In applying Euclidean geometry to pre-relativistic mechanics a further indeterminateness enters through the choice of the co ordinate system : the state of motion of the co-ordinate system is arbitrary to a certain degree, namely, in that substitutions of the co-ordinates of the form x'=x—vt not allow co-ordinate systems to be applied of which the states of motion were different from those expressed in these equations. In this sense we speak of "inertial systems." In these f avoured inertial systems we are confronted with a new property of space so far as geometrical relations are concerned. Regarded more accurately, this is not a property of space alone but of the f our dimensional continuum consisting of time and space conjointly.
Appearance of Time.—At this point time enters explicitly into our discussion for the first time. In their applications space (place) and time always occur together. Every event that hap pens in the world is determined by the space-co-ordinates x, y, z, and the time-co-ordinate t. Thus the physical description was four-dimensional right from the beginning. But this four-dimen sional continuum seemed to resolve itself into the three-dimen sional continuum of space and the one-dimensional continuum of time. This apparent resolution owed its origin to the illusion that the meaning of the concept "simultaneity" is self-evident, and this illusion arises from the fact that we receive news of near events almost instantaneously owing to the agency of light.
This faith in the absolute significance of simultaneity was destroyed by the law regulating the propagation of light in empty space or, respectively, by the Maxwell-Lorentz electrodynamics. Two infinitely near points can be connected by means of a light signal if the relation = holds for them. It further follows that ds has a value which, for arbitrarily chosen infinitely near space-time points, is independ ent of the particular inertial system selected. In agreement with this we find that for passing from one inertial system to another, linear equations of transformation hold which do not in general leave the time-values of the events unchanged. It thus became manifest that the four-dimensional continuum of space cannot be split up into a time-continuum and a space-continuum except in an arbitrary way. This invariant quantity ds may be measured by means of measuring-rods and clocks.
Four-dimensional Geometry.—On the invariant ds a four dimensional geometry may be built up which is in a large measure analogous to Euclidean geometry in three dimensions. In this
way physics becomes a sort of statics in a four-dimensional con tinuum. Apart from the difference in the number of dimensions the latter continuum is distinguished from that of Euclidean geometry in that may be greater or less than zero. Corre sponding to this we differentiate between time-like and space-like line-elements. The boundary between them is marked out by the element of the "light-cone" which starts out from every point. If we consider only elements which belong to the same time-value, we have These elements ds may have real counterparts in distances at rest and, as before, Euclidean geometry holds for these elements.
For the empirical law of the equality of inertial and gravita tional mass led us to interpret the state of the continuum, in so far as it manifests itself with reference to a non-inertial system, as a gravitational field and to treat non-inertial systems as equivalent to inertial systems. Referred to such a system, which is connected with the inertial system by a non-linear transforma tion of the co-ordinates, the metrical invariant assumes the general form : sum is to be taken over the indices for all combinations r 1, 12, • • • 44. The variability of the g,p's is equivalent to the exist ence of a gravitational field. If the gravitational field is suffi ciently general it is not possible at all to find an inertial system, that is, a co-ordinate system with reference to which may be expressed in the simple form given above: but in this case, too, there is in the infinitesimal neighbourhood of a space-time point a local system of reference for which the last-mentioned simple form for ds holds. This state of the facts leads to a type of geometry which Riemann's genius created more than half a century before the advent of the general theory of relativity of which Riemann divined the high importance for physics.