Gravitation and Light 337 



difference to the cosmos if t is everywhere increased by the same 

 constant: therefore the scale of time must be everywhere the same 

 — which excludes any possibility of local scales of time, A change 

 of origin of measurement for time is not the same as progress of 

 events in time, unless the scale of time is everywhere the same. 



The matter may be put from a different angle as follows. To 

 obtain the time of transit of a ray from P to Q it is not possible 

 to add elements of heterogeneous local times such as 8^*. What 

 can be done is to find the true underlying time of transit. If this 

 homogeneous true time is delayed at the start, at one end of the 

 path at P, it is delayed by an equal amount at arrival at the other 

 end, as the equations of transit do not involve this time explicitly: 

 hence apparent times at the two ends are delayed not by equal 

 amounts, but by amounts inversely as their local scales, so that 

 a ray cannot (as has been impKed) transmit apparent time along 

 its path. 



The alternative development is, as above, that 8ct^ being the 

 underlying unchanging standard there are local scales of time, and 

 local scales of length which may involve direction, and therefore 

 also of velocity (including that of the rays) which is their quotient. 

 The path of a ray from point to point is determined by making 

 the number of wave-lengths from the one to the other minimum, 

 that is by Sjds/X = : but Ss and A are both altered to the same 

 scale; thus there is no alteration due to gravitation in the varia- 

 tional equation determining the ray-path, so that it would suffer 

 no deflection. The essential feature in the argument is that, 

 whether rays may be regarded as the limiting case of free orbits 

 or not, their specification has been postulated so that the ray- 

 velocities correspond in the same way as all other velocities in 

 the two frames. 



Appendix. — On Space and Time. 



Let us try for a closer realization of these abstract positions. 

 The Gauss-E,iemann theory for an ordinary curved surface will be 

 wide enough to serve as an illustration. The theory involves 

 coordinates p, q: they must represent something. The very least 

 we can do for them is to regard the surface as twofold extension 

 dotted over with points, so that the coordinates express their 

 order of arrangement according to some plan of counting them 

 with respect to this extension in which they lie. There is no metric 

 idea at all in this numeration, and nothing to distinguish one 

 surface from another. Now bring in an infinitesimal unchanging 



* Yet it is just such elements of quasi-time d.v^ that are added together, ■infra 

 p. 343. It is the so-called shifting clock-time and absolute time running parallel 

 that are the source of all this confusion. 



23—2 



