and $\nu$ , this equation can hold true for any choice of the vectors ( $A^{\mu}$ ) and $dx_{\nu}$ only if the expression in the brackets equals zero for all combinations of the indices. By cyclically switching the indices $\mu$ , $\nu$ , $\alpha$ , we get a total of three equations, from which we derive, considering the symmetrical property of the $\Gamma_{\mu\nu}^{\alpha}$ , $\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] = g_{\alpha\beta} \Gamma_{\mu\nu}^{\beta}, \qquad \text{(68)}$ [Pg 75] where, following Christoffel, the abbreviation has been used, $\left[{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}}\right] = \tfrac{1}{2}\left( \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} \right). \qquad \text{(69)}$

If we multiply (68) by $g^{\alpha\sigma}$ and sum over the $\alpha$ , we obtain $\Gamma_{\mu\nu}^{\alpha} = \tfrac{1}{2} g^{\sigma\alpha}\left( \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} \right) = \left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}, \qquad \text{(70)}$ in which $\left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}$ is the Christoffel symbol of the second kind. Thus the quantities $\Gamma$ are deduced from the $g_{\mu\nu}$ . Equations (67) and (70) are the foundation for the following discussion.

If we multiply (68) by $g^{\alpha\sigma}$ and sum over the $\alpha$ , we get $\Gamma_{\mu\nu}^{\alpha} = \tfrac{1}{2} g^{\sigma\alpha}\left( \frac{\partial g_{\mu\alpha}}{\partial x_{\nu}} + \frac{\partial g_{\nu\alpha}}{\partial x_{\mu}} - \frac{\partial g_{\mu\nu}}{\partial x_{\alpha}} \right) = \left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}, \qquad \text{(70)}$ where $\left\{{\genfrac{}{}{0pt}{}{\mu\nu}{\alpha}}\right\}$ is the Christoffel symbol of the second kind. So the quantities $\Gamma$ are derived from the $g_{\mu\nu}$ . Equations (67) and (70) form the basis for the following discussion.

Co-variant Differentiation of Tensors. If ( $A^{\mu} + \delta A^{\mu}$ ) is the vector resulting from an infinitesimal parallel displacement from $P_{1}$ to $P_{2}$ , and ( $A^{\mu} + dA^{\mu}$ ) the vector $A^{\mu}$ at the point $P_{2}$ , then the difference of these two, $dA^{\mu} - \delta A^{\mu} = \left( \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma},$ is also a vector. Since this is the case for an arbitrary choice of the $dx_{\sigma}$ , it follows that ${A^{\mu}}_{;\, \sigma} = \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} \qquad \text{(71)}$ is a tensor, which we designate as the co-variant derivative of the tensor of the first rank (vector). Contracting this tensor, we obtain the divergence of the contra-variant tensor $A^{\mu}$ . In this we must observe that according to (70), $\Gamma_{\mu\sigma}^{\sigma} = \tfrac{1}{2} g^{\sigma\alpha} \frac{\partial g_{\sigma\alpha}}{\partial x_{\mu}} = \frac{1}{\sqrt{g}}\, \frac{\partial \sqrt{g}}{\partial x_{\mu}}. \qquad \text{(72)}$ [Pg 76] If we put, further, $A^{\mu} \sqrt{g} = \mathfrak A^{\mu}, \qquad \text{(73)}$ a quantity designated by Weyl as the contra-variant tensor density [16] of the first rank, it follows that, $\mathfrak A = \frac{\partial \mathfrak A^{\mu}}{\partial x_{\mu}} \qquad \text{(74)}$ is a scalar density.

Co-variant Differentiation of Tensors. If ( $A^{\mu} + \delta A^{\mu}$ ) is the vector that results from an infinitesimal parallel shift from $P_{1}$ to $P_{2}$ , and ( $A^{\mu} + dA^{\mu}$ ) represents the vector $A^{\mu}$ at the point $P_{2}$ , then the difference between these two, $dA^{\mu} - \delta A^{\mu} = \left( \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha}\right) dx_{\sigma},$ is also a vector. Since this holds for any choice of the $dx_{\sigma}$ , it follows that ${A^{\mu}}_{;\, \sigma} = \frac{\partial A^{\mu}}{\partial x_{\sigma}} + \Gamma_{\sigma\alpha}^{\mu} A^{\alpha} \qquad \text{(71)}$ is a tensor, which we call the co-variant derivative of the first-rank tensor (vector). By contracting this tensor, we get the divergence of the contra-variant tensor $A^{\mu}$ . Here, we must note that according to (70), $\Gamma_{\mu\sigma}^{\sigma} = \tfrac{1}{2} g^{\sigma\alpha} \frac{\partial g_{\sigma\alpha}}{\partial x_{\mu}} = \frac{1}{\sqrt{g}}\, \frac{\partial \sqrt{g}}{\partial x_{\mu}}. \qquad \text{(72)}$ [Pg 76] If we further set, $A^{\mu} \sqrt{g} = \mathfrak A^{\mu}, \qquad \text{(73)}$ a quantity referred to by Weyl as the contra-variant tensor density [16] of the first rank, it follows that, $\mathfrak A = \frac{\partial \mathfrak A^{\mu}}{\partial x_{\mu}} \qquad \text{(74)}$ is a scalar density.

[16]This expression is justified, in that $A^{\mu}\sqrt{g}\, dx = \mathbf A^{\mu}\, dx$ has a tensor character. Every tensor, when multiplied by $\sqrt{g}$ , changes into a tensor density. We employ capital Gothic letters for tensor densities.

[16]This expression makes sense because $A^{\mu}\sqrt{g}\, dx = \mathbf A^{\mu}\, dx$ has a tensor character. Every tensor, when multiplied by $\sqrt{g}$ , becomes a tensor density. We use capital Gothic letters for tensor densities.

We get the law of parallel displacement for the co-variant vector $B_{\mu}$ by stipulating that the parallel displacement shall be effected in such a way that the scalar $\phi = A^{\mu} B_{\mu}$ remains unchanged, and that therefore $A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu}$ vanishes for every value assigned to ( $A^{\mu}$ ). We therefore get $\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}. \qquad \text{(75)}$

We derive the law of parallel displacement for the covariant vector $B_{\mu}$ by specifying that the parallel displacement should be carried out in such a way that the scalar $\phi = A^{\mu} B_{\mu}$ remains constant. This means that $A^{\mu}\, \delta B_{\mu} + B_{\mu}\, \delta A^{\mu}$ must equal zero for every value of ( $A^{\mu}$ ). Thus, we find that $\delta B_{\mu} = \Gamma_{\mu\sigma}^{\alpha} A_{\alpha}\, dx_{\sigma}. \qquad \text{(75)}$

From this we arrive at the co-variant derivative of the co-variant vector by the same process as that which led to (71), $B_{\mu;\, \sigma} = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}. \qquad \text{(76)}$ [Pg 77] By interchanging the indices $\mu$ and $\beta$ , and subtracting, we get the skew-symmetrical tensor, $\phi_{\mu\sigma} = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \frac{\partial B_{\sigma}}{\partial x_{\mu}}. \qquad \text{(77)}$

From this, we get the covariant derivative of the covariant vector using the same method as in (71), $B_{\mu;\, \sigma} = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \Gamma_{\mu\sigma}^{\alpha} B_{\alpha}. \qquad \text{(76)}$ [Pg 77] By swapping the indices $\mu$ and $\beta$ , and subtracting, we obtain the skew-symmetric tensor, $\phi_{\mu\sigma} = \frac{\partial B_{\mu}}{\partial x_{\sigma}} - \frac{\partial B_{\sigma}}{\partial x_{\mu}}. \qquad \text{(77)}$

For the co-variant differentiation of tensors of the second and higher ranks we may use the process by which (75) was deduced. Let, for example, ( $A_{\sigma\tau}$ ) be a co-variant tensor of the second rank. Then $A_{\sigma\tau} E^{\sigma} F^{\tau}$ is a scalar, if $E$ and $F$ are vectors. This expression must not be changed by the $\delta$ -displacement; expressing this by a formula, we get, using (67), $\delta A_{\sigma\tau}$ , whence we get the desired co-variant derivative, $A_{\sigma\tau;\, \rho} = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}. \qquad \text{(78)}$

For the co-variant differentiation of tensors of the second and higher ranks, we can use the method by which (75) was derived. For example, let ( $A_{\sigma\tau}$ ) be a co-variant tensor of the second rank. Then $A_{\sigma\tau} E^{\sigma} F^{\tau}$ is a scalar if $E$ and $F$ are vectors. This expression must remain unchanged by the $\delta$ -displacement; expressing this with a formula, we have, using (67), $\delta A_{\sigma\tau}$ , from which we obtain the desired co-variant derivative, $A_{\sigma\tau;\, \rho} = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha\tau} - \Gamma_{\tau\rho}^{\alpha} A_{\sigma\alpha}. \qquad \text{(78)}$

In order that the general law of co-variant differentiation of tensors may be clearly seen, we shall write down two co-variant derivatives deduced in an analogous way: $\begin{align*} A_{\sigma;\, \rho}^{\tau} &= \frac{\partial A_{\sigma}^{\tau}}{\partial x_{\rho}} - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha},\, \qquad & &\text{(79)} \\ {A^{\sigma\tau}}_{;\, \rho} &= \frac{\partial A^{\sigma\tau}}{\partial x_{\rho}} + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}. \qquad & &\text{(80)} \end{align*}$ The general law of formation now becomes evident. From these formulae we shall deduce some others which are of interest for the physical applications of the theory.

To clearly understand the general law of co-variant differentiation of tensors, we will present two co-variant derivatives derived in a similar manner: $\begin{align*} A_{\sigma;\, \rho}^{\tau} &= \frac{\partial A_{\sigma}^{\tau}}{\partial x_{\rho}} - \Gamma_{\sigma\rho}^{\alpha} A_{\alpha}^{\tau} + \Gamma_{\alpha\rho}^{\tau} A_{\sigma}^{\alpha},\, \qquad & &\text{(79)} \\ {A^{\sigma\tau}}_{;\, \rho} &= \frac{\partial A^{\sigma\tau}}{\partial x_{\rho}} + \Gamma_{\alpha\rho}^{\sigma} A^{\alpha\tau} + \Gamma_{\alpha\rho}^{\tau} A^{\sigma\alpha}. \qquad & &\text{(80)} \end{align*}$ The general law of formation is now clear. From these formulas, we will derive some others that are relevant for the physical applications of the theory.

In case $A_{\sigma\tau}$ is skew-symmetrical, we obtain the tensor $A_{\sigma\tau\rho} = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + \frac{\partial A_{\tau\rho}}{\partial x_{\sigma}} + \frac{\partial A_{\rho\sigma}}{\partial x_{\tau}}, \qquad \text{(81)}$ [Pg 78] which is skew-symmetrical in all pairs of indices, by cyclic interchange and addition.

If $A_{\sigma\tau}$ is skew-symmetrical, we get the tensor $A_{\sigma\tau\rho} = \frac{\partial A_{\sigma\tau}}{\partial x_{\rho}} + \frac{\partial A_{\tau\rho}}{\partial x_{\sigma}} + \frac{\partial A_{\rho\sigma}}{\partial x_{\tau}}, \qquad \text{(81)}$ [Pg 78] which is skew-symmetrical in all pairs of indices through cyclic interchange and addition.

If, in (78), we replace $A_{\sigma\tau}$ by the fundamental tensor, $g_{\sigma\tau}$ , then the right-hand side vanishes identically; an analogous statement holds for (80) with respect to $g^{\sigma\tau}$ ; that is, the co-variant derivatives of the fundamental tensor vanish. That this must be so we see directly in the local system of co-ordinates.

If, in (78), we replace $A_{\sigma\tau}$ with the fundamental tensor, $g_{\sigma\tau}$ , then the right-hand side becomes zero; a similar statement applies for (80) concerning $g^{\sigma\tau}$ ; that is, the covariant derivatives of the fundamental tensor are zero. This is evident when we look directly at the local coordinate system.

In case $A^{\sigma\tau}$ is skew-symmetrical, we obtain from (80), by contraction with respect to $\tau$ and $\rho$ , $\mathfrak A^{\sigma} = \frac{\partial \mathfrak A^{\sigma\tau}}{\partial x_{\tau}}. \qquad \text{(82)}$

In case $A^{\sigma\tau}$ is skew-symmetrical, we get from (80), by contracting with respect to $\tau$ and $\rho$ , $\mathfrak A^{\sigma} = \frac{\partial \mathfrak A^{\sigma\tau}}{\partial x_{\tau}}. \qquad \text{(82)}$

In the general case, from (79) and (80), by contraction with respect to $\tau$ and $\rho$ , we obtain the equations, $\begin{align*} \mathfrak A_{\sigma} &= \frac{\partial \mathfrak A_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak A_{\alpha}^{\beta}, \qquad & &\text{(83)} \\ \mathfrak A^{\sigma} &= \frac{\partial \mathfrak A^{\sigma\alpha}}{\partial x_{\alpha}} + \Gamma_{\alpha\beta}^{\sigma} \mathfrak A^{\alpha\beta}. \qquad & &\text{(84)} \end{align*}$

In the general case, from (79) and (80), by contracting with respect to $\tau$ and $\rho$ , we derive the equations, $\begin{align*} \mathfrak A_{\sigma} &= \frac{\partial \mathfrak A_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak A_{\alpha}^{\beta}, \qquad & &\text{(83)} \\ \mathfrak A^{\sigma} &= \frac{\partial \mathfrak A^{\sigma\alpha}}{\partial x_{\alpha}} + \Gamma_{\alpha\beta}^{\sigma} \mathfrak A^{\alpha\beta}. \qquad & &\text{(84)} \end{align*}$

The Riemann Tensor. If we have given a curve extending from the point $P$ to the point $G$ of the continuum, then a vector $A^{\mu}$ , given at $P$ , may, by a parallel displacement, be moved along the curve to $G$ . If the continuum is Euclidean (more generally, if by a suitable choice of co-ordinates the $g_{\mu\nu}$ , are constants) then the vector obtained at $G$ as a result of this displacement does not depend upon the choice of the curve joining $P$ and $G$ . But otherwise, the result depends upon the path of the displacement.

The Riemann Tensor. If we have a curve extending from the point $P$ to the point $G$ , then a vector $A^{\mu}$ , defined at $P$ , can be moved along the curve to $G$ through a parallel displacement. If the continuum is Euclidean (more generally, if we choose coordinates such that the $g_{\mu\nu}$ , are constants), then the vector obtained at $G$ from this displacement does not depend on the curve chosen between $P$ and $G$ . However, in other cases, the result does depend on the path taken during the displacement.

FIG.4.

In this case, therefore, a vector suffers a change, $\Delta A^{\mu}$ (in its direction, not its magnitude), when it is carried from a [Pg 79] point $P$ of a closed curve, along the curve, and back to P. We shall now calculate this vector change: $\Delta A^{\mu} = \oint \delta A^{\mu}.$ As in Stokes' theorem for the line integral of a vector around a closed curve, this problem may be reduced to the integration around a closed curve with infinitely small linear dimensions; we shall limit ourselves to this case.

In this situation, a vector experiences a change, $\Delta A^{\mu}$ (in its direction, not its magnitude), when it is moved from a [Pg 79] point $P$ of a closed curve, along the curve, and back to P. We will now calculate this vector change: $\Delta A^{\mu} = \oint \delta A^{\mu}.$ Similar to Stokes' theorem for the line integral of a vector around a closed curve, this problem can be simplified to integration around a closed curve with infinitely small linear dimensions; we will focus on this scenario.

We have, first, by (67), $\Delta A^{\mu} = -\oint \Gamma_{\alpha\beta}^{\mu} A^{\alpha}\, dx_{\beta}.$

In this, $\Gamma_{\alpha\beta}^{\mu}$ is the value of this quantity at the variable point $G$ of the path of integration. If we put $\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P}$ [Pg 80] and denote the value of $\Gamma_{\alpha\beta}^{\mu}$ at $P$ by $\overline{\Gamma_{\alpha\beta}^{\mu}}$ then we have, with sufficient accuracy, $\Gamma_{\alpha\beta}^{\mu} = \overline{\Gamma_{\alpha\beta}^{\mu}} + \frac{\overline{\partial \Gamma_{\alpha\beta}^{\mu}}}{\partial x_{\nu}}\, \xi^{\nu}.$

In this, $\Gamma_{\alpha\beta}^{\mu}$ is the value of this quantity at the point $G$ along the integration path. If we set $\xi^{\mu} = (x_{\mu})_{G} - (x_{\mu})_{P}$ [Pg 80] and denote the value of $\Gamma_{\alpha\beta}^{\mu}$ at $P$ as $\overline{\Gamma_{\alpha\beta}^{\mu}}$ then we can say, with sufficient accuracy, $\Gamma_{\alpha\beta}^{\mu} = \overline{\Gamma_{\alpha\beta}^{\mu}} + \frac{\overline{\partial \Gamma_{\alpha\beta}^{\mu}}}{\partial x_{\nu}}\, \xi^{\nu}.$

Let, further, $A^{\alpha}$ be the value obtained from $\overline{A^{\alpha}}$ by a parallel displacement along the curve from $P$ to $G$ . It may now easily be proved by means of (67) that $A^{\mu}$ - $\overline{A^{\mu}}$ is infinitely small of the first order, while, for a curve of infinitely small dimensions of the first order, $\Delta A^{\mu}$ is infinitely small of the second order. Therefore there is an error of only the second order if we put $A^{\alpha} = \overline{A^{\alpha}} - \overline{\Gamma_{\sigma\tau}^{\alpha}}\; \overline{A^{\sigma}}\; \overline{\xi^{\tau}}.$

Let’s say $A^{\alpha}$ is the value obtained from $\overline{A^{\alpha}}$ by moving parallel along the curve from $P$ to $A^{\mu}$ - $\overline{A^{\mu}}$ is an infinitely small quantity of the first order, while for a curve of infinitely small dimensions of the first order, $\Delta A^{\mu}$ is an infinitely small quantity of the second order. Hence, there is only a second-order error if we assume $A^{\alpha} = \overline{A^{\alpha}} - \overline{\Gamma_{\sigma\tau}^{\alpha}}\; \overline{A^{\sigma}}\; \overline{\xi^{\tau}}.$

If we introduce these values of $\Gamma_{\alpha\beta}^{\mu}$ and $A^{\alpha}$ into the integral, we obtain, neglecting all quantities of a higher order of small quantities than the second, $\Delta A^{\mu} = - \left(\frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) A^{\sigma} \oint \xi^{\alpha}\, d\xi^{\beta}. \qquad \text{(85)}$ The quantity removed from under the sign of integration refers to the point $P$ . Subtracting $\dfrac{1}{2} d(\xi^{\alpha} \xi^{\beta})$ from the integrand, we obtain $\tfrac{1}{2} \oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}).$ This skew-symmetrical tensor of the second rank, $f^{\alpha\beta}$ , characterizes the surface element bounded by the curve in magnitude and position. If the expression in the brackets in (85) were skew-symmetrical with respect to the indices $\alpha$ and $\beta$ , we could [Pg 81] conclude its tensor character from (85). We can accomplish this by interchanging the summation indices $\alpha$ and $\beta$ in (85) and adding the resulting equation to (85). We obtain $2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}, \qquad \text{(86)}$ in which $R_{\sigma\alpha\beta}^{\mu} = - \frac{\partial \Gamma_{\sigma\alpha}^{\mu}}{\partial x_{\beta}} + \frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}. \qquad \text{(87)}$

If we plug in these values of $\Gamma_{\alpha\beta}^{\mu}$ and $A^{\alpha}$ into the integral, we get, ignoring all higher-order small quantities beyond the second order, $\Delta A^{\mu} = - \left(\frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}\right) A^{\sigma} \oint \xi^{\alpha}\, d\xi^{\beta}. \qquad \text{(85)}$ The quantity taken out from under the integral sign refers to the point $P$ . By subtracting $\dfrac{1}{2} d(\xi^{\alpha} \xi^{\beta})$ from the integrand, we obtain $\tfrac{1}{2} \oint (\xi^{\alpha}\, d\xi^{\beta} - \xi^{\beta}\, d\xi^{\alpha}).$ This skew-symmetric tensor of rank two, $f^{\alpha\beta}$ , describes the surface element defined by the curve in both size and position. If the expression in the brackets in (85) was skew-symmetric with respect to the indices $\alpha$ and $\beta$ , we could [Pg 81] conclude its tensor nature from (85). We can achieve this by swapping the summation indices $\alpha$ and $\beta$ in (85) and adding the resulting equation to (85). We obtain $2\Delta A^{\mu} = -R_{\sigma\alpha\beta}^{\mu} A^{\sigma} f^{\alpha\beta}, \qquad \text{(86)}$ in which $R_{\sigma\alpha\beta}^{\mu} = - \frac{\partial \Gamma_{\sigma\alpha}^{\mu}}{\partial x_{\beta}} + \frac{\partial \Gamma_{\sigma\beta}^{\mu}}{\partial x_{\alpha}} + \Gamma_{\rho\alpha}^{\mu} \Gamma_{\sigma\beta}^{\rho} - \Gamma_{\rho\beta}^{\mu} \Gamma_{\sigma\alpha}^{\rho}. \qquad \text{(87)}$

The tensor character of $R_{\sigma\alpha\beta}^{\mu}$ follows from (86); this is the Riemann curvature tensor of the fourth rank, whose properties of symmetry we do not need to go into. Its vanishing is a sufficient condition (disregarding the reality of the chosen co-ordinates) that the continuum is Euclidean.

The tensor nature of $R_{\sigma\alpha\beta}^{\mu}$ is derived from (86); this represents the Riemann curvature tensor of the fourth rank, and we won't delve into its symmetry properties. Its being zero is a sufficient condition (ignoring the reality of the selected coordinates) for the continuum to be Euclidean.

By contraction of the Riemann tensor with respect to the indices $\mu$ , $\beta$ , we obtain the symmetrical tensor of the second rank, $R_{\mu\nu} = - \frac{\partial \Gamma_{\mu\nu}^{\alpha}}{\partial x_{\alpha}} + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + \frac{\partial \Gamma_{\mu\alpha}^{\alpha}}{\partial x_{\nu}} - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}. \qquad \text{(88)}$ The last two terms vanish if the system of co-ordinates is so chosen that $g = \text{constant}$ . From $R_{\mu\nu}$ , we can form the scalar, $R = g^{\mu\nu} R_{\mu\nu}. \qquad \text{(89)}$

By contracting the Riemann tensor with respect to the indices $\mu$ , $\beta$ , we get the symmetrical tensor of the second rank, $R_{\mu\nu} = - \frac{\partial \Gamma_{\mu\nu}^{\alpha}}{\partial x_{\alpha}} + \Gamma_{\mu\beta}^{\alpha} \Gamma_{\nu\alpha}^{\beta} + \frac{\partial \Gamma_{\mu\alpha}^{\alpha}}{\partial x_{\nu}} - \Gamma_{\mu\nu}^{\alpha}\Gamma_{\alpha\beta}^{\beta}. \qquad \text{(88)}$ The last two terms disappear if the coordinate system is chosen so that $g = \text{constant}$ . From $R_{\mu\nu}$ , we can create the scalar, $R = g^{\mu\nu} R_{\mu\nu}. \qquad \text{(89)}$

Straightest Geodetic Lines. A line may be constructed in such a way that its successive elements arise from each other by parallel displacements. This is the natural generalization of the straight line of the Euclidean geometry. For such a line, we have $\delta \left(\frac{dx_{\mu}}{ds}\right) = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}.$ [Pg 82] The left-hand side is to be replaced by $\dfrac{d^{2} x_{\mu}}{ds^{2}}$ ,[17] so that we have $\frac{d^{2} x_{\mu}}{ds^{2}} + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0. \qquad \text{(90)}$ We get the same line if we find the line which gives a stationary value to the integral $\int ds\quad\text{or}\quad \int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}}$ between two points (geodetic line).

Straightest Geodetic Lines. A line can be created in a way that its consecutive parts connect to each other through parallel shifts. This is the natural extension of the straight line found in Euclidean geometry. For such a line, we have $\delta \left(\frac{dx_{\mu}}{ds}\right) = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, dx_{\beta}.$ [Pg 82] The left side should be replaced by $\dfrac{d^{2} x_{\mu}}{ds^{2}}$ ,[17] so we have $\frac{d^{2} x_{\mu}}{ds^{2}} + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0. \qquad \text{(90)}$ We arrive at the same line if we identify the line that gives a stationary value to the integral $\int ds\quad\text{or}\quad \int \sqrt{g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}}$ between two points (geodetic line).

[17]The direction vector at a neighbouring point of the curve results, by a parallel displacement along the line element ( $dx_{\beta}$ ), from the direction vector of each point considered.

[17]The direction vector at a nearby point on the curve comes from the direction vector of each point being considered, through a parallel shift along the line element ( $dx_{\beta}$ ).

[Pg 83]

LECTURE IV

THE GENERAL THEORY OF RELATIVITY
(continued)

WE are now in possession of the mathematical apparatus which is necessary to formulate the laws of the general theory of relativity. No attempt will be made in this presentation at systematic completeness, but single results and possibilities will be developed progressively from what is known and from the results obtained. Such a presentation is most suited to the present provisional state of our knowledge.

WE now have the mathematical tools needed to express the laws of the general theory of relativity. This presentation won’t aim for complete systematic coverage, but will gradually build on individual results and possibilities derived from what is known and the results we’ve achieved. This approach is most fitting for our current provisional understanding.

A material particle upon which no force acts moves, according to the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with real time co-ordinate) this is a real straight line. The natural, that is, the simplest, generalization of the straight line which is plausible in the system of concepts of Riemann's general theory of invariants is that of the straightest, or geodetic, line. We shall accordingly have to assume, in the sense of the principle of equivalence, that the motion of a material particle, under the action only of inertia and gravitation, is described by the equation, $\frac{d^{2} x_{\mu}}{ds^{2}} + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0. \qquad \text{(90)}$ In fact, this equation reduces to that of a straight line if all the components, $\Gamma_{\alpha\beta}^{\mu}$ , of the gravitational field vanish.

A material particle that isn't influenced by any force moves, based on the principle of inertia, uniformly in a straight line. In the four-dimensional continuum of the special theory of relativity (with a real time coordinate), this is a true straight line. The most straightforward and reasonable generalization of the straight line within the framework of Riemann's general theory of invariants is that of the straightest or geodetic line. Therefore, we need to assume, in line with the principle of equivalence, that the movement of a material particle, influenced only by inertia and gravity, is described by the equation, $\frac{d^{2} x_{\mu}}{ds^{2}} + \Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} = 0. \qquad \text{(90)}$ Actually, this equation simplifies to that of a straight line if all the components, $\Gamma_{\alpha\beta}^{\mu}$ , of the gravitational field are zero.

How are these equations connected with Newton's equations of motion? According to the special theory of relativity, the $g_{\mu\nu}$ as well as the $g^{\mu\nu}$ , have the values, with respect to an inertial [Pg 84] system (with real time co-ordinate and suitable choice of the sign of $ds^{2}$ ), $\left. \begin{array}{*{4}{>{\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right\}. \qquad \text{(91)}$ The equations of motion then become $\frac{d^{2} x_{\mu}}{ds^{2}} = 0.$ We shall call this the "first approximation" to the $g_{\mu\nu}$ -field. In considering approximations it is often useful, as in the special theory of relativity, to use an imaginary $x_{4}$ -co-ordinate, as then the $g_{\mu\nu}$ . to the first approximation, assume the values $\left. \begin{array}{*{4}{>{\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right\}. \qquad \text{(91a)}$ These values may be collected in the relation $g_{\mu\nu} = -\delta_{\mu\nu}.$ To the second approximation we must then put $g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, \qquad \text{(92)}$ where the $\gamma_{\mu\nu}$ are to be regarded as small of the first order. [Pg 85]

How are these equations related to Newton's equations of motion? According to the special theory of relativity, the $g_{\mu\nu}$ and the $g^{\mu\nu}$ have values in relation to an inertial [Pg 84] system (with real time coordinates and an appropriate choice of the sign of $ds^{2}$ ), $\left. \begin{array}{*{4}{>{\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & 1 \end{array} \right\}. \qquad \text{(91)}$ The equations of motion then simplify to $\frac{d^{2} x_{\mu}}{ds^{2}} = 0.$ We’ll refer to this as the "first approximation" of the $g_{\mu\nu}$ -field. When considering approximations, it is often helpful, as in the special theory of relativity, to use an imaginary $x_{4}$ -coordinate, since then the $g_{\mu\nu}$ to the first approximation takes the values $\left. \begin{array}{*{4}{>{\quad}r}} -1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{array} \right\}. \qquad \text{(91a)}$ These values can be collected in the relation $g_{\mu\nu} = -\delta_{\mu\nu}.$ For the second approximation, we need to write $g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, \qquad \text{(92)}$ where the $\gamma_{\mu\nu}$ are to be considered small and of the first order. [Pg 85]

Both terms of our equation of motion are then small of the first order. If we neglect terms which, relatively to these, are small of the first order, we have to put $\begin{gather*} ds^{2} = -{dx_{\nu}}^{2} = dl^{2} (1 - q^{2}), \qquad & &\text{(93)}\\ \Gamma_{\alpha\beta}^{\mu} = -\delta_{\mu\sigma} [^{\alpha\beta}_{\,\delta}] = -[^{\alpha\beta}_{\,\mu}] = \frac{1}{2} \left( \frac{\partial \gamma_{\alpha\beta}}{\partial x_{\mu}} - \frac{\partial \gamma_{\alpha\mu}}{\partial x_{\beta}} - \frac{\partial \gamma_{\beta\mu}}{\partial x_{\alpha}}\right). \qquad & &\text{(94)} \end{gather*}$ We shall now introduce an approximation of a second kind. Let the velocity of the material particles be very small compared to that of light. Then $ds$ will be the same as the time differential, $dl$ . Further, $\dfrac{dx_{1}}{ds}$ , $\dfrac{dx_{2}}{ds}$ , $\dfrac{dx_{3}}{ds}$ will vanish compared to $\dfrac{dx_{4}}{ds}$ . We shall assume, in addition, that the gravitational field varies so little with the time that the derivatives of the $\gamma_{\mu\nu}$ by $x_{4}$ may be neglected. Then the equation of motion (for $\mu$ = 1,2,3) reduces to $\frac{d^{2} x_{\mu}}{dl^{2}} = \frac{\partial}{\partial x_{\mu}} \left(\frac{\gamma_{44}}{2}\right). \qquad \text{(90a)}$ This equation is identical with Newton's equation of motion for a material particle in a gravitational field, if we identify $\left(\dfrac{\gamma_{44}}{2}\right)$ with the potential of the gravitational field; whether or not this is allowable, naturally depends upon the field equations of gravitation, that is, it depends upon whether or not this quantity satisfies, to a first approximation, the same laws of the field as the gravitational potential in Newton's theory. A glance at (90) and (90a) shows that the $\Gamma_{\alpha\beta}^{\mu}$ actually do play the rôle of the intensity of the gravitational field. These quantities do not have a tensor character.

Both terms in our equation of motion are small of the first order. If we ignore terms that are relatively small of the first order, we need to set $\begin{gather*} ds^{2} = -{dx_{\nu}}^{2} = dl^{2} (1 - q^{2}), \qquad & &\text{(93)}\\ \Gamma_{\alpha\beta}^{\mu} = -\delta_{\mu\sigma} [^{\alpha\beta}_{\,\delta}] = -[^{\alpha\beta}_{\,\mu}] = \frac{1}{2} \left( \frac{\partial \gamma_{\alpha\beta}}{\partial x_{\mu}} - \frac{\partial \gamma_{\alpha\mu}}{\partial x_{\beta}} - \frac{\partial \gamma_{\beta\mu}}{\partial x_{\alpha}}\right). \qquad & &\text{(94)} \end{gather*}$ Now, we will introduce a second kind of approximation. Let the speed of the material particles be very small compared to the speed of light. Then $ds$ will be the same as the time differential, $dl$ . Additionally, $\dfrac{dx_{1}}{ds}$ , $\dfrac{dx_{2}}{ds}$ , $\dfrac{dx_{3}}{ds}$ will become negligible compared to $\dfrac{dx_{4}}{ds}$ . We will also assume that the gravitational field varies very little with time, so the derivatives of the $\gamma_{\mu\nu}$ with respect to $x_{4}$ can be ignored. Then the equation of motion (for $\mu$ = 1,2,3) simplifies to $\frac{d^{2} x_{\mu}}{dl^{2}} = \frac{\partial}{\partial x_{\mu}} \left(\frac{\gamma_{44}}{2}\right). \qquad \text{(90a)}$ This equation is the same as Newton's equation of motion for a material particle in a gravitational field, if we equate $\left(\dfrac{\gamma_{44}}{2}\right)$ with the potential of the gravitational field; whether this is valid depends on the field equations of gravitation, meaning it depends on whether this quantity obeys, to a first approximation, the same laws of the field as the gravitational potential in Newton's theory. A look at (90) and (90a) shows that the $\Gamma_{\alpha\beta}^{\mu}$ indeed plays the role of the intensity of the gravitational field. These quantities do not have a tensor character.

Equations (90) express the influence of inertia and gravitation upon the material particle. The unity of inertia and gravitation [Pg 86] is formally expressed by the fact that the whole left-hand side of (90) has the character of a tensor (with respect to any transformation of co-ordinates), but the two terms taken separately do not have tensor character, so that, in analogy with Newton's equations, the first term would be regarded as the expression for inertia, and the second as the expression for the gravitational force.

Equations (90) describe how inertia and gravity affect a material particle. The unity of inertia and gravity is shown by the fact that the entire left side of (90) behaves like a tensor (regardless of any coordinate transformation), while the two terms on their own do not have a tensor nature. Thus, drawing an analogy with Newton's equations, the first term could be seen as representing inertia, and the second term as representing the gravitational force. [Pg 86]

We must next attempt to find the laws of the gravitational field. For this purpose, Poisson's equation, $\Delta\phi = 4\pi K\rho$ of the Newtonian theory must serve as a model. This equation has its foundation in the idea that the gravitational field arises from the density $\rho$ of ponderable matter. It must also be so in the general theory of relativity. But our investigations of the special theory of relativity have shown that in place of the scalar density of matter we have the tensor of energy per unit volume. In the latter is included not only the tensor of the energy of ponderable matter, but also that of the electromagnetic energy. We have seen, indeed, that in a more complete analysis the energy tensor can be regarded only as a provisional means of representing matter. In reality, matter consists of electrically charged particles, and is to be regarded itself as a part, in fact, the principal part, of the electromagnetic field. It is only the circumstance that we have not sufficient knowledge of the electromagnetic field of concentrated charges that compels us, provisionally, to leave undetermined in presenting the theory, the true form of this tensor. From this point of view our problem now is to introduce a tensor, $T_{\mu\nu}$ . of the second rank, [Pg 87] whose structure we do not know provisionally, and which includes in itself the energy density of the electromagnetic field and of ponderable matter; we shall denote this in the following as the "energy tensor of matter."

We next need to find the laws of the gravitational field. For this, we will use Poisson's equation, $\Delta\phi = 4\pi K\rho$ from Newtonian theory as a model. This equation is based on the idea that the gravitational field comes from the density $\rho$ of physical matter. This should also hold true in general relativity. However, our exploration of special relativity has shown that instead of the scalar density of matter, we need to consider the energy tensor per unit volume. This tensor not only includes the energy of physical matter but also that of electromagnetic energy. We have found that in a more detailed analysis, the energy tensor can only be seen as a temporary way to represent matter. Essentially, matter is made up of electrically charged particles and is actually a part, primarily, of the electromagnetic field itself. The reason we aren't able to define the true form of this tensor in our theory is due to our limited understanding of the electromagnetic field's behavior in concentrated charges. Therefore, our current goal is to introduce a second-rank tensor, $T_{\mu\nu}$ , whose structure we don't yet know, and which encompasses the energy density of both the electromagnetic field and physical matter; we will refer to this as the "energy tensor of matter." [Pg 87]

According to our previous results, the principles of momentum and energy are expressed by the statement that the divergence of this tensor vanishes (47c). In the general theory of relativity, we shall have to assume as valid the corresponding general co-variant equation. If ( $T_{\mu\nu}$ ) denotes the co-variant energy tensor of matter, $\mathfrak T_{\sigma}^{\nu}$ the corresponding mixed tensor density, then, in accordance with (83), we must require that $0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta} \qquad \text{(95)}$ be satisfied. It must be remembered that besides the energy density of the matter there must also be given an energy density of the gravitational field, so that there can be no talk of principles of conservation of energy and momentum for matter alone. This is expressed mathematically by the presence of the second term in (95), which makes it impossible to conclude the existence of an integral equation of the form of (49). The gravitational field transfers energy and momentum to the "matter," in that it exerts forces upon it and gives it energy; this is expressed by the second term in (95).

According to our previous results, the principles of momentum and energy are shown by the fact that the divergence of this tensor is zero (47c). In the general theory of relativity, we need to consider the corresponding general covariant equation as valid. If ( $T_{\mu\nu}$ ) represents the covariant energy tensor of matter, and $\mathfrak T_{\sigma}^{\nu}$ is the corresponding mixed tensor density, then, according to (83), we must require that $0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta} \qquad \text{(95)}$ be met. It’s important to remember that in addition to the energy density of matter, there must also be an energy density of the gravitational field, so we can't talk about conservation of energy and momentum for matter alone. This is mathematically expressed by the second term in (95), which makes it impossible to deduce the existence of an integral equation like (49). The gravitational field transfers energy and momentum to "matter" by exerting forces on it and imparting energy; this is shown by the second term in (95).

If there is an analogue of Poisson's equation in the general theory of relativity, then this equation must be a tensor equation for the tensor $g_{\mu\nu}$ of the gravitational potential; the energy tensor of matter must appear on the right-hand side of this equation. On the left-hand side of the equation there must be a differential tensor in the $g_{\mu\nu}$ . We have to find this differential [Pg 88] tensor. It is completely determined by the following three conditions:—

If there's a version of Poisson's equation in general relativity, then it must be a tensor equation involving the tensor $g_{\mu\nu}$ representing the gravitational potential; the energy tensor of matter needs to be on the right side of this equation. On the left side, there should be a differential tensor in the $g_{\mu\nu}$ . We need to identify this differential tensor. It is fully determined by the following three conditions:—

1. It may contain no differential coefficients of the $g_{\mu\nu}$ higher than the second.

1. It might not have any differential coefficients of the $g_{\mu\nu}$ above the second order.

2. It must be linear and homogeneous in these second differential coefficients.

2. It has to be linear and uniform in these second differential coefficients.

3. Its divergence must vanish identically.

3. Its divergence must completely disappear.

The first two of these conditions are naturally taken from Poisson's equation. Since it may be proved mathematically that all such differential tensors can be formed algebraically (i.e. without differentiation) from Riemann's tensor, our tensor must be of the form $R_{\mu\nu} + ag_{\mu\nu} R,$ in which $R_{\mu\nu}$ and $R$ are defined by (88) and (89) respectively. Further, it may be proved that the third condition requires a to have the value $-\dfrac{1}{2}$ . For the law of the gravitational field we therefore get the equation $R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}. \qquad \text{(96)}$ Equation (95) is a consequence of this equation. $\kappa$ denotes a constant, which is connected with the Newtonian gravitation constant.

The first two of these conditions come directly from Poisson's equation. It can be mathematically proven that all such differential tensors can be produced algebraically (i.e., without differentiation) from Riemann's tensor, so our tensor must take the form $R_{\mu\nu} + ag_{\mu\nu} R,$ where $R_{\mu\nu}$ and $R$ are defined by (88) and (89) respectively. Furthermore, it can be shown that the third condition requires a to equal $-\dfrac{1}{2}$ . Thus, we derive the equation for the law of the gravitational field: $R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R = - \kappa T_{\mu\nu}. \qquad \text{(96)}$ Equation (95) follows from this equation. $\kappa$ represents a constant related to the Newtonian gravitational constant.

In the following I shall indicate the features of the theory which are interesting from the point of view of physics, using as little as possible of the rather involved mathematical method. It must first be shown that the divergence of the left-hand side actually vanishes. The energy principle for matter may be expressed, by (83), $0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta}, \qquad \text{(97)}$ [Pg 89] in which $\mathfrak T_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}.$ The analogous operation, applied to the left-hand side of (96), will lead to an identity.

In the following, I will highlight the features of the theory that are relevant to physics, while minimizing the use of the complex mathematical methods involved. First, we need to demonstrate that the divergence of the left-hand side actually disappears. The energy principle for matter can be expressed as (83), $0 = \frac{\partial \mathfrak T_{\sigma}^{\alpha}}{\partial x_{\alpha}} - \Gamma_{\sigma\beta}^{\alpha} \mathfrak T_{\alpha}^{\beta}, \qquad \text{(97)}$ [Pg 89] where $\mathfrak T_{\sigma}^{\alpha} = T_{\sigma\tau} g^{\tau\alpha} \sqrt{-g}.$ The same operation applied to the left-hand side of (96) will result in an identity.

In the region surrounding each world-point there are systems of co-ordinates for which, choosing the $x_{4}$ -co-ordinate imaginary, at the given point, $g_{\mu\nu} = g^{\mu\nu} = -\delta_{\mu\nu} = \begin{cases} -1 & \text{if $\mu = \nu$}, \\ \,\,\,\,0 & \text{if $\mu \neq \nu$}, \end{cases}$ and for which the first derivatives of the $g_{\mu\nu}$ and the $g^{\mu\nu}$ vanish. We shall verify the vanishing of the divergence of the left-hand side at this point. At this point the components $\Gamma_{\sigma\beta}^{\alpha}$ vanish, so that we have to prove the vanishing only of $\frac{\partial}{\partial x_{\sigma}} \left[ \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) \right].$ Introducing (88) and (70) into this expression, we see that the only terms that remain are those in which third derivatives of the $g^{\mu\nu}$ enter. Since the $g_{\mu\nu}$ are to be replaced by $-\delta_{\mu\nu}$ , we obtain, finally, only a few terms which may easily be seen to cancel each other. Since the quantity that we have formed has a tensor character, its vanishing is proved for every other system of co-ordinates also, and naturally for every other four-dimensional point. The energy principle of matter (97) is thus a mathematical consequence of the field equations (96).

In the area around each world-point, there are coordinate systems where, by selecting the $x_{4}$ -coordinate as imaginary at the specific point, $g_{\mu\nu} = g^{\mu\nu} = -\delta_{\mu\nu} = \begin{cases} -1 & \text{if $\mu = \nu$}, \\ \,\,\,\,0 & \text{if $\mu \neq \nu$}, \end{cases}$ and where the first derivatives of the $g_{\mu\nu}$ and $g^{\mu\nu}$ are zero. We will verify that the divergence of the left-hand side is zero at this point. Here, the components $\Gamma_{\sigma\beta}^{\alpha}$ are zero, so we only need to show that $\frac{\partial}{\partial x_{\sigma}} \left[ \sqrt{-g}\, g^{\nu\sigma} (R_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} R) \right].$ By substituting (88) and (70) into this expression, we find that only terms including the third derivatives of the $g^{\mu\nu}$ remain. As the $g_{\mu\nu}$ are replaced with $-\delta_{\mu\nu}$ , we ultimately get only a few terms that easily cancel each other out. Since the quantity we formed has a tensor character, its zero value is verified for any other coordinate system as well, and of course for every other four-dimensional point. Thus, the energy principle of matter (97) is a mathematical outcome of the field equations (96).

In order to learn whether the equations (96) are consistent with experience, we must, above all else, find out whether they [Pg 90] lead to the Newtonian theory as a first approximation. For this purpose we must introduce various approximations into these equations. We already know that Euclidean geometry and the law of the constancy of the velocity of light are valid, to a certain approximation, in regions of a great extent, as in the planetary system. If, as in the special theory of relativity, we take the fourth co-ordinate imaginary, this means that we must put $g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, \qquad \text{(98)}$ in which the $\gamma_{\mu\nu}$ are so small compared to 1 that we can neglect the higher powers of the $\gamma_{\mu\nu}$ and their derivatives. If we do this, we learn nothing about the structure of the gravitational held, or of metrical space of cosmical dimensions, but we do learn about the influence of neighbouring masses upon physical phenomena.

To determine if the equations (96) are consistent with reality, we first need to find out if they lead to Newtonian theory as an initial approximation. To do this, we need to apply various approximations to these equations. We already know that Euclidean geometry and the law of the constant speed of light are valid, to some extent, in large areas, like the planetary system. If we treat the fourth coordinate as imaginary, as in the special theory of relativity, it means we must set $g_{\mu\nu} = -\delta_{\mu\nu} + \gamma_{\mu\nu}, \qquad \text{(98)}$ where the $\gamma_{\mu\nu}$ are so small compared to 1 that we can ignore the higher powers of the $\gamma_{\mu\nu}$ and their derivatives. By doing this, we don't gain any insights into the structure of the gravitational field or the metric space of cosmic dimensions, but we do learn about the influence of nearby masses on physical phenomena.

Before carrying through this approximation we shall transform (96). We multiply (96) by $g^{\mu\nu}$ , summed over the $\mu$ and $\nu$ observing the relation which follows from the definition of the $g^{\mu\nu}$ , $g_{\mu\nu} g^{\mu\nu} = 4,$ we obtain the equation $R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T.$ If we put this value of $R$ in (96) we obtain $R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) = -\kappa T_{\mu\nu}^{*}. \qquad \text{(96a)}$ When the approximation which has been mentioned is carried out, we obtain for the left-hand side, $-\tfrac{1}{2}\left( \frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + \frac{\partial^{2} \gamma_{\alpha\alpha}}{\partial x_{\mu}\, \partial x_{\nu}} - \frac{\partial^{2} \gamma_{\mu\alpha}}{\partial x_{\nu}\, \partial x_{\alpha}} - \frac{\partial^{2} \gamma_{\nu\alpha}}{\partial x_{\mu}\, \partial x_{\alpha}} \right)$ [Pg 91] or $-\tfrac{1}{2}\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + \tfrac{1}{2} \frac{\partial}{\partial x_{\nu}}\left( \frac{{\partial \gamma'}_{\mu\alpha}}{\partial x_{\alpha}} \right) + \tfrac{1}{2} \frac{\partial}{\partial x_{\mu}}\left( \frac{{\partial \gamma'}_{\nu\alpha}}{\partial x_{\alpha}} \right),$ in which has been put ${\gamma'}_{\mu\nu} = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}. \qquad \text{(99)}$

Before we proceed with this approximation, we will transform (96). We multiply (96) by $g^{\mu\nu}$ , summing over the $\mu$ and $\nu$ and observing the relationship that follows from the definition of the $g^{\mu\nu}$ , $g_{\mu\nu} g^{\mu\nu} = 4,$ we get the equation $R = \kappa g^{\mu\nu} T_{\mu\nu} = \kappa T.$ If we substitute this value of $R$ into (96), we obtain $R_{\mu\nu} = -\kappa (T_{\mu\nu} - \tfrac{1}{2}g_{\mu\nu} T) = -\kappa T_{\mu\nu}^{*}. \qquad \text{(96a)}$ When we make the previously mentioned approximation, we find for the left side, $-\tfrac{1}{2}\left( \frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + \frac{\partial^{2} \gamma_{\alpha\alpha}}{\partial x_{\mu}\, \partial x_{\nu}} - \frac{\partial^{2} \gamma_{\mu\alpha}}{\partial x_{\nu}\, \partial x_{\alpha}} - \frac{\partial^{2} \gamma_{\nu\alpha}}{\partial x_{\mu}\, \partial x_{\alpha}} \right)$ [Pg 91] or $-\tfrac{1}{2}\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} + \tfrac{1}{2} \frac{\partial}{\partial x_{\nu}}\left( \frac{{\partial \gamma'}_{\mu\alpha}}{\partial x_{\alpha}} \right) + \tfrac{1}{2} \frac{\partial}{\partial x_{\mu}}\left( \frac{{\partial \gamma'}_{\nu\alpha}}{\partial x_{\alpha}} \right),$ where we have defined ${\gamma'}_{\mu\nu} = \gamma_{\mu\nu} - \tfrac{1}{2}\gamma_{\sigma\sigma}\delta_{\mu\nu}. \qquad \text{(99)}$

We must now note that equation (96) is valid for any system of co-ordinates. We have already specialized the system of co-ordinates in that we have chosen it so that within the region considered the $g_{\mu\nu}$ differ infinitely little from the constant values $-\delta_{\mu\nu}$ . But this condition remains satisfied in any infinitesimal change of co-ordinates, so that there are still four conditions to which the $\gamma_{\mu\nu}$ may be subjected, provided these conditions do not conflict with the conditions for the order of magnitude of the $\gamma_{\mu\nu}$ . We shall now assume that the system of co-ordinates is so chosen that the four relations— $0 = \frac{{\partial\gamma'}_{\mu\nu}}{\partial x_{\nu}} = \frac{\partial\gamma_{\mu\nu}}{\partial x_{\nu}} - \tfrac{1}{2} \frac{\partial\gamma_{\sigma\sigma}}{\partial x_{\mu}} \qquad \text{(100)}$ are satisfied. Then (96a) takes the form $\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}. \qquad \text{(96b)}$ These equations may be solved by the method, familiar in electrodynamics, of retarded potentials; we get, in an easily understood notation, $\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}. \qquad \text{(101)}$ [Pg 92]

We should point out that equation (96) holds true for any coordinate system. We have already narrowed down the coordinate system by selecting one where, within the region in question, the $g_{\mu\nu}$ only slightly differ from the constant values $-\delta_{\mu\nu}$ . However, this condition remains valid with any infinitesimal change of coordinates, meaning there are still four conditions that the $\gamma_{\mu\nu}$ can follow, as long as these conditions do not contradict the order of magnitude for the $\gamma_{\mu\nu}$ . Now, we will assume that the coordinate system is chosen such that the four relationships— $0 = \frac{{\partial\gamma'}_{\mu\nu}}{\partial x_{\nu}} = \frac{\partial\gamma_{\mu\nu}}{\partial x_{\nu}} - \tfrac{1}{2} \frac{\partial\gamma_{\sigma\sigma}}{\partial x_{\mu}} \qquad \text{(100)}$ are met. Then (96a) becomes $\frac{\partial^{2} \gamma_{\mu\nu}}{{\partial x_{\alpha}}^{2}} = 2\kappa T_{\mu\nu}^{*}. \qquad \text{(96b)}$ These equations can be solved using the method known from electrodynamics, involving retarded potentials; we arrive at a straightforward notation, $\gamma_{\mu\nu} = -\frac{\kappa}{2\pi} \int \frac{T_{\mu\nu}^{*}(x_{0}, y_{0}, z_{0}, t - r)}{r}\, dV_{0}. \qquad \text{(101)}$ [Pg 92]

In order to see in what sense this theory contains the Newtonian theory, we must consider in greater detail the energy tensor of matter. Considered phenomenologically, this energy tensor is composed of that of the electromagnetic field and of matter in the narrower sense. If we consider the different parts of this energy tensor with respect to their order of magnitude, it follows from the results of the special theory of relativity that the contribution of the electromagnetic field practically vanishes in comparison to that of ponderable matter. In our system of units, the energy of one gram of matter is equal to 1, compared to which the energy of the electric fields may be ignored, and also the energy of deformation of matter, and even the chemical energy. We get an approximation that is fully sufficient for our purpose if we put $\left. \begin{aligned} T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}, \\ ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. \end{aligned} \right\} \qquad \text{(102)}$ In this, $\sigma$ is the density at rest, that is, the density of the ponderable matter, in the ordinary sense, measured with the aid of a unit measuring rod, and referred to a Galilean system of co-ordinates moving with the matter.

To understand how this theory encompasses the Newtonian theory, we need to examine the energy tensor of matter in more detail. From a phenomenological perspective, this energy tensor includes both the electromagnetic field and matter in a stricter sense. If we look at the different components of this energy tensor in terms of their magnitude, the results from the special theory of relativity indicate that the contribution from the electromagnetic field is negligible when compared to that of tangible matter. In our system of units, the energy of one gram of matter is set to 1, making the energy from electric fields, the energy from deformation of matter, and even chemical energy insignificant by comparison. We can simplify things sufficiently for our purposes as follows: $\left. \begin{aligned} T^{\mu\nu} &= \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}, \\ ds^{2} &= g_{\mu\nu}\, dx_{\mu}\, dx_{\nu}. \end{aligned} \right\} \qquad \text{(102)}$ Here, $\sigma$ represents the density at rest, which means the density of tangible matter in the usual sense, measured using a standard measuring rod, and referenced to a Galilean coordinate system that moves with the matter.

We observe, further, that in the co-ordinates we have chosen, we shall make only a relatively small error if we replace the $g_{\mu\nu}$ by $-\delta_{\mu\nu}$ , so that we put $ds^{2} = -\sum {dx_{\mu}}^{2}. \qquad \text{(102a)}$

We also notice that in the coordinates we've chosen, we'll make only a relatively small error if we replace the $g_{\mu\nu}$ with $-\delta_{\mu\nu}$ , so that we write $ds^{2} = -\sum {dx_{\mu}}^{2}. \qquad \text{(102a)}$

The previous developments are valid however rapidly the masses which generate the field may move relatively to our chosen system of quasi-Galilean co-ordinates. But in astronomy [Pg 93] we have to do with masses whose velocities, relatively to the co-ordinate system employed, are always small compared to the velocity of light, that is, small compared to 1, with our choice of the unit of time. We therefore get an approximation which is sufficient for nearly all practical purposes if in (101) we replace the retarded potential by the ordinary (non-retarded) potential, and if, for the masses which generate the field, we put $\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad \frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}. \qquad \text{(103)}$ Then we get for $T^{\mu\nu}$ and $T_{\mu\nu}$ the values $\left. \begin{array}{*{3}{>{\qquad}r}>{\quad}r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\sigma \end{array} \right\}. \qquad \text{(104)}$ For $T$ we get the value $\sigma$ , and, finally, for $T_{\mu\nu}^{*}$ the values, $\left. \begin{array}{*{3}{>{\qquad}c}>{\quad}r} \dfrac{\sigma}{2} & 0 & 0 & 0 \\ 0 & \dfrac{\sigma}{2} & 0 & 0 \\ 0 & 0 & \dfrac{\sigma}{2} & 0 \\ 0 & 0 & 0 & -\dfrac{\sigma}{2} \end{array} \right\}. \qquad \text{(104a)}$

The previous developments hold true regardless of how fast the masses that create the field may move in relation to our selected quasi-Galilean coordinate system. However, in astronomy, we deal with masses whose speeds, compared to the coordinate system used, are always small in relation to the speed of light, meaning they are small compared to 1, given our choice of time unit. Therefore, we obtain an approximation that is adequate for almost all practical purposes if we replace the retarded potential in (101) with the regular (non-retarded) potential, and if we set for the masses that generate the field, $\frac{dx_{1}}{ds} = \frac{dx_{2}}{ds} = \frac{dx_{3}}{ds} = 0,\quad \frac{dx_{4}}{ds} = \frac{\sqrt{-1}\, dl}{dl} = \sqrt{-1}. \qquad \text{(103)}$ Then for $T^{\mu\nu}$ and $T_{\mu\nu}$ we get the values $\left. \begin{array}{*{3}{>{\qquad}r}>{\quad}r} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\sigma \end{array} \right\}. \qquad \text{(104)}$ For $T$ we get the value $\sigma$ , and, finally, for $T_{\mu\nu}^{*}$ the values, $\left. \begin{array}{*{3}{>{\qquad}c}>{\quad}r} \dfrac{\sigma}{2} & 0 & 0 & 0 \\ 0 & \dfrac{\sigma}{2} & 0 & 0 \\ 0 & 0 & \dfrac{\sigma}{2} & 0 \\ 0 & 0 & 0 & -\dfrac{\sigma}{2} \end{array} \right\}. \qquad \text{(104a)}$

We thus get, from (101), $\left. \begin{aligned} \gamma_{11} = \gamma_{22} = \gamma_{33} &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ \gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \end{aligned} \right\} \qquad \text{(101a)}$ [Pg 94] while all the other $\gamma_{\mu\nu}$ , vanish. The least of these equations, in connexion with equation (90a), contains Newton's theory of gravitation. If we replace $l$ by $ct$ we get $\frac{d^{2} x_{\mu}}{dt^{2}} = \frac{\kappa c^{2}}{8\pi}\, \frac{\partial}{\partial x_{\mu}} \left\{ \int \frac{\sigma\, dV_{0}}{r} \right\}. \qquad \text{(90b)}$ We see that the Newtonian gravitation constant $K$ , is connected with the constant $\kappa$ that enters into our field equations by the relation $K = \frac{\kappa c^{2}}{8\pi}. \qquad \text{(105)}$ From the known numerical value of $K$ , it therefore follows that $\kappa = \frac{8\pi K}{c^{2}} = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} = 1.86·10^{-27}. \qquad \text{(105a)}$ From (101) we see that even in the first approximation the structure of the gravitational field differs fundamentally from that which is consistent with the Newtonian theory; this difference lies in the fact that the gravitational potential has the character of a tensor and not a scalar. This was not recognized in the past because only the component $g_{44}$ , to a first approximation, enters the equations of motion of material particles.

We get the following from (101): $\left. \begin{aligned} \gamma_{11} = \gamma_{22} = \gamma_{33} &= -\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ \gamma_{44} &= +\frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \end{aligned} \right\} \qquad \text{(101a)}$ [Pg 94] while all the other $\gamma_{\mu\nu}$ , disappear. The simplest of these equations, along with equation (90a), contains Newton's theory of gravitation. If we replace $l$ with $ct$ we get $\frac{d^{2} x_{\mu}}{dt^{2}} = \frac{\kappa c^{2}}{8\pi}\, \frac{\partial}{\partial x_{\mu}} \left\{ \int \frac{\sigma\, dV_{0}}{r} \right\}. \qquad \text{(90b)}$ This shows that the Newtonian gravitational constant $K$ , is linked to the constant $\kappa$ that appears in our field equations by the relation $K = \frac{\kappa c^{2}}{8\pi}. \qquad \text{(105)}$ From the known numerical value of $K$ , it follows that $\kappa = \frac{8\pi K}{c^{2}} = \frac{8\pi·6.67·10^{-8}}{9·10^{20}} = 1.86·10^{-27}. \qquad \text{(105a)}$ From (101) we see that even in the first approximation the structure of the gravitational field is fundamentally different from what is described by Newtonian theory; this difference is that the gravitational potential has the nature of a tensor rather than a scalar. This was not recognized in the past because only the component $g_{44}$ , to a first approximation, enters the equations of motion for material particles.

In order now to be able to judge the behaviour of measuring rods and clocks from our results, we must observe the following. According to the principle of equivalence, the metrical relations of the Euclidean geometry are valid relatively to a Cartesian system of reference of infinitely small dimensions, and in a suitable state of motion (freely falling, and without rotation). We can make the same statement for local systems of co-ordinates [Pg 95] which, relatively to these, have small accelerations, and therefore for such systems of co-ordinates as are at rest relatively to the one we have selected. For such a local system, we have, for two neighbouring point events, $ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} = - dS^{2} + dT^{2},$ where $dS$ is measured directly by a measuring rod and $dT$ by a clock at rest relatively to the system; these are the naturally measured lengths and times. Since $ds^{2}$ , on the other hand, is known in terms of the co-ordinates $x_{\nu}$ employed in finite regions, in the form $ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu},$ we have the possibility of getting the relation between naturally measured lengths and times, on the one hand, and the corresponding differences of co-ordinates, on the other hand. As the division into space and time is in agreement with respect to the two systems of co-ordinates, so when we equate the two expressions for $ds^{2}$ we get two relations. If, by (101a), we put $\begin{aligned} ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ \qquad + \mspace{26mu}\left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}, \end{aligned}$ we obtain, to a sufficiently close approximation, $\left. \begin{aligned} &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ &\qquad \begin{aligned} &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}, \\ dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. \end{aligned} \end{aligned} \right\} \qquad \text{(106)}$ [Pg 96]

To judge the behavior of measuring rods and clocks based on our results, we need to keep the following in mind. The principle of equivalence tells us that the geometric relationships of Euclidean geometry hold true relative to a Cartesian reference system of infinitely small dimensions, in an appropriate state of motion (freely falling and without rotation). The same applies to local coordinate systems that have small accelerations relative to these, and also for those that are at rest relative to our chosen system. For such a local system, for two neighboring point events, we have: $ds^{2} = - {dX_{1}}^{2} - {dX_{2}}^{2} - {dX_{3}}^{2} + dT^{2} = - dS^{2} + dT^{2},$ where $dS$ is directly measured by a measuring rod and $dT$ by a clock at rest relative to the system; these are the naturally measured lengths and times. Since $ds^{2}$ is expressed in terms of the coordinates $x_{\nu}$ used in finite regions, in the form $ds^{2} = g_{\mu\nu}\, dx_{\mu}\, dx_{\nu},$ we can establish the relationship between naturally measured lengths and times on one side and the corresponding coordinate differences on the other. Since the division into space and time is consistent across both coordinate systems, when we set the two expressions for $ds^{2}$ equal, we derive two relationships. If we express it as (101a), $\begin{aligned} ds^{2} = -\left(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) \\ \qquad + \mspace{26mu}\left(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl^{2}, \end{aligned}$ we arrive, to a close approximation, at $\left. \begin{aligned} &\sqrt{{dX_{1}}^{2} + {dX_{2}}^{2} + {dX_{3}}^{2}} \\ &\qquad \begin{aligned} &= \left(1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) \sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}, \\ dT &= \left(1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}\right) dl. \end{aligned} \end{aligned} \right\} \qquad \text{(106)}$ [Pg 96]

The unit measuring rod has therefore the length, $1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}$ in respect to the system of co-ordinates we have selected. The particular system of co-ordinates we have selected insures that this length shall depend only upon the place, and not upon the direction. If we had chosen a different system of co-ordinates this would not be so. But however we may choose a system of co-ordinates, the laws of configuration of rigid rods do not agree with those of Euclidean geometry; in other words, we cannot choose any system of co-ordinates so that the co-ordinate differences, $\Delta x_{1}$ , $\Delta x_{2}$ , $\Delta x_{3}$ , corresponding to the ends of a unit measuring rod, oriented in any way, shall always satisfy the relation $\Delta x_{1}^{2} + \Delta x_{2}^{2} + \Delta x_{3}^{2} = 1$ . In this sense space is not Euclidean, but "curved." It follows from the second of the relations above that the interval between two beats of the unit clock ( $dT$ = 1) corresponds to the "time" $1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}$ in the unit used in our system of co-ordinates. The rate of a clock is accordingly slower the greater is the mass of the ponderable matter in its neighbourhood. We therefore conclude that spectral lines which are produced on the sun's surface will be displaced towards the red, compared to the corresponding lines produced on the earth, by about 2 • 10^-6 of their wave-lengths. At first, this important consequence of the theory appeared to conflict with experiment; but results obtained during the past year seem to make the existence of this effect more probable, and [Pg 97] it can hardly be doubted that this consequence of the theory will be confirmed within the next year.

The unit measuring rod has a length of, $1 - \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}$ based on the coordinate system we've chosen. This specific coordinate system ensures that this length depends only on the location, not the direction. If we had chosen a different coordinate system, that wouldn't be the case. However, no matter how we choose a coordinate system, the configuration rules for rigid rods don't align with those of Euclidean geometry; in other words, we can't select any coordinate system where the coordinate differences, $\Delta x_{1}$ , $\Delta x_{2}$ , $\Delta x_{3}$ , corresponding to the ends of a unit measuring rod, positioned in any way, will always satisfy the relation $\Delta x_{1}^{2} + \Delta x_{2}^{2} + \Delta x_{3}^{2} = 1$ . In this sense, space is not Euclidean, but "curved." From the second of the relationships above, we see that the interval between two ticks of the unit clock ( $dT$ = 1) corresponds to the "time" $1 + \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}$ in the unit used in our coordinate system. The ticking of a clock is therefore slower when there is more mass of matter nearby. We conclude that spectral lines produced on the sun's surface will shift towards the red compared to the corresponding lines produced on Earth, by about 2 • 10^-6 of their wavelengths. Initially, this significant outcome of the theory appeared to contradict experimental results; however, findings from the past year suggest that the existence of this effect is becoming more likely, and [Pg 97] it is almost certain that this consequence of the theory will be confirmed in the next year.

Another important consequence of the theory, which can be tested experimentally, has to do with the path of rays of light. In the general theory of relativity also the velocity of light is everywhere the same, relatively to a local inertial system. This velocity is unity in our natural measure of time. The law of the propagation of light in general co-ordinates is therefore, according to the general theory of relativity, characterized, by the equation $ds^{2} = 0$ To within the approximation which we are using, and in the system of co-ordinates which we have selected, the velocity of light is characterized, according to (106), by the equation $\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}.$ The velocity of light $L$ , is therefore expressed in our co-ordinates by $\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}. \qquad \text{(107)}$ We can therefore draw the conclusion from this, that a ray of light passing near a large mass is deflected. If we imagine the sun, of mass $M$ concentrated at the origin of our system of co-ordinates, then a ray of fight, travelling parallel to the $x_{3}$ -axis. in the $x_{1}-x_{3}$ plane, at a distance $\Delta$ from the origin, will be deflected, in all, by an amount $\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\partial L}{\partial x_{1}}\, dx_{3}$ [Pg 98] towards the sun. On performing the integration we get $\alpha = \frac{\kappa M}{2\pi\Delta}. \qquad \text{(108)}$

Another important outcome of the theory, which can be tested experimentally, involves the path of rays of light. In the general theory of relativity, the speed of light is always the same relative to a local inertial system. This speed is set as one in our natural time measurement. Thus, the law governing the propagation of light in general coordinates is represented, according to the general theory of relativity, by the equation $ds^{2} = 0$ Within the approximation we are using, and in the coordinate system we have chosen, the speed of light is represented according to (106) by the equation $\biggl(1 + \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) ({dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}) = \biggl(1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}\biggr) dl^{2}.$ The speed of light $L$ is therefore expressed in our coordinates by $\frac{\sqrt{{dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2}}}{dl} = 1 - \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}. \qquad \text{(107)}$ From this, we can conclude that a ray of light passing near a large mass is deflected. If we imagine the sun, with mass $M$ concentrated at the origin of our coordinate system, then a ray of light, traveling parallel to the $x_{3}$ -axis in the $x_{1}-x_{3}$ plane, at a distance $\Delta$ from the origin, will be deflected, overall, by an amount $\alpha = \int_{-\infty}^{+\infty} \frac{1}{L}\, \frac{\partial L}{\partial x_{1}}\, dx_{3}$ [Pg 98] towards the sun. After performing the integration, we find $\alpha = \frac{\kappa M}{2\pi\Delta}. \qquad \text{(108)}$

The existence of this deflection, which amounts to 1.7'' for $\Delta$ equal to the radius of the sun, was confirmed, with remarkable accuracy, by the English Solar Eclipse Expedition in 1919, and most careful preparations have been made to get more exact observational data at the solar eclipse in 1922. It should be noted that this result, also, of the theory is not influenced by our arbitrary choice of a system of co-ordinates.

The presence of this deflection, which measures 1.7'' for $\Delta$ equal to the radius of the sun, was confirmed with impressive accuracy by the English Solar Eclipse Expedition in 1919, and thorough preparations have been made to gather more precise observational data during the solar eclipse in 1922. It's worth mentioning that this result from the theory is not affected by our arbitrary choice of a coordinate system.

This is the place to speak of the third consequence of the theory which can be tested by observation, namely, that which concerns the motion of the perihelion of the planet Mercury. The secular changes in the planetary orbits are known with such accuracy that the approximation we have been using is no longer sufficient for a comparison of theory and observation. It is necessary to go back to the general field equations (96). To solve this problem I made use of the method of successive approximations. Since then, however, the problem of the central symmetrical statical gravitational field has been completely solved by Schwarzschild and others; the derivation given by H. Weyl in his book, "Raum-Zeit-Materie," is particularly elegant. The calculation can be simplified somewhat if we do not go back directly to the equation (96), but base it upon a principle of variation that is equivalent to this equation. I shall indicate the procedure only in so far as is necessary for understanding the method. [Pg 99]

This is where we talk about the third consequence of the theory that can be tested by observation, specifically regarding the motion of Mercury's perihelion. The long-term changes in planetary orbits are known with such precision that the approximation we’ve been using is no longer good enough for comparing theory and observation. We need to return to the general field equations (96). I used the method of successive approximations to solve this issue. However, since then, the problem of the centrally symmetrical static gravitational field has been completely resolved by Schwarzschild and others; the derivation provided by H. Weyl in his book, "Raum-Zeit-Materie," is especially elegant. The calculation can be simplified a bit if we don’t directly refer back to equation (96), but instead base it on a principle of variation that is equivalent to this equation. I will outline the procedure only as necessary for understanding the method. [Pg 99]

In the case of a statical field, $ds^{2}$ must have the form $\left. \begin{aligned} ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}, \\ d\sigma^{2} &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}, \end{aligned} \right\} \qquad \text{(109)}$ where the summation on the right-hand side of the last equation is to be extended over the space variables only. The central symmetry of the field requires the $\gamma_{\mu\nu}$ , to be of the form, $\gamma_{\alpha\beta} = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}; \qquad \text{(110)}$ $f^{2}$ , $\mu$ and $\lambda$ are functions of $r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}$ only. One of these three functions can be chosen arbitrarily, because our system of co-ordinates is, a priori, completely arbitrary; for by a substitution $\begin{align*} {x'}_{4} &= x_{4}, \\ {x'}_{\alpha} &= F(r) x_{\alpha}, \end{align*}$ we can always insure that one of these three functions shall be an assigned function of $r$ '. In place of (110) we can therefore put, without limiting the generality, $\gamma_{\alpha\beta} = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}. \qquad \text{(110a)}$

In the case of a static field, $ds^{2}$ should take the form $\left. \begin{aligned} ds^{2} &= -d\sigma^{2} + f^{2}\, {dx_{4}}^{2}, \\ d\sigma^{2} &= \sum_{\text{$1$--$3$}} \gamma_{\alpha\beta}\, dx_{\alpha}\, dx_{\beta}, \end{aligned} \right\} \qquad \text{(109)}$ where the sum on the right side of the last equation is to be taken over the spatial variables only. The symmetrical nature of the field means that the $\gamma_{\mu\nu}$ must be in the form of $\gamma_{\alpha\beta} = \mu \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}; \qquad \text{(110)}$ $f^{2}$ , $\mu$ and $\lambda$ are functions of $r = \sqrt{{x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2}}$ only. One of these three functions can be chosen freely, since our coordinate system is a priori completely arbitrary; by using a substitution $\begin{align*} {x'}_{4} &= x_{4}, \\ {x'}_{\alpha} &= F(r) x_{\alpha}, \end{align*}$ we can always make sure that one of these three functions will be a designated function of $r$ '. Therefore, instead of (110), we can write, without losing generality, $\gamma_{\alpha\beta} = \delta_{\alpha\beta} + \lambda x_{\alpha} x_{\beta}. \qquad \text{(110a)}$

In this way the $g_{\mu\nu}$ are expressed in terms of the two quantities $\lambda$ and $f$ . These are to be determined as functions of $r$ , by introducing them into equation (96), after first calculating [Pg 100] the $\Gamma_{\mu\nu}^{\sigma}$ from (109) and (110a). We have $\left. \begin{aligned} \Gamma_{\alpha\beta}^{\sigma} &= \tfrac{1}{2} \frac{x_{\sigma}}{r} · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} {1 + \lambda r^{2}}\,\, \text{(for $\alpha, \beta, \sigma = 1, 2, 3)$}, \\ \Gamma_{44}^{4} &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad \text{(for $\alpha, \beta = 1, 2, 3)$}, \\ \Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\partial f^{2}}{\partial x_{\alpha}},\quad \Gamma_{44}^{\alpha} = -\tfrac{1}{2}{g^{\alpha\beta}}\, \frac{\partial f^{2}}{\partial {x_{\beta}}}. \end{aligned} \right\} \qquad \text{(110b)}$

In this way, the $g_{\mu\nu}$ are expressed in terms of the two quantities $\lambda$ and $r$ , by plugging them into equation (96), after first calculating [Pg 100] the $\Gamma_{\mu\nu}^{\sigma}$ from (109) and (110a). We have $\left. \begin{aligned} \Gamma_{\alpha\beta}^{\sigma} &= \tfrac{1}{2} \frac{x_{\sigma}}{r} · \frac{\lambda' x_{\alpha} x_{\beta} + 2\lambda r\, \delta_{\alpha\beta}} {1 + \lambda r^{2}}\,\, \text{(for $\alpha, \beta, \sigma = 1, 2, 3)$}, \\ \Gamma_{44}^{4} &= \Gamma_{4\beta}^{\alpha} = \Gamma_{\alpha\beta}^{4} = 0\quad \text{(for $\alpha, \beta = 1, 2, 3)$}, \\ \Gamma_{4\alpha}^{4} &= \tfrac{1}{2} f^{-2}\, \frac{\partial f^{2}}{\partial x_{\alpha}},\quad \Gamma_{44}^{\alpha} = -\tfrac{1}{2}{g^{\alpha\beta}}\, \frac{\partial f^{2}}{\partial {x_{\beta}}}. \end{aligned} \right\} \qquad \text{(110b)}$

With the help of these results, the field equations furnish Schwarzschild's solution: $ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ \frac{dr^{2}} {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) \right], \qquad \text{(109a)}$ in which we have put $\left. \begin{aligned} x_{4} &= l, \\ x_{1} &= r \sin\theta \sin\phi, \\ x_{2} &= r \sin\theta \cos\phi, \\ x_{3} &= r \cos\theta, \\ A &= \frac{\kappa M}{4\pi}. \end{aligned} \right\} \qquad \text{(109b)}$

With these results, the field equations provide Schwarzschild's solution: $ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \left[ \frac{dr^{2}} {1 - \dfrac{A}{r}} + r^{2} (\sin^{2}\theta\, d\phi^{2} + d\theta^{2}) \right], \qquad \text{(109a)}$ where we have defined $\left. \begin{aligned} x_{4} &= l, \\ x_{1} &= r \sin\theta \sin\phi, \\ x_{2} &= r \sin\theta \cos\phi, \\ x_{3} &= r \cos\theta, \\ A &= \frac{\kappa M}{4\pi}. \end{aligned} \right\} \qquad \text{(109b)}$

$M$ denotes the sun's mass, centrally symmetrically placed about the origin of co-ordinates; the solution (109) is valid only outside of this mass, where all the $T_{\mu\nu}$ vanish. If the motion of the planet takes place in the $x_{1}-x_{2}$ plane then we must replace (109a) by $ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}. \qquad \text{(109c)}$ [Pg 101]

$M$ represents the mass of the sun, which is symmetrically centered around the origin of the coordinates. The solution (109) is only applicable outside of this mass, where all the $T_{\mu\nu}$ values are zero. If the planet's motion occurs in the $x_{1}-x_{2}$ plane, we need to modify (109a) to $ds^{2} = \left(1 - \frac{A}{r}\right) dl^{2} - \frac{dr^{2}}{1 - \dfrac{A}{r}} - r^{2}\, d\phi^{2}. \qquad \text{(109c)}$ [Pg 101]

The calculation of the planetary motion depends upon equation (90). From the first of equations (110b) and (90) we get, for the indices 1, 2, 3, $\frac{d}{ds} \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0,$ or, if we integrate, and express the result in polar co-ordinates, $r^{2} \frac{d\phi}{ds} = \text{constant}. \qquad \text{(111)}$

The calculation of planetary motion is based on equation (90). From the first of equations (110b) and (90), we get, for the indices 1, 2, 3, $\frac{d}{ds} \left(x_{\alpha} \frac{dx_{\beta}}{ds} - x_{\beta} \frac{dx_{\alpha}}{ds}\right) = 0,$ or, if we integrate and express the result in polar coordinates, $r^{2} \frac{d\phi}{ds} = \text{constant}. \qquad \text{(111)}$

From (90), for $\mu$ = 4, we get $0 = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} {\, \frac{dl}{ds}} = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} {\, \frac{dl}{ds}}.$ From this, after multiplication by $f^{2}$ and integration, we have $f^{2} \frac{dl}{ds} = \text{constant}. \qquad \text{(112)}$

From (90), for $\mu$ = 4, we get $0 = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{dx_{\alpha}}\, \frac{dx_{\alpha}}{ds} {\, \frac{dl}{ds}} = \frac{d^{2} l}{ds^{2}} + \frac{1}{f^{2}}\, \frac{df^{2}}{ds} {\, \frac{dl}{ds}}.$ From this, after multiplying by $f^{2}$ and integrating, we have $f^{2} \frac{dl}{ds} = \text{constant}. \qquad \text{(112)}$

In (109c), (111) and (112) we have three equations between the four variables $s$ , $r$ , $l$ and $\phi$ , from which the motion of the planet may be calculated in the same way as in classical mechanics. The most important result we get from this is a secular rotation of the elliptic orbit of the planet in the same sense as the revolution of the planet, amounting in radians per revolution to $\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}, \qquad \text{(113)}$ [Pg 102] where $\begin{align*} a &= \text{the semi-major axis of the planetary orbit in centimetres.} \\ e &= \text{the numerical eccentricity.} \\ c &= 3·10^{+10}, \text{the velocity of light}\, \mathit{in\, vacuo}. \\ T &= \text{the period of revolution in seconds.} \end{align*}$ This expression furnishes the explanation of the motion of the perihelion of the planet Mercury, which has been known for a hundred years (since Leverrier), and for which theoretical astronomy has hitherto been unable satisfactorily to account.

In (109c), (111), and (112), we have three equations involving four variables $s$ , $r$ , $l$ and $\phi$ , from which we can calculate the motion of the planet similar to classical mechanics. The key result we derive from this is a gradual rotation of the planet's elliptical orbit in the same direction as its revolution, which amounts to $\frac{24 \pi^{3} a^{2}}{(1 - e^{2}) c^{2} T^{2}}, \qquad \text{(113)}$ [Pg 102] where $\begin{align*} a &= \text{the semi-major axis of the planetary orbit in centimeters.} \\ e &= \text{the numerical eccentricity.} \\ c &= 3·10^{+10}, \text{the speed of light in a vacuum.} \\ T &= \text{the period of revolution in seconds.} \end{align*}$ This equation explains the motion of Mercury's perihelion, which has been known for a hundred years (since Leverrier), and theoretical astronomy has yet to satisfactorily account for it.

There is no difficulty in expressing Maxwell's theory of the electromagnetic field in terms of the general theory of relativity; this is done by application of the tensor formation (81), (82) and (77). Let $\phi_{\mu}$ be a tensor of the first rank, to be denoted as an electromagnetic 4-potential; then an electromagnetic field tensor may be defined by the relations, $\phi_{\mu\nu} = \frac{\partial \phi_{\mu}}{\partial x_{\nu}} - \frac{\partial \phi_{\nu}}{\partial x_{\mu}}. \qquad \text{(114)}$ The second of Maxwell's systems of equations is then defined by the tensor equation, resulting from this, $\frac{\partial \phi_{\mu\nu}}{\partial x_{\rho}} + \frac{\partial \phi_{\nu\rho}}{\partial x_{\mu}} + \frac{\partial \phi_{\rho\mu}}{\partial x_{\nu}} = 0, \qquad \text{(114a)}$ and the first of Maxwell's systems of equations is defined by the tensor-density relation $\frac{\partial \mathfrak F^{\mu\nu}}{\partial x_{\nu}} = \mathfrak J^{\mu}, \qquad \text{(115)}$ [Pg 103] in which $\begin{align*} \mathfrak F^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}, \\ \mathfrak J^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. \end{align*}$ If we introduce the energy tensor of the electromagnetic field into the right-hand side of (96), we obtain (115), for the special case $\mathfrak J^{\mu}$ = 0, as a consequence of (96) by taking the divergence. This inclusion of the theory of electricity in the scheme of the general theory of relativity has been considered arbitrary and unsatisfactory by many theoreticians. Nor can we in this way conceive of the equilibrium of the electricity which constitutes the elementary electrically charged particles. A theory in which the gravitational field and the electromagnetic field enter as an essential entity would be much preferable. H. Weyl, and recently Th. Kaluza, have discovered some ingenious theorems along this direction; but concerning them, I am convinced that they do not bring us nearer to the true solution of the fundamental problem. I shall not go into this further, but shall give a brief discussion of the so-called cosmological problem, for without this, the considerations regarding the general theory of relativity would, in a certain sense, remain unsatisfactory.

There’s no difficulty in expressing Maxwell's theory of the electromagnetic field using the general theory of relativity; this is done by applying the tensor formulations (81), (82), and (77). Let $\phi_{\mu}$ be a first-rank tensor, referred to as an electromagnetic 4-potential; then an electromagnetic field tensor can be defined by the following relationships, $\phi_{\mu\nu} = \frac{\partial \phi_{\mu}}{\partial x_{\nu}} - \frac{\partial \phi_{\nu}}{\partial x_{\mu}}. \qquad \text{(114)}$ The second of Maxwell's equations is then defined by the tensor equation derived from this, $\frac{\partial \phi_{\mu\nu}}{\partial x_{\rho}} + \frac{\partial \phi_{\nu\rho}}{\partial x_{\mu}} + \frac{\partial \phi_{\rho\mu}}{\partial x_{\nu}} = 0, \qquad \text{(114a)}$ while the first of Maxwell's equations is characterized by the tensor-density relation $\frac{\partial \mathfrak F^{\mu\nu}}{\partial x_{\nu}} = \mathfrak J^{\mu}, \qquad \text{(115)}$ [Pg 103] where $\begin{align*} \mathfrak F^{\mu\nu} &= \sqrt{-g}\, g^{\mu\nu} g^{\nu\tau} \phi_{\sigma\tau}, \\ \mathfrak J^{\mu} &= \sqrt{-g}\, \rho \frac{dx_{\nu}}{ds}. \end{align*}$ If we include the energy tensor of the electromagnetic field in the right side of (96), we arrive at (115) for the special case $\mathfrak J^{\mu}$ = 0, as a consequence of (96) by taking the divergence. This incorporation of electrical theory into the framework of general relativity has been deemed arbitrary and unsatisfactory by many theorists. Additionally, this approach doesn’t allow us to understand the equilibrium of electricity, which constitutes elementary electrically charged particles. A theory where the gravitational field and the electromagnetic field are treated as essential components would be much more desirable. H. Weyl, and more recently Th. Kaluza, have made some clever discoveries in this direction; however, I believe they do not bring us closer to the true solution of the fundamental problem. I won’t delve further into this but will briefly discuss the so-called cosmological problem, as without it, the discussions surrounding the general theory of relativity would, in some way, remain unsatisfactory.

Our previous considerations, based upon the field equations (96), had for a foundation the conception that space on the whole is Galilean-Euclidean, and that this character is disturbed only by masses embedded in it. This conception was certainly justified as long as we were dealing with spaces of the order of magnitude of those that astronomy has to do with. But whether portions of the universe, however large they may be, are quasi-Euclidean, is a wholly different question. We can [Pg 104] make this clear by using an example from the theory of surfaces which we have employed many times. If a portion of a surface is observed by the eye to be practically plane, it does not at all follow that the whole surface has the form of a plane; the surface might just as well be a sphere, for example, of sufficiently large radius. The question as to whether the universe as a whole is non-Euclidean was much discussed from the geometrical point of view before the development of the theory of relativity. But with the theory of relativity, this problem has entered upon a new stage, for according to this theory the geometrical properties of bodies are not independent, but depend upon the distribution of masses.

Our earlier discussions, based on the field equations (96), were grounded in the idea that space is generally Galilean-Euclidean, and that this nature is disrupted only by the masses within it. This idea was valid as long as we were dealing with spaces similar to those in astronomy. However, whether parts of the universe, no matter how large, are quasi-Euclidean is a completely different matter. We can illustrate this with an example from surface theory that we've used many times. If a section of a surface appears flat to the eye, it doesn’t necessarily mean the entire surface is flat; for instance, the surface could be a sphere with a large enough radius. The question of whether the universe as a whole is non-Euclidean was widely debated from a geometric perspective before the theory of relativity emerged. With the advent of relativity, this issue has taken on a new dimension, since, according to the theory, the geometric properties of objects are not independent but depend on how masses are distributed.

If the universe were quasi-Euclidean, then Mach was wholly wrong in his thought that inertia, as well as gravitation, depends upon a kind of mutual action between bodies. For in this case, with a suitably selected system of co-ordinates, the $g_{\mu\nu}$ would be constant at infinity, as they are in the special theory of relativity, while within finite regions the $g_{\mu\nu}$ would differ from these constant values by small amounts only, with a suitable choice of co-ordinates, as a result of the influence of the masses in finite regions. The physical properties of space would not then be wholly independent, that is, uninfluenced by matter, but in the main they would be, and only in small measure, conditioned by matter. Such a dualistic conception is even in itself not satisfactory; there are, however, some important physical arguments against it, which we shall consider.

If the universe were somewhat Euclidean, then Mach was completely mistaken in thinking that inertia, like gravitation, relies on some sort of mutual interaction between bodies. In this scenario, with a properly chosen coordinate system, the $g_{\mu\nu}$ would remain constant at infinity, just as they do in the special theory of relativity. Meanwhile, within finite regions, the $g_{\mu\nu}$ would only slightly differ from these constant values, given an appropriate choice of coordinates due to the influence of the masses in those finite spaces. The physical characteristics of space wouldn’t be completely independent, meaning they would be affected by matter, but mostly they would be determined by it only to a small extent. This dualistic view is inherently unsatisfactory; however, there are also some significant physical arguments against it that we will discuss.

The hypothesis that the universe is infinite and Euclidean at infinity, is, from the relativistic point of view, a complicated hypothesis. In the language of the general theory of relativity it demands that the Riemann tensor of the fourth rank $R_{iklm}$ [Pg 105] shall vanish at infinity, which furnishes twenty independent conditions, while only ten curvature components $R_{\mu\nu}$ , enter into the laws of the gravitational field. It is certainly unsatisfactory to postulate such a far-reaching limitation without any physical basis for it.

The idea that the universe is infinite and flat at infinity is, from a relativistic perspective, quite complex. In the language of general relativity, this requires that the Riemann tensor of the fourth rank $R_{iklm}$ [Pg 105] must be zero at infinity, which creates twenty independent conditions, while only ten curvature components $R_{\mu\nu}$ are involved in the laws of the gravitational field. It's definitely unsatisfactory to impose such a significant restriction without any physical justification for it.

But in the second place, the theory of relativity makes it appear probable that Mach was on the right road in his thought that inertia depends upon a mutual action of matter. For we shall show in the following that, according to our equations, inert masses do act upon each other in the sense of the relativity of inertia, even if only very feebly. What is to be expected along the line of Mach's thought?

But secondly, the theory of relativity suggests that Mach was correct in thinking that inertia is influenced by the interaction of matter. We will demonstrate in the following sections that, according to our equations, inert masses do exert influence on each other in terms of the relativity of inertia, even if it's very slight. What should we anticipate based on Mach's ideas?

1. The inertia of a body must increase when ponderable masses are piled up in its neighbourhood.

1. The inertia of an object must increase when heavy masses are stacked up nearby.

2. A body must experience an accelerating force when neighbouring masses are accelerated, and, in fact, the force must be in the same direction as the acceleration.

2. A body has to feel an accelerating force when nearby masses are accelerated, and, in fact, the force must be in the same direction as the acceleration.

3. A rotating hollow body must generate inside of itself a "Coriolis field," which deflects moving bodies in the sense of the rotation, and a radial centrifugal field as well.

3. A rotating hollow object has to create a "Coriolis field" inside it that deflects moving objects in the direction of the rotation, along with a radial centrifugal field too.

We shall now show that these three effects, which are to be expected in accordance with Mach's ideas, are actually present according to our theory, although their magnitude is so small that confirmation of them by laboratory experiments is not to be thought of. For this purpose we shall go back to the equations of motion of a material particle (90), and carry the approximations somewhat further than was done in equation (90a). [Pg 106]

We will now demonstrate that these three effects, which align with Mach's ideas, are indeed present according to our theory, although their size is so small that confirming them through laboratory experiments isn't feasible. To do this, we will revisit the equations of motion for a material particle (90) and extend the approximations a bit further than in equation (90a). [Pg 106]

First, we consider $\gamma_{44}$ as small of the first order. The square of the velocity of masses moving under the influence of the gravitational force is of the same order, according to the energy equation. It is therefore logical to regard the velocities of the material particles we are considering, as well as the velocities of the masses which generate the field, as small, of the order $\dfrac{1}{2}$ . We shall now carry out the approximation in the equations that arise from the field equations (101) and the equations of motion (90) so far as to consider terms, in the second member of (90), that are linear in those velocities. Further, we shall not put $ds$ and $dl$ equal to each other, but, corresponding to the higher approximation, we shall put $ds = \sqrt{g_{44}}\, dl = \left(1 - \frac{\gamma_{44}}{2}\right) dl.$ From (90) we obtain, at first, $\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} \left(1 + \frac{\gamma_{44}}{2}\right). \qquad \text{(116)}$

First, we consider $\gamma_{44}$ as small of the first order. The square of the velocity of masses moving under the influence of gravitational force is of the same order, according to the energy equation. It is therefore logical to treat the velocities of the material particles we're considering, as well as the velocities of the masses that generate the field, as small, of the order $\dfrac{1}{2}$ . We will now approximate the equations that arise from the field equations (101) and the equations of motion (90) to the extent of considering terms, in the second member of (90), that are linear in those velocities. Additionally, we will not equate $ds$ and $dl$ to each other, but, corresponding to the higher approximation, we will set $ds = \sqrt{g_{44}}\, dl = \left(1 - \frac{\gamma_{44}}{2}\right) dl.$ From (90) we initially get, $\frac{d}{dl}\left[\left(1 + \frac{\gamma_{44}}{2}\right) \frac{dx_{\mu}}{dl}\right] = -\Gamma_{\alpha\beta}^{\mu} \frac{dx_{\alpha}}{dl}\, \frac{dx_{\beta}}{dl} \left(1 + \frac{\gamma_{44}}{2}\right). \qquad \text{(116)}$

From (101) we get, to the approximation sought for, $\left. \begin{aligned} -\gamma_{11} = -\gamma_{22} = -\gamma_{33} &= \gamma_{44} = \frac{\kappa}{4\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ \gamma_{4\alpha} &= -\frac{i \kappa}{2} \int \frac{\sigma \dfrac{dx_{\alpha}}{ds}\, dV_{0}}{r}, \\ \gamma_{\alpha\beta} &= 0, \end{aligned} \right\} \qquad \text{(117)}$ in which, in (117), $\alpha$ and $\beta$ denote the space indices only. [Pg 107]

On the right-hand side of (116) we can replace 1 + $\left(\dfrac{\gamma_{44}}{2}\right)$ by 1 and $-\Gamma_{\alpha\beta}^{\mu}$ by $\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right]$ . It is easy to see, in addition, that to this degree of approximation we must put $\begin{align*} \left[{{\genfrac{}{}{0pt}{}{44}{\mu}}}\right] &= -\tfrac{1}{2} \frac{\partial \gamma_{44}}{\partial x_{\mu}} + \frac{\partial \gamma_{4\mu}}{\partial x_{4}},\\ \left[{{\genfrac{}{}{0pt}{}{\alpha 4}{\mu}}}\right] &= \tfrac{1}{2} \left(\frac{\partial \gamma_{4\mu}}{\partial x_{\alpha}} - \frac{\partial \gamma_{4\alpha}}{\partial x_{\mu}}\right), \\ \left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right] &= 0, \end{align*}$ in which $\alpha$ , $\beta$ and $\mu$ denote space indices. We therefore obtain from (116), in the usual vector notation, $\DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\rot}{rot} \left. \begin{aligned} \frac{d}{dl}\bigl[(1 + \overline{\sigma}) \mathbf v \bigr] &= \grad \overline{\sigma} + \frac{\partial \mathfrak A}{\partial l} + [\rot \mathfrak A, \mathbf v], \\ \overline{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ \mathfrak A &= \frac{\kappa}{2 \pi} \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}. \end{aligned} \right\} \qquad \text{(118)}$

On the right side of (116), we can replace 1 + $\left(\dfrac{\gamma_{44}}{2}\right)$ with 1 and $-\Gamma_{\alpha\beta}^{\mu}$ with $\left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right]$ . It is easy to see that, for this level of approximation, we need to write $\begin{align*} \left[{{\genfrac{}{}{0pt}{}{44}{\mu}}}\right] &= -\tfrac{1}{2} \frac{\partial \gamma_{44}}{\partial x_{\mu}} + \frac{\partial \gamma_{4\mu}}{\partial x_{4}},\\ \left[{{\genfrac{}{}{0pt}{}{\alpha 4}{\mu}}}\right] &= \tfrac{1}{2} \left(\frac{\partial \gamma_{4\mu}}{\partial x_{\alpha}} - \frac{\partial \gamma_{4\alpha}}{\partial x_{\mu}}\right), \\ \left[{{\genfrac{}{}{0pt}{}{\alpha\beta}{\mu}}}\right] &= 0, \end{align*}$ where $\alpha$ , $\beta$ and $\mu$ represent space indices. Therefore, we derive from (116), in the usual vector notation, $\DeclareMathOperator{\grad}{grad} \DeclareMathOperator{\rot}{rot} \left. \begin{aligned} \frac{d}{dl}\bigl[(1 + \overline{\sigma}) \mathbf v \bigr] &= \grad \overline{\sigma} + \frac{\partial \mathfrak A}{\partial l} + [\rot \mathfrak A, \mathbf v], \\ \overline{\sigma} &= \frac{\kappa}{8\pi} \int \frac{\sigma\, dV_{0}}{r}, \\ \mathfrak A &= \frac{\kappa}{2 \pi} \int \frac{\sigma \dfrac{dx_{\alpha}}{dl}\, dV_{0}}{r}. \end{aligned} \right\} \qquad \text{(118)}$

The equations of motion, (118), show now, in fact, that

The equations of motion, (118), now actually show that

1. The inert mass is proportional to 1 + $\overline{\sigma}$ and therefore increases when ponderable masses approach the test body.

1. The inert mass is proportional to 1 + $\overline{\sigma}$ and so it increases when physical masses get closer to the test body.

2. There is an inductive action of accelerated masses, of the same sign, upon the test body. This is the term $\dfrac{\partial \mathfrak A}{\partial l}$ . [Pg 108]

2. There is an inductive effect from accelerated masses, of the same type, on the test object. This is referred to as $\dfrac{\partial \mathfrak A}{\partial l}$ . [Pg 108]

3. A material particle, moving perpendicularly to the axis of rotation inside a rotating hollow body, is deflected in the sense of the rotation (Coriolis field). The centrifugal action, mentioned above, inside a rotating hollow body, also follows from the theory, as has been shown by Thirring.[18]

3. A material particle moving perpendicular to the axis of rotation inside a rotating hollow body is deflected in the direction of the rotation (Coriolis effect). The centrifugal force mentioned earlier within a rotating hollow body also comes from the theory, as demonstrated by Thirring.[18]

[18]That the centrifugal action must be inseparably connected with the existence of the Coriolis field may be recognized, even without calculation, in the special case of a co-ordinate system rotating uniformly relatively to an inertial system; our general co-variant equations naturally must apply to such a case.

[18]It can be understood that the centrifugal force is closely linked to the presence of the Coriolis field, even without doing any calculations, in the specific situation where a coordinate system is rotating uniformly in relation to an inertial system; our general co-variant equations should naturally apply in this scenario.

Although all of these effects are inaccessible to experiment, because $\kappa$ is so small, nevertheless they certainly exist according to the general theory of relativity. We must see in them a strong support for Mach's ideas as to the relativity of all inertial actions. If we think these ideas consistently through to the end we must expect the whole inertia, that is, the whole $g_{\mu\nu}$ -field, to be determined by the matter of the universe, and not mainly by the boundary conditions at infinity.

Although we can’t experiment with all these effects because $\kappa$ is so small, they definitely exist according to the general theory of relativity. We should view them as strong support for Mach's ideas about the relativity of all inertial actions. If we think through these ideas completely, we must expect the entire inertia, meaning the entire $g_{\mu\nu}$ -field, to be shaped by the matter of the universe, rather than mainly by the boundary conditions at infinity.

For a satisfactory conception of the $g_{\mu\nu}$ -field of cosmical dimensions, the fact seems to be of significance that the relative velocity of the stars is small compared to the velocity of light. It follows from this that, with a suitable choice of co-ordinates, $g_{44}$ is nearly constant in the universe, at least, in that part of the universe in which there is matter. The assumption appears natural, moreover, that there are stars in all parts of the universe, so that we may well assume that the inconstancy of $g_{44}$ depends only upon the circumstance that matter is not distributed continuously, but is concentrated in single celestial bodies and systems of bodies. If we are willing to ignore these more local [Pg 109] non-uniformities of the density of matter and of the $g_{\mu\nu}$ -field, in order to learn something of the geometrical properties of the universe as a whole, it appears natural to substitute for the actual distribution of masses a continuous distribution, and furthermore to assign to this distribution a uniform density $\sigma$ . In this imagined universe all points with space directions will be geometrically equivalent; with respect to its space extension it will have a constant curvature, and will be cylindrical with respect to its $x_{4}$ -co-ordinate. The possibility seems to be particularly satisfying that the universe is spatially bounded and thus, in accordance with our assumption of the constancy of $\sigma$ , is of constant curvature, being either spherical or elliptical; for then the boundary conditions at infinity which are so inconvenient from the standpoint of the general theory of relativity, may be replaced by the much more natural conditions for a closed surface.

To understand the $g_{\mu\nu}$ -field on a cosmic scale, it’s important to note that the relative speed of stars is small compared to the speed of light. This indicates that, with an appropriate choice of coordinates, $g_{44}$ is almost constant throughout the universe, at least in regions that contain matter. It seems reasonable to assume that stars are distributed throughout the universe, so we can suggest that any variations in $g_{44}$ are due to the non-continuous distribution of matter, which is concentrated in discrete celestial bodies and systems. If we overlook these localized irregularities in matter density and the $g_{\mu\nu}$ -field, to gain insight into the overall geometrical properties of the universe, it makes sense to replace the actual mass distribution with a continuous one and assume it has a uniform density $x_{4}$ -coordinate. It’s especially pleasing to consider that the universe could be spatially bounded and, based on our assumption of a constant $\sigma$ density, could also exhibit constant curvature, being either spherical or elliptical. This way, the problematic boundary conditions at infinity, as problematic in general relativity, could instead be substituted with more natural conditions applicable to a closed surface.

According to what has been said, we are to put $ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, \qquad \text{(119)}$ in which the indices $\mu$ and $\nu$ run from 1 to 3 only. The $\gamma_{\mu\nu}$ will be such functions of $x_{1}$ , $x_{2}$ , $x_{3}$ as correspond to a three-dimensional continuum of constant positive curvature. We must now investigate whether such an assumption can satisfy the field equations of gravitation.

According to what has been said, we need to put $ds^{2} = {dx_{4}}^{2} - \gamma_{\mu\nu}\, dx_{\mu}\, dx_{\nu}, \qquad \text{(119)}$ where the indices $\mu$ and $\nu$ range from 1 to 3 only. The $\gamma_{\mu\nu}$ will be functions of $x_{1}$ , $x_{2}$ , $x_{3}$ that correspond to a three-dimensional continuum of constant positive curvature. We now need to investigate whether this assumption can meet the field equations of gravitation.

In order to be able to investigate this, we must first find what differential conditions the three-dimensional manifold of constant curvature satisfies. A spherical manifold of three dimensions, [Pg 110] embedded in a Euclidean continuum of four dimensions,[19] is given by the equations $\begin{align*} {x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}, \\ {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. \end{align*}$ By eliminating $x_{4}$ , we get $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}.$

To investigate this, we first need to determine the differential conditions that the three-dimensional manifold with constant curvature satisfies. A three-dimensional spherical manifold embedded in a four-dimensional Euclidean space is described by the equations [Pg 110] [19] $\begin{align*} {x_{1}}^{2} + {x_{2}}^{2} + {x_{3}}^{2} + {x_{4}}^{2} &= a^{2}, \\ {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + {dx_{4}}^{2} &= ds^{2}. \end{align*}$ By removing $x_{4}$ , we have $ds^{2} = {dx_{1}}^{2} + {dx_{2}}^{2} + {dx_{3}}^{2} + \frac{(x_{1}\, dx_{1} + x_{2}\, dx_{2} + x_{3}\, dx_{3})^{2}} {a^{2} - {x_{1}}^{2} - {x_{2}}^{2} - {x_{3}}^{2}}.$

[19]The aid of a fourth space dimension has naturally no significance except that of a mathematical artifice.

[19]The idea of a fourth spatial dimension doesn't have any real meaning apart from being a mathematical trick.

As far as terms of the third and higher degrees in the $x_{\nu}$ , we can put, in the neighbourhood of the origin of co-ordinates, $ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) dx_{\mu}\, dx_{\nu}.$

As for the terms of the third and higher degrees in the $x_{\nu}$ , we can write, near the origin of coordinates, $ds^{2} = \left(\delta_{\mu\nu} + \frac{x_{\mu} x_{\nu}}{a^{2}}\right) dx_{\mu}\, dx_{\nu}.$

Inside the brackets are the $g_{\mu\nu}$ of the manifold in the neighbourhood of the origin. Since the first derivatives of the $g_{\mu\nu}$ , and therefore also the $\Gamma_{\mu\nu}^{\sigma}$ , vanish at the origin, the calculation of the $R_{\mu\nu}$ for this manifold, by (88), is very simple at the origin. We have $R_{\mu\nu} = -\frac{2}{a^{2}} \delta_{\mu\nu} = \frac{2}{a^{2}} g_{\mu\nu}.$

Inside the brackets are the $g_{\mu\nu}$ of the manifold near the origin. Since the first derivatives of the $g_{\mu\nu}$ , and therefore also the $\Gamma_{\mu\nu}^{\sigma}$ , are zero at the origin, calculating the $R_{\mu\nu}$ for this manifold, as in (88), is quite straightforward at the origin. We have $R_{\mu\nu} = -\frac{2}{a^{2}} \delta_{\mu\nu} = \frac{2}{a^{2}} g_{\mu\nu}.$

Since the relation $R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}$ is universally co-variant, and since all points of the manifold are geometrically equivalent, this relation holds for every system of co-ordinates, and everywhere in the manifold. In order to avoid confusion with [Pg 111] the four-dimensional continuum, we shall, in the following, designate quantities that refer to the three-dimensional continuum by Greek letters, and put $P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}. \qquad \text{(120)}$

Since the relationship $R_{\mu\nu} = \dfrac{2}{a^{2}} g_{\mu\nu}$ is universally covariant, and since all points of the manifold are geometrically equivalent, this relationship applies to every coordinate system, and everywhere in the manifold. To avoid confusion with [Pg 111] the four-dimensional continuum, we will, from now on, use Greek letters to indicate quantities that refer to the three-dimensional continuum and denote $P_{\mu\nu} = -\frac{2}{a^{2}} \gamma_{\mu\nu}. \qquad \text{(120)}$

We now proceed to apply the field equations (96) to our special case. From (119) we get for the four-dimensional manifold, $\left. \begin{aligned} R_{\mu\nu} &= P_{\mu\nu} \quad \text{for the indices 1 to 3}, \\ R_{14} &= R_{24} = R_{34} = R_{44} = 0. \end{aligned} \right\} \qquad \text{(121)}$

We now move on to apply the field equations (96) to our specific case. From (119), we obtain for the four-dimensional manifold, $\left. \begin{aligned} R_{\mu\nu} &= P_{\mu\nu} \quad \text{for the indices 1 to 3}, \\ R_{14} &= R_{24} = R_{34} = R_{44} = 0. \end{aligned} \right\} \qquad \text{(121)}$

For the right-hand side of (96) we have to consider the energy tensor for matter distributed like a cloud of dust. According to what has gone before we must therefore put $T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}$ specialized for the case of rest. But in addition, we shall add a pressure term that may be physically established as follows. Matter consists of electrically charged particles. On the basis of Maxwell's theory these cannot be conceived of as electromagnetic fields free from singularities. In order to be consistent with the facts, it is necessary to introduce energy terms, not contained in Maxwell's theory, so that the single electric particles may hold together in spite of the mutual repulsions between their elements, charged with electricity of one sign. For the sake of consistency with this fact, Poincaré has assumed a pressure [Pg 112] to exist inside these particles which balances the electrostatic repulsion. It cannot, however, be asserted that this pressure vanishes outside the particles. We shall be consistent with this circumstance if, in our phenomenological presentation, we add a pressure term. This must not, however, be confused with a hydrodynamical pressure, as it serves only for the energetic presentation of the dynamical relations inside matter. In this sense we put $T_{\mu\nu} = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} - g_{\mu\nu} p. \qquad \text{(122)}$

For the right side of (96), we need to consider the energy tensor for matter distributed like a cloud of dust. According to what we've discussed, we should use $T^{\mu\nu} = \sigma \frac{dx_{\mu}}{ds}\, \frac{dx_{\nu}}{ds}$ specifically for the case when at rest. However, we will also add a pressure term that can be validated as follows. Matter is made up of electrically charged particles. Based on Maxwell's theory, these cannot be viewed as electromagnetic fields free of singularities. To align with the facts, it is necessary to include energy terms not part of Maxwell's theory so that individual electric particles can stay together despite the mutual repulsions among their components, which carry electricity of the same charge. To maintain consistency with this reality, Poincaré proposed that there is pressure inside these particles that balances the electrostatic repulsion. However, it cannot be claimed that this pressure disappears outside the particles. We will be consistent with this scenario if, in our phenomenological presentation, we include a pressure term. This should not be mistaken for hydrodynamic pressure, as it only serves to represent the energy dynamics inside the matter. In this context, we write $T_{\mu\nu} = g_{\mu\sigma} g_{\nu\beta} \frac{dx_{\alpha}}{ds}\, \frac{dx_{\beta}}{ds} - g_{\mu\nu} p. \qquad \text{(122)}$

In our special case we have, therefore, to put $\begin{align*} T_{\mu\nu} &= \gamma_{\mu\nu} p \quad \text{(for}\,\,\mu\,\, \text{and}\,\,\,\nu \,\,\text{from 1 to 3)}, \\ T_{44} &= \sigma - p, \\ T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p = \sigma - 4p. \end{align*}$ Observing that the field equation (96) may be written in the form $R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T),$ we get from (96) the equations, $\begin{align*} +\frac{2}{a^{2}} \gamma_{\mu\nu} &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}, \\ 0 &= -\kappa \left(\frac{\sigma}{2} + p\right). \end{align*}$ From this follows $\left. \begin{aligned} p &= -\frac{\sigma}{2}, \\ a &= \sqrt{\frac{2}{\kappa\sigma}}. \end{aligned} \right\} \qquad \text{(123)}$

In our specific case, we need to establish $\begin{align*} T_{\mu\nu} &= \gamma_{\mu\nu} p \quad \text{(for}\,\,\mu\,\, \text{and}\,\,\,\nu \,\,\text{from 1 to 3)}, \\ T_{44} &= \sigma - p, \\ T &= -\gamma^{\mu\nu} \gamma_{\mu\nu} p + \sigma - p = \sigma - 4p. \end{align*}$ Noting that the field equation (96) can be expressed as $R_{\mu\nu} = -\kappa(T_{\mu\nu} - \tfrac{1}{2} g_{\mu\nu} T),$ we derive from (96) the equations, $\begin{align*} +\frac{2}{a^{2}} \gamma_{\mu\nu} &= \kappa \left(\frac{\sigma}{2} - p\right) \gamma_{\mu\nu}, \\ 0 &= -\kappa \left(\frac{\sigma}{2} + p\right). \end{align*}$ From this, it follows $\left. \begin{aligned} p &= -\frac{\sigma}{2}, \\ a &= \sqrt{\frac{2}{\kappa\sigma}}. \end{aligned} \right\} \qquad \text{(123)}$

If the universe is quasi-Euclidean, and its radius of curvature therefore infinite, then σ would vanish. But it is improbable that [Pg 113] the mean density of matter in the universe is actually zero; this is our third argument against the assumption that the universe is quasi-Euclidean. Nor does it seem possible that our hypothetical pressure can vanish; the physical nature of this pressure can be appreciated only after we have a better theoretical knowledge of the electromagnetic field. According to the second of equations (123) the radius, $a$ , of the universe is determined in terms of the total mass, $M$ , of matter, by the equation $a = \frac{M\kappa}{4\pi^{2}}. \qquad \text{(124)}$ The complete dependence of the geometrical upon the physical properties becomes clearly apparent by means of this equation.

If the universe is almost Euclidean and its radius of curvature is therefore infinite, then σ would disappear. However, it's unlikely that the average density of matter in the universe is actually zero; this is our third point against the idea that the universe is almost Euclidean. It also doesn't seem feasible for our hypothetical pressure to vanish; we can only fully understand the nature of this pressure after we gain a better theoretical understanding of the electromagnetic field. According to the second of equations (123), the radius, $a$ , of the universe is determined based on the total mass, $M$ , of matter, by the equation $a = \frac{M\kappa}{4\pi^{2}}. \qquad \text{(124)}$ The complete dependence of the geometric properties on the physical characteristics becomes clearly evident through this equation.

Thus we may present the following arguments against the conception of a space-infinite, and for the conception of a space-bounded, universe:—

Thus we can make the following arguments against the idea of an infinite universe and in favor of the idea of a bounded universe:—

1. From the standpoint of the theory of relativity, the condition for a closed surface is very much simpler than the corresponding boundary condition at infinity of the quasi-Euclidean structure of the universe.

1. From the perspective of the theory of relativity, the requirement for a closed surface is much simpler than the related boundary condition at infinity of the quasi-Euclidean structure of the universe.

2. The idea that Mach expressed, that inertia depends upon the mutual action of bodies, is contained, to a first approximation, in the equations of the theory of relativity; it follows from these equations that inertia depends, at least in part, upon mutual actions between masses. As it is an unsatisfactory assumption to make that inertia depends in part upon mutual actions, and in part upon an independent property of space, Mach's idea gains in probability. But this idea of Mach's corresponds only to a finite universe, bounded in space, and not to a quasi-Euclidean, infinite universe. From the standpoint of epistemology [Pg 114] it is more satisfying to have the mechanical properties of space completely determined by matter, and this is the case only in a space-bounded universe.

2. Mach's idea that inertia is influenced by the interactions between bodies is reflected, at least in a basic way, in the equations of relativity theory. These equations suggest that inertia is partly determined by the interactions between masses. Since it's not a satisfying assumption to think of inertia as being influenced by both mutual interactions and an independent property of space, Mach's idea becomes more likely. However, Mach's concept only applies to a finite, space-bounded universe and not to an infinite, quasi-Euclidean universe. From an epistemological perspective, it's more satisfying to have the mechanical properties of space completely defined by matter, which only happens in a universe with boundaries. [Pg 114]

3. An infinite universe is possible only if the mean density of matter in the universe vanishes. Although such an assumption is logically possible, it is less probable than the assumption that there is a finite mean density of matter in the universe. [Pg 115]

3. An infinite universe is possible only if the average density of matter in the universe is zero. While this assumption is logically possible, it's less likely than the assumption that there is a finite average density of matter in the universe. [Pg 115]

INDEX

Accelerated masses, inductive
action of, 108
Addition and subtraction of
tensors, 14
—theorem of velocities, 38

Accelerated masses, inductive
action of, __A_TAG_PLACEHOLDER_0__
Adding and subtracting
tensors, __A_TAG_PLACEHOLDER_0__
—theorem of velocities, 38

Biot-Savart force, 44

Biot-Savart law, __A_TAG_PLACEHOLDER_0__

Centrifugal force, 64
Clocks, moving, 38
Compressible viscous fluid, 22
Concept of space, 3
—time, 28
Conditions of orthogonality, 7
Congruence, theorems of, 3
Conservation principles, 54
Continuum, four-dimensional, 31
Contraction of tensors, 14
Contra-variant vectors, 69
—tensors, 71
Co-ordinates, preferred systems
of, 8
Co-variance of equation of
continuity, 21
Co-variant, 12 et seq.
—vector, 68
Criticism of principle of inertia, 62
Criticisms of theory of
relativity, 29
Curvilinear co-ordinates, 65

Centrifugal force, 64
Clocks, in motion, 38
Compressible viscous fluid, 22
Concept of space, 3
—time, 28
Conditions of orthogonality, 7
Congruence, theorems of, 3
Conservation principles, 54
Four-dimensional continuum, 31
Contraction of tensors, 14
Contra-variant vectors, 69
—tensors, 71
Preferred coordinate systems
of, __A_TAG_PLACEHOLDER_0__
Co-variance of the equation of
continuity, __A_TAG_PLACEHOLDER_0__
Co-variant, 12 et seq.
—vector, 68
Criticism of the principle of inertia, 62
Criticisms of the theory of
relativity, __A_TAG_PLACEHOLDER_0__
Curvilinear coordinates, 65

Differentiation of tensors, 73, 76
Displacement of spectral lines, 97

Energy and mass, 45, 49
—tensor of electromagnetic
field, 50
—of matter, 54
Equation of continuity, co-variance
of, 21
Equations of motion of material
particle, 50
Equivalence of mass and
energy, 49
Equivalent spaces of reference, 25
Euclidean geometry, 4

Energy and mass, 45, 49
—tensor of electromagnetic
field, __A_TAG_PLACEHOLDER_0__
—of matter, 54
Equation of continuity, co-variance
of, __A_TAG_PLACEHOLDER_0__
Equations of motion of material
particle, __A_TAG_PLACEHOLDER_0__
Equivalence of mass and
energy, __A_TAG_PLACEHOLDER_0__
Equivalent spaces of reference, 25
Euclidean geometry, 4

Finiteness of universe, 105
Fizeau, 28
Four-dimensional continuum, 31
Four-vector, 41
Fundamental tensor, 71

Finiteness of the universe, 105
Fizeau, 28
Four-dimensional continuum, 31
Four-vector, 41
Fundamental tensor, 71

Galilean regions, 62
—transformation, 27
Gauss, 65
Geodetic lines, 82
Geometry, Euclidean, 4
Gravitation constant, 95
Gravitational mass, 60

Homogeneity of space, 17
Hydrodynamical equations, 54
Hypotheses of pre-relativity
physics, 73

Homogeneity of space, 17
Hydrodynamical equations, 54
Hypotheses of pre-relativity
physics, __A_TAG_PLACEHOLDER_0__

Inductive action of accelerated
masses, 108
Inert and gravitational mass, equality
of, 60
Invariant, 9 et seq.
Isotropy of space, 17

Inductive action of accelerated
masses, __A_TAG_PLACEHOLDER_0__
Inert and gravitational mass, equality
of, __A_TAG_PLACEHOLDER_0__
Invariant, 9 et seq.
Isotropy of space, 17

Kaluza, 104

Kaluza, __A_TAG_PLACEHOLDER_0__

Levi-Civita, 73
Light-cone, 41
Light ray, path of, 98
Light-time, 33
Linear orthogonal
transformation, 7
Lorentz electromotive force, 44
—transformation, 31

Levi-Civita, 73
Light cone, 41
Path of a light ray, 98
Light-time, 33
Linear orthogonal
transformation, __A_TAG_PLACEHOLDER_0__
Lorentz electromotive force, 44
—transformation, 31

Mach, 59, 105, 106, 109, 114
Mass and Energy, 45, 49
—equality of gravitational and
inert, 60
—gravitational, 60
Maxwell's equations, 23
Mercury, perihelion of, 99, 103
Michelson and Morley, 28
Minkowski, 32
Motion of particle, equations of, 50
Moving measuring rods and
clocks, 38
Multiplication of tensors, 14

Mach, 59, 105, 106, 109, 114
Mass and Energy, 45, 49
—equivalence of gravitational and
inertial, __A_TAG_PLACEHOLDER_0__
—gravitational, 60
Maxwell's equations, 23
Mercury, perihelion of, 99, 103
Michelson and Morley, 28
Minkowski, 32
Motion of particles, equations of, 50
Moving measuring rods and
clocks, __A_TAG_PLACEHOLDER_0__
Multiplication of tensors, 14

Newtonian gravitation
constant, 95

Newton's law of gravitation
constant, __A_TAG_PLACEHOLDER_0__

Operations on tensors, 13 et seq.
Orthogonal transformations, linear, 7
Orthogonality, conditions of, 7

Operations on tensors, 13 and others.
Orthogonal transformations, linear, 7
Orthogonality, conditions of, 7

Path of light ray, 98
Perihelion of Mercury, 99, 103
Poisson's equation, 87
Preferred systems of
co-ordinates, 8
Pre-relativity physics, hypotheses
of, 26
Principle of equivalence, 61
—inertia, criticism of, 62
Principles of conservation, 54

Path of light ray, 98
Perihelion of Mercury, 99, 103
Poisson's equation, 87
Preferred systems of
coordinates, __A_TAG_PLACEHOLDER_0__
Pre-relativity physics, hypotheses
of, __A_TAG_PLACEHOLDER_0__
Principle of equivalence, 61
—inertia, criticism of, 62
Principles of conservation, 54

Radius of Universe, 113
Rank of tensor, 13
Ray of light, path of, 98
Reference, space of, 3
Riemann, 68
—tensor, 79, 82, 105
Rods (measuring) and clocks in
motion, 38
Rotation, 63

Radius of Universe, 113
Rank of tensor, 13
Path of light, 98
Space reference, 3
Riemann, 68
—tensor, 79, 82, 105
Measuring rods and clocks in
motion, __A_TAG_PLACEHOLDER_0__
Rotation, 63

Simultaneity, 17, 29
Sitter, 28
Skew-symmetrical tensor, 15
Solar Eclipse expedition (1919), 99
Space, concept of, 2
—Homogeneity of, 17
—Isotropy of, 17
Spaces of reference, 3
—equivalence of, 25
Special Lorentz transformation, 34
Spectral lines, displacement of, 97
Straightest lines, 82
Stress tensor, 22
Symmetrical tensor, 15
Systems of co-ordinates,
preferred, 8

Simultaneity, 17, 29
Observer, 28
Skew-symmetric tensor, 15
Solar Eclipse expedition (1919), 99
Concept of space, 2
—Homogeneity of, 17
—Isotropy of, 17
Reference frames, 3
—equivalence of, 25
Special Lorentz transformation, 34
Displacement of spectral lines, 97
Straightest paths, 82
Stress tensor, 22
Symmetric tensor, 15
Coordinate systems,
preferred, 8

Tensor, 12 et seq., 68 et seq.
—Addition and subtraction of, 14
—Contraction of, 14
—Fundamental, 71
—Multiplication of, 14
—operations, 13 et seq.
—Rank of, 13
—Symmetrical and
Skew-symmetrical, 15
Tensors, formation by
differentiation, 73
Theorem for addition of
velocities, 38
Theorems of congruence, 3
Theory of relativity, criticisms
of, 29
Thirring, 109
Time-concept, 28
Time-space concept, 31
Transformation, Galilean, 27
—Linear orthogonal, 7

Tensor, 12 et seq., 68 et seq.
—Addition and subtraction of, 14
—Contraction of, 14
—Fundamental, 71
—Multiplication of, 14
—operations, 13 et seq.
—Rank of, 13
—Symmetrical and
Skew-symmetric, __A_TAG_PLACEHOLDER_0__
Tensors, formation by
differentiation, __A_TAG_PLACEHOLDER_0__
Theorem for addition of
velocities, __A_TAG_PLACEHOLDER_0__
Theorems of congruence, 3
Theory of relativity, criticisms
of, __A_TAG_PLACEHOLDER_0__
Thirring, 109
Time concept, 28
Time-space concept, 31
Transformation, Galilean, 27
—Linear orthogonal, 7

Universe, Finiteness of, 105
—Radius of, 113

Universe, Finiteness of, __A_TAG_PLACEHOLDER_0__
—Radius of, __A_TAG_PLACEHOLDER_1__

Vector, co-variant, 69
—contra-variant, 69
Velocities, addition theorem of, 38
Viscous compressible fluid, 22

Weyl, 73, 99, 104

Weyl, __A_TAG_PLACEHOLDER_0__, __A_TAG_PLACEHOLDER_1__, __A_TAG_PLACEHOLDER_2__

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TRANSCRIBER'S NOTES

Transcriber's Notes

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Minor typographical corrections and formatting changes have been made without comment.

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THE MEANING OF RELATIVITY

CONTENTS

THE MEANING OF RELATIVITY

LECTURE I SPACE AND TIME IN PRE-RELATIVITY PHYSICS

LECTURE II THE THEORY OF SPECIAL RELATIVITY

LECTURE III THE GENERAL THEORY OF RELATIVITY

LECTURE IV THE GENERAL THEORY OF RELATIVITY (continued)

INDEX

THE MEANING OF
RELATIVITY

LECTURE I

SPACE AND TIME IN PRE-RELATIVITY PHYSICS

LECTURE II

THE THEORY OF SPECIAL RELATIVITY

LECTURE III

THE GENERAL THEORY OF RELATIVITY

LECTURE IV

THE GENERAL THEORY OF RELATIVITY
(continued)