APPENDIX IITHE CURVATURES OF SPACE-TIME
IT may be of interest to explain the significance of Einstein’s gravitational equations more fully. We will restrict our attention to the equations of the general theory in the case of an infinite universe. These equations may be written:Here it must be noticed thatandare letters for which the four numbers representing the four dimensions of space-time must be substituted in pairs. There exist, therefore, a number of gravitational equations—sixteen in all. When we discard those which are mere repetitions of the others, hence which yield us no new information, we find that this number reduces to ten.
Now we have already a general idea of the significance of these equations. The left-hand side gives us the curvatures of space-time from point to point as measured in our mesh-system, and the right-hand side gives us the various aspects of matter, energy and momentum situated at the same points where the curvature is being calculated. If we agree to select the nearest possible approach to a Galilean mesh-system, and for reasons of simplicity agree to treat it as a Galilean system, we are able to write out the components of the energy-tensor of matter in this mesh-system, and as a result we obtain:This gives us all the equations, since the six we have omitted to write out, namely,are mere repetitions of
We have thus written out the ten equations. In them,represents the density of the matter at the point considered;,,,its component velocities at the point considered, and,,,etc., the strains and stresses existing in its interior. If we are dealing with non-coherent matter, we may ignore these strains and stresses.
We see that the curvatures of space-time are connected with the various conditions of the matter. If we heat it, we increase its energy, its molecules vibrate faster, and the curvatures of space-time, hence the field of gravitation, are affected correspondingly. We may also note that whereas the density of mass of the matter at a point, namely,,affects the curvature,the momentum components of the matter,,affect the curvatures,,,and itsvis vivacomponents,,affect the curvatures;;.Whereas one type of curvature is produced by mass, another is produced by momentum, and still another by energy.
Of course, these mathematical equations merely express relationships; and it is impossible to deduce from them alone whether it is mass, momentum andvis vivawhich produce the respective space-time curvatures, or whether it is these space-time curvatures which arise in some mysterious way and are interpreted by our senses as mass, momentum andvis viva. Eddington, as we know, prefers the second attitude, whereas the majority of thinkers prefer the first.
It is to be noted that if we adopt Eddington’s views, a velocity (say,), being equivalent to,appears as a ratio of two of the space-time curvatures; hence the co-presence of two special types of space-time curvature would reveal itself to our consciousness as a velocity. Inasmuch as a velocity implies the passage of time, it would seem as though this mysterious passage might be connected in some way with certain of the space-time curvatures. It would be but a step to assume that our entire perceptual world might eventually be reduced to these curvatures. We do not insist on this aspect of the question, first, because we are not certain that they correspond to Eddington’s views, and, secondly, because the theory of relativity has, thus far at least, been unable to interpret electromagnetic phenomena in terms of space-time curvatures.
But there is an interesting idea that Eddington suggests in his book, “Space, Time and Gravitation.” He argues that we are now in a position to understand why it is that velocity is always relative, that is to say, is meaningless otherwise than in relationship to matter. The fact is that the curvatures of space-time, such as,or,connoteandrespectively,and not simply.The presence of these curvatures connotes, therefore, matter together with velocity, and no curvature taken by itself defines bare velocity without matter.
Acceleration, on the other hand, is connected with the metrical’s of space-time; and these, of course, subsist regardless of the presence or absence of matter, since they represent aspects of the structure of absolute space-time itself. We thus understand how it comes that acceleration manifests itself as absolute. Mach’s views lead us to ascribe the existence of these’s of acceleration not to space-time itself, but to the totality of the matter of the universe. Under this aspect the-distribution acquires the likeness of a species of ether conditioned by matter, which in turn conditions the behaviour of bodies in space and the evolution of phenomena in time. It should be emphasised, however, that rash philosophical conclusions on these and kindred subjects should be avoided. In the present state of our knowledge, even the most competent of scientists are far from having succeeded in solving all these arduous problems.