DEC 10 1920 THE CONNECTICUT ACADEMY VOLUME 23] SEPTEMBER 1920 [PAGES 383-416 The Principle of General CONNECTICUT ACADEMY OF ARTS AND SCIENCES YALE UNIVERSITY PRESS THE PRINCIPLE OF GENERAL RELATIVITY AND EINSTEIN'S THEORY OF GRAVITATION Einstein's theory of gravitation has presented difficulties to many readers on account of the complicated analysis involved. The object of this paper is to present the mathematical part of the theory in as simple a form as possible, and at the same time to translate Einstein's tensor analysis into a notation more closely resembling the vector analysis of Gibbs, in the hope of making this very remarkable theory more readily intelligible to the average physicist. (a) THE PRINCIPLE OF GENERAL RELATIVITY. A moving element is defined as a point-whether in a material body or at the head of a light signal-which can be continuously identified. Every physical measurement, when analyzed, is found to consist of observations of coincidences of two or more moving elements. Thus the measurement of temperature by means of the mercury thermometer consists in noting a coincidence between the top of the mercury column and a certain division on the scale beside it, or, more strictly, a coincidence between the light rays coming to the observer's eye from these two points. A physical phenomenon, then, may be defined as a coincidence in time and space of two or more moving elements. In order to specify the place and time of a physical phenomenon, it is necessary to have a set of reference elements each of which locates a point in space, and with each of which is associated a device. for assigning numerical values to the times of occurrence of successive events at that point. A reference system is an assemblage of such reference points filling all space for all time. A physical phenomenon is specified relative to a given reference system by the identification of the reference point at which it happened, and the time at that point of the occurrence. The determination of a physical law involves the measurement of space and time intervals between a number of phenomena. Consequently a device is needed for measuring the distance be 385 tween two points in a reference system, and synchronizing time at these points. In order to be of value in comparing coincidences, this device must be subject to the following conditions, though otherwise its nature is quite arbitrary: (a) Two points which are in synchronism with a third are also in synchronism with each other. (b) The distance between two points is independent of the time at which it is measured. Evidently the motion of a moving element may be described relative to an infinite number of reference systems, which are themselves in various states of motion relative to one another. However, these systems are not, in general, of the same significance. For let A, B and C be three systems from which the motion of the moving element P may be observed. Suppose it is found that the motion of P relative to A is conditioned by that of B, but is independent of that of C. In such case the motion of P is said to be related to B, which is known as a related reference system. C, on the other hand, is an unrelated or ideal reference system. Thus for the motion of a shot, the gun from which it is fired constitutes a related reference system. The velocity of a sound wave is determined, not by the motion of the source, but by the characteristics of the medium through which it passes. Hence in this case, the source determines an ideal reference system, while the medium is a related one. In the case of propagation of a disturbance through empty space it has been recognized for a long time that the source does not constitute a related reference system. The Michelson-Morley experiment has shown the same to be true of the "ether." Hence it may be inferred that For the motion of an effect which travels through empty space, such as a light wave or one of the moving elements which form an electromagnetic or a gravitational field, there is no related reference system. If this statement be granted, the question arises as to how the law connecting a group of related physical phenomena may differ when determined in terms of the space and time intervals of different reference systems. Evidently, If a law governing physical phenomena which are conditioned solely by those effects which travel through empty space, is deter |