Space, Time, Matter

Hermann Weyl

• Publisher: Dover Publications
• Year: 2013
• ISBN:
  9780486318424
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

`The standard treatise on the general theory of relativity.` — Nature
`Whatever the future may bring, Professor Weyl's book will remain a classic of physics.` — British Journal for Philosophy and Science
Reflecting the revolution in scientific and philosophic thought which accompanied the Einstein relativity theories, Dr. Weyl has probed deeply into the notions of space, time, and matter. A rigorous examination of the state of our knowledge of the world following these developments is undertaken with this guiding principle: that although further scientific thought may take us far beyond our present conception of the world, we may never again return to the previous narrow and restricted scheme.
Although a degree of mathematical sophistication is presupposed, Dr. Weyl develops all the tensor calculus necessary to his exposition. He then proceeds to an analysis of the concept of Euclidean space and the spatial conceptions of Riemann. From this the nature of the amalgamation of space and time is derived. This leads to an exposition and examination of Einstein's general theory of relativity and the concomitant theory of gravitation. A detailed investigation follows devoted to gravitational waves, a rigorous solution of the problem of one body, laws of conservation, and the energy of gravitation. Dr. Weyl's introduction of the concept of tensor-density as a magnitude of quantity (contrasted with tensors which are considered to be magnitudes of intensity) is a major step toward a clearer understanding of the relationships among space, time, and matter.

CHAPTER I

EUCLIDEAN SPACE. ITS MATHEMATICAL FORMULATION AND
ITS RÔLE IN PHYSICS

§ 1. Deduction of the Elementary Conceptions of Space from
that of Equality

JUST as we fixed the present moment (“now”) as a geometrical point in time, so we fix an exact “here,” a point in space, as the first element of continuous spatial extension, which, like time, is infinitely divisible. Space is not a one-dimensional continuum like time. The principle by which it is continuously extended cannot be reduced to the simple relation of “earlier” or “later ”. We shall refrain from inquiring what relations enable us to grasp this continuity conceptually. On the other hand, space, like time, is a form of phenomena. Precisely the same content, identically the same thing, still remaining what it is, can equally well be at some place in space other than that at which it is actually. The new portion of Space S′ then occupied by it is equal to that portion S which it actually occupied. S and S′ are said to be congruent. To every point P of S there corresponds one definite homologous point P′ of S′ which, after the above displacement to a new position, would be surrounded by exactly the same part of the given content as that which surrounded P originally. We shall call this “transformation” (in virtue of which the point P′ corresponds to the point P) a congruent transformation. Provided that the appropriate subjective conditions are satisfied the given material tiling would seem to us after the displacement exactly the same as before. There is reasonable justification for believing that a rigid body, when placed in two positions successively, realises this idea of the equality of two portions of space ; by a rigid body we mean one which, however it be moved or treated, can always be made to appear the same to us as before, if we take up the appropriate position with respect to it. I shall evolve the scheme of geometry from the conception of equality combined with that of continuous connection—of which the latter offers great difficulties to analysis —and shall show in a superficial sketch how all fundamental conceptions of geometry may be traced back to them. My real object in doing so will be to single out translations among possible congruent transformations. Starting from the conception of translation I shall then develop Euclidean geometry along strictly axiomatic lines.

First of all the straight line. Its distinguishing feature is that it is determined by two of its points. Any other line can, even when two of its points are kept fixed, be brought into another position by a congruent transformation (the test of straightness).

Thus, if A and B are two different points, the straight line g = AB includes every point which becomes transformed into itself by all those congruent transformations which transform AB into themselves. (In familiar language, the straight line lies evenly between its points.) Expressed kinematically, this is tantamount to saying that we regard the straight line as an axis of rotation. It is homogeneous and a linear continuum just like time. Any arbitrary point on it divides it into two parts, two “rays”. If B lies on one of these parts and C on the other, then A is said to be between B and C and the points of one part lie to the right of A, the points of the other part to the left. (The choice as to which is right or left is determined arbitrarily.) The simplest fundamental facts which are implied by the conception “between” can be formulated as exactly and completely as a geometry which is to be built up by deductive processes demands. For this reason we endeavour to trace back all conceptions of continuity to the conception “between,” i.e. to the relation “A is a point of the straight line BC and lies between B and C” (this is the reverse of the real intuitional relation). Suppose A′ to be a point on g to the right of A, then A′ also divides the line g into two parts. We call that to which A belongs the left-hand side. If, however, A′ lies to the left of A the position is reversed. With this convention, analogous relations hold not only for A and A′ but also for any two points of a straight line. The points of a straight line are ordered by the terms left and right in precisely the same way as points of time by the terms earlier and later.

Left and right are equivalent. There is one congruent transformation which leaves A fixed, but which interchanges the two halves into which A divides the straight line. Every finite portion of straight line AB may be superposed upon itself in such a way that it is reversed (i.e. so that B falls on A, and A falls on B). On the other hand, a congruent transformation which transforms A into itself, and all points to the right of A into points to the right of A, and all...

Table of contents

• Cover
• Title Page
• Copyright Page
• Preface
• Translator’s Note
• Contents
• Introduction
• Chapter I: Euclidean Space. Its Mathematical Form and Its Rôle in Physics.
• Chapter II: The Metrical Continuum
• Chapter III: Relativity of Space and Time
• Chapter IV: General Theory of Relativity
• Appendix I
• Appendix II
• Bibliographical References
• Index