The theory which is sketched in the following pages forms the most wide-going generalization conceivable of what is at present known as "the theory of Relativity;" this latter theory I differentiate from the former "Special Relativity theory," and suppose it to be known. The generalization of the Relativity theory has been made much easier through the form given to the special Relativity theory by Minkowski, which mathematician was the first to recognize clearly the formal equivalence of the space like and time-like co-ordinates, and who made use of it in the building up of the theory. The mathematical apparatus useful for the general relativity theory, lay already complete in the "Absolute Differential Calculus", which were based on the researches of , and on the non-Euclidean manifold, and which have been shaped into a system by and , and already applied to the problems of theoretical physics. I have in part B of this communication developed in the simplest and clearest manner, all the supposed mathematical auxiliaries, not known to Physicists, which will be useful for our purpose, so that, a study of the mathematical literature is not necessary for an understanding of this paper. Finally in this place I thank my friend , by whose help I was not only spared the study of the mathematical literature pertinent to this subject, but who also aided me in the researches on the field equations of gravitation.
A. Principal considerations about the Postulate of Relativity.
§ 1. Remarks on the Special Relativity Theory.
The special relativity theory rests on the following postulate which also holds valid for the ian mechanics.
If a co-ordinate system K be so chosen that when referred to it the physical laws hold in their simplest forms, these laws would be also valid when referred to another system of co-ordinates K′ which is subjected to an uniform translational motion relative to K. We call this postulate "The Special Relativity Principle". By the word special, it is signified that the principle is limited to the case, when K′ has uniform translatory motion with reference to K, but the equivalence of K and K′ does not extend to the case of non-uniform motion of K′ relative to K.
The Special Relativity Theory does not differ from the classical mechanics through the assumption of this postulate, but only through the postulate of the constancy of light-velocity in vacuum which, when combined with the special relativity postulate, gives in a well-known way, the relativity of synchronism as well as the Lorentz transformation, with all the relations between moving rigid bodies and clocks.
The modification which the theory of space and time has undergone through the special relativity theory, is indeed a profound one, but a weightier point remains untouched. According to the special relativity theory, the theorems of geometry are to be looked upon as the laws about any possible relative positions of solid bodies at rest, and more generally the theorems of kinematics, as theorems which describe the relation between measurable bodies and clocks. Consider two material points of a solid body at rest; then according to these conceptions there corresponds to these points a wholly definite extent of length, independent of kind, position, orientation and time of the body.
Similarly let us consider two positions of the pointers of a clock which is at rest with reference to a co-ordinate system; then to these positions, there always corresponds a time-interval of a definite length, independent of time and place. It would be soon shown that the general relativity theory can not hold fast to this simple physical significance of space and time.
§ 2. About the reasons which explain the extension of the relativity-postulate.