The Meaning of Relativity

Including the Relativistic Theory of the Non-Symmetric Field - Fifth Edition

Albert Einstein

• Publisher: Princeton University Press
• Year: 2014
• Print ISBN: 9780691120270, 9780691080079
• eBook ISBN: 9781400851874
• Edition: 5
• Topic: Physical Sciences
• Subtopic: Relativity in Physics

In 1921, five years after the appearance of his comprehensive paper on general relativity and twelve years before he left Europe permanently to join the Institute for Advanced Study, Albert Einstein visited Princeton University, where he delivered the Stafford Little Lectures for that year. These four lectures constituted an overview of his then-controversial theory of relativity. Princeton University Press made the lectures available under the title The Meaning of Relativity, the first book by Einstein to be produced by an American publisher. As subsequent editions were brought out by the Press, Einstein included new material amplifying the theory. A revised version of the appendix "Relativistic Theory of the Non-Symmetric Field," added to the posthumous edition of 1956, was Einstein's last scientific paper.

APPENDIX II

RELATIVISTIC THEORY OF THE NON-SYMMETRIC FIELD

BEFORE starting with the subject proper I am first going to discuss the “strength” of systems of field equations in general. This discussion is of intrinsic interest quite apart from the particular theory presented here. For a deeper understanding of our problem, however, it is almost indispensable.

ON THE “COMPATIBILITY” AND THE “STRENGTH” OF SYSTEMS OF FIELD EQUATIONS

Given certain field variables and a system of field equations for them, the latter will not in general determine the field completely. There still remain certain free data for a solution of the field equations. The smaller the number of free data consistent with the system of field equations, the “stronger” is the system. It is clear that in the absence of any other viewpoint from which to select the equations, one will prefer a “stronger” system to a less strong one. It is our aim to find a measure for this strength of systems of equations. It will turn out that such a measure can be defined which will even enable us to compare with each other the strengths of systems whose field variables differ with respect to number and kind.

We shall present the concepts and methods involved here in examples of increasing complexity, restricting ourselves to four-dimensional fields, and in the course of these examples we shall successively introduce the relevant concepts.

First example: The scalar wave equation*

Here the system consists of only one differential equation for one field variable. We assume ϕ to be expanded in a Taylor series in the neighborhood of a point P (which presupposes the analytic character of ϕ). The totality of its coefficients describes then the function completely. The number of nth order coefficients (that is, the nth order derivatives of ϕ at the point P) is equal to

(abbreviated

), and all these coefficients could be freely chosen if the differential equation did not imply certain relations between them. Since the equation is of second order, these relations are found by (n − 2) fold differentiation of the equation. We thus obtain for the nth order coefficients

conditions. The number of nth order coefficients remaining free is therefore

This number is positive for any n. Hence, if the free coefficients for all orders smaller than n have been fixed, the conditions for the coefficients of order n can always be satisfied without changing the coefficients already chosen.

Analogous reasoning can be applied to systems consisting of several equations. If the number of free nth order coefficients does not become smaller than zero, we call the system of equations absolutely compatible. We shall restrict ourselves to such systems of equations. All systems known to me which are used in physics are of this kind.

Let us now rewrite equation (1). We have

where z₁ = +6.

If we restrict ourselves to large values of n, we may neglect the terms

etc. in the parenthesis, and we obtain for (1) asymptotically

We call z₁ the “coefficient of freedom,” which in our case has the value 6. The larger this coefficient, the weaker is the corresponding system of equations.

Second example: Maxwell’s equations for empty space

ϕ^ik results from the antisymmetric tensor ϕ_ik by raising the covariant indices with the help of

These are 4 + 4 field equations for six field variables. Among these eight equations, there exist two identities. If the left-hand sides of the field equations are denoted by Gⁱ and H_ikl respectively, the identities have the form

In this case we reason as follows.

The Taylor expansion of the six field components furnishes

coefficients of the nth order. The co...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Introduction
• A Note on the Fifth Edition
• Space and Time in Pre-Relativity Physics
• The Theory of Special Relativity
• The General Theory of Relativity
• The General Theory of Relativity (Continued)
• Appendix for the Second Edition
• Appendix II. Relativistic Theory of the Non-Symmetric Field
• Index