270 DOC. 148 S P A C E - T I M E 610 SPACE-TIME 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 arbitary to a certain degree, namely, in that substitutions of the co-ordinates of the form x' = x vt y '-y z ' = Z also appear possible. On the other hand, earlier mechanics did 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 favoured- 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 four- 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 happens 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- dimensional continuum seemed to resolve itself into the three- dimensional continuum of space and the one-dimensional con- tinuum 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 electro- dynamics. Two infinitely near points can be connected by means of a light-signal if the relation ds* = c*dt*—dx*—dy*—dz* *= o 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 ds2 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 clement of the “light-cone” ds2 = o which starts out from every point. If we consider only elements which belong to the same time-value, we have ds*—dx*+dy*-f dz* These elements ds may have real counterparts in distances at rest and, as before, Euclidean geometry holds for these elements. Effect of Relativity, Special and General.— This is the modifica- tion which the doctrine of space and time has undergone through the restricted theory of relativity. The doctrine of space has been still further modified by the general theory of relativity, because this theory denies that the three-dimensional spatial section of the space-time continuum is Euclidean in character. Therefore it asserts that Euclidean geometry does not hold for the relative positions of bodies that are continuously in contact. 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 ds2 assumes the general form:— ds* ^ ^ g M rdxHdx, where the gμν's arc functions of the co-ordinates and where the sum is to be taken over the indices for all combinations 11, 12, . . . 44. The variability of the gμν's is equivalent to the existence of a gravitational field. If the gravitational field is sufficiently general it is not possible at all to find an inertial system, that is, a co-ordinate system with reference to which ds2 may be expressed in the simple form given above:— ds*—c*dt*—dx*—dy*—dz* 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. Riemann's Geometry.— Riemann’s geometry of an n-dimension- al space bears the same relation to Euclidean geometry of an n- dimensional space as the general geometry of curved surfaces bears to the geometry of the plane. For the infinitesimal neigh- bourhood of a point on a curved surface there is a local co- ordinate system in which the distance ds between two infinitely near points is given by the equation ds*-dx*+dy* For any arbitrary (Gaussian) co-ordinate-system, however, an expression of the form ds*=gndx*+ 2g,idx1dx,+g22dx2* holds in a finite region of the curved surface. If the gμν's are given as functions of x1 and x2 the surface is then fully deter- mined geometrically. For from this formula we can calculate for every combination of two infinitely near points on the surface the length ds of the minute rod connecting them and with the help of this formula all networks that can be constructed on the surface with these little rods can be calculated. In particular, the curvature” at every point of the surface can be calculated this is the quantity that expresses to what extent and in what way the laws regulating the positions of the minute rods in the immediate vicinity of the point under consideration deviate from those of the geometry of the plane. This theory of surfaces by Gauss has been extended by Riemann to continua of any arbitrary number of dimensions and has thus paved the way for the general theory of relativity. For it was shown above that corresponding to two infinitely near space-time points there is a number ds which can be obtained by measure- ment with rigid measuring-rods and clocks (in the case of time- like elements, indeed, with a clock alone). This quantity occurs in the mathematical theory in place of the length of the minute rods in three-dimensional geometry. The curves for which ∫ds has stationary values determine the paths of material points and rays of light in the gravitational field, and the “curvature” of space is dependent on the matter distributed over space. Just as in Euclidean geometry the space-concept refers to the position-possibilities of rigid bodies, so in the general theory of relativity the space-time-concept refers to the behaviour of rigid bodies and clocks. But the space-time-continuum differs from the space-continuum in that the laws regulating the behaviour of these objects (clocks and measuring-rods) depend on where they happen to be. The continuum (or the quantities that describe it) enters explicitly into the laws of nature, and con- versely these properties of the continuum are determined by physical factors. The relations that connect space and time can no longer be kept distinct from physics proper. [12] [13] [14] [15] [16] [17]
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