THE SPACE-TIME MANIFOLD OF RELATIVITY. THE NON-EUCLIDEAN GEOMETRY OF MECHANICS AND ELECTROMAGNETICS.

BY EDWIN B. WILSON AND GILBERT N. LEWIS.

Introduction.

1. The concept of space has different meanings to different persons according to their experience in abstract reasoning. On the one hand is the common space, which for the educated person has been formulated in the three dimensional geometry of Euclid. On the other hand the mathematician has become accustomed to extend the concept of space to any manifold of which the properties are completely determined, as in Euclidean geometry, by a system of self-consistent postulates. Most of these highly ingenious geometries cannot be expected to be of service in the discussion of physical phenomena.

Until recently the physicist has found the three dimensional space of Euclid entirely adequate to his needs, and has therefore been inclined to attribute to it a certain reality. It is, however, inconsistent with the philosophic spirit of our time to draw a sharp distinction between that which is real and that which is convenient,1 and it would be dogmatic to assert that no discoveries of physics might render so convenient as to be almost imperative the modification or extension of our present system of geometry. Indeed it seemed to Minkowski that such a change was already necessitated by the facts which led to the formulation of the Principle of Relativity.

2. The possibility of associating three dimensional space and one dimensional time to form a four dimensional manifold has doubtless occurred to many; but as long as space and time were assumed to be wholly independent, such a union seemed purely artificial. The idea of abandoning once for all this assumption of independence, although fore-shadowed in Lorentz's use of local time, was first clearly stated by

1 See, for example, H. Poincaré, La Science et l'Hypothèse.

Einstein. The theorems of the principle of relativity which correlate space and time appeared, however, far less bizarre and unnatural when Minkowski showed that they were merely theorems in a four dimensional geometry.

Suppose that a student of ordinary space, habituated to the interpretation of geometry with the aid of a definite horizontal plane and vertical axis, should suddenly discover that all the essential geometrical properties of interest to him could be expressed by reference to a new plane, inclined to the horizontal, and a new axis inclined to the vertical. Whereas formerly he had attributed special significance to heights on the one hand and to horizontal extension on the other, he would now recognize that these were purely conventional and that the fundamental properties were those such as distance and angle, which remain invariant in the change to a new system of reference.

Let us now consider a four dimensional manifold formed by adjoining to the familiar x, y, z axes of space a t axis of time. Any point in this manifold will represent a definite place at a definite time. Space then appears as a sort of cross section through this manifold, comprising all points of a given time. For convenience we may temporarily ignore one of the dimensions of space, say z, and discuss the three dimensional manifold of x, y, t. This means that we will consider only positions and motions in a plane. The locus in time of a particle which does not change its position in space, that is, of a particle at rest, will be a straight line parallel to the t axis. Uniform rectilinear motion of a particle will then be represented by a straight line inclined to the t axis.

3. If we adopt the view that uniform motion is only relative, we may with equal right consider the second particle at rest and the first particle in motion. In this case the locus of the second particle must be taken as a new time axis. What corresponding change this will necessitate in our spacial system of reference will depend entirely upon the kind of geometry that we are led to adopt in order to make the geometrical invariants of the transformation correspond to the fundamental physical invariants whose occurrence in mechanics and electromagnetics has led to the principle of relativity.

It is immediately evident that if uniform motion is to be represented by straight lines, the statement that all motion is relative shows that the transformation must be of such a character as to carry straight lines into straight lines. In other words, the transformation must be linear. Further we must assume that the origin of our space and time axes is entirely arbitrary.

The further characteristics of this transformation must be determined by a study of the important physical invariants. Fundamental among these invariants is the velocity of light, which by the second postulate of the principle of relativity must be the same to all observers. Hence any line in our four dimensional manifold which represents motion with the velocity of light must bear the same relation to every set of reference axes. This is a condition which certainly cannot be fulfilled by any transformation of axes to which we are accustomed in real Euclidean space. It is indeed a condition sufficient to determine the properties of that non-Euclidean geometry which we are to investigate.

Minkowski, in his two papers on relativity, used two different methods. In his first and elaborate treatment of the subject he introduced the imaginary unit V-1 in such a way that the lines which represent motion with the velocity of light become the imaginary invariant lines familiar to mathematicians who discuss the real and imaginary geometry of Euclidean space. In this way, however, the points of the manifold which represent a particle in position and time become imaginary; the transformations are imaginary; the whole method becomes chiefly analytical. In his second, a brief paper, Minkowski makes use of certain geometrical constructions which have their simplest interpretation only in a non-Euclidean geometry. 4. It is the purpose of the present work to develop the four dimensional non-Euclidean geometry which is demanded by the principle of relativity, and to show that the laws of electromagnetics and mechanics not only can be simply interpreted in this way but also are for the most part mere theorems in this geometry.

In the first sections we shall develop in some detail the non-Euclidean geometry in two dimensions. For it is only by a thorough comprehension of this simpler case that it is possible to proceed into the more difficult domains involving three and four dimensions. This part of the paper will be continued by a discussion of vectors and the vector notation that will be employed. At this point it is possible in a few simple cases to show the applications of the non-Euclidean geometry to problems in kinematics and mechanics.

The sections devoted to three dimensions will be occupied largely with numerous analytical developments of the vector algebra, many of which are directly applicable not only in space of higher dimensions

2 Gesammelte Abhandlungen von Hermann Minkowski, Vol. 2, pp. 352404 and pp. 431-444.

but also in Euclidean space. We are led further to a consideration of certain vectors of singular character. The study of the singular plane leads to the brief consideration of another interesting and important non-Euclidean plane geometry.

Passing to the general case of four dimensions we shall meet further new types of vectors, and shall attempt even here to facilitate as far as is possible the visualization of the geometrical results. We shall continue further the analytical development, and in particular consider the properties of the differential operator quad. In this connection a very general and important equation for the transformation of integrals is obtained. The idea of the geometric vector field will then be introduced, and the properties of these fields will be taken up in detail.

The subject of electromagnetics and mechanics is prefaced with a short discussion of the possibility of replacing conceptually continuous and discontinuous distributions by one another, and we shall point out that in one important case such a transformation is impossible. The science of electromagnetics is treated both from the point of view of the point charge and from that of the continuous distribution. In both cases it is shown that the field of potential and the field of force are merely the geometrical fields previously mentioned, except for a constant multiplier. Particular attention is given to the field of an accelerated electron,3 and in this field we find that the vectors of singular properties play an important rôle. With the aid of these vectors the problem of electromagnetic energy is discussed. The science of mechanics, which is treated in a fragmentary way in some preceding sections, is now given a more general treatment, and the conservation laws of momentum, mass and energy are shown to be special deductions from a single general law stating the constancy of a certain four dimensional vector, which we have called the vector of extended momentum. Finally it is pointed out that this last vector gives rise to geometric vector fields which can be identified with the

3 There seems to be a widespread impression that the principle of relativity is inadequate to deal with problems involving acceleration. But the essential idea of relativity can be expressed by the statement that there are certain vectors in the geometry of four dimensions which are independent of any arbitrary choice of the axes of space and time. Those problems which involve acceleration will be shown to possess no greater inherent difficulties than those that involve only uniform motion. It is, moreover, especially to be emphasized that the methods which are to be employed in this paper necessitate none of the approximations that are commonly employed in electromagnetic theory. Such terms as "quasi-stationary," for example, will not

be used.

fields of gravitational potential and gravitational force. Moreover, it is shown that these fields are identical in mathematical form with the electromagnetic fields, and that all the equations of the electromagnetic field must be directly applicable to the gravitational.

In an appendix a few rules for the use of Gibbs's dyadics, which have occasionally been employed in the text, are stated. And a brief discussion of some of the mathematical aspects of our plane nonEuclidean geometry is given.

THE NON-EUCLIDEAN GEOMETRY IN TWO DIMENSIONS.

Translation or the Parallel Transformation.

5. In discussing a non-Euclidean geometry various methods of procedure are available; a set of postulates may be laid down, or the differential method of Riemann may be followed, or the theory of groups may be used as by Lie, or (if the geometry falls under the general projective type, as is here the case) the projective measure of length and angle may be made the basis. For our present purpose we need not restrict ourselves to any one of these; but since the first is familiar to all, we shall employ it as far as convenience permits. Some of the other methods will, however, be briefly discussed in the appendix, §§ 64, 65.

With a view to simplicity we shall at first limit the discussion to the case of a plane. Points and lines will be taken as undefined, and most of the relations connecting them will be the same as in Euclidean plane geometry. Thus:

4

1°. Through two points one and only one line can be drawn. 2o. Two lines intersect in one and only one point, except that 3o. Through any point not on a given line one and only one parallel (non-intersecting) line can be drawn.

4°. The line shall be regarded as a continuous array of points in open order.

6. In regard to congruence or "free mobility" it is important to proceed more circumspectly than did Euclid. The transformations of Euclidean geometry may be divided into translations and rotations, of which the former alone are the same for our geometry. It seems desirable, therefore, to discuss first and in some detail the postulates

4 We make no claim of completeness or independence for these postulates, which are designed primarily to show the points of similarity or dissimilarity between our geometry and the Euclidean. A like remark may be made with respect to proofs of theorems.