100 Dr. L. Silberstein on General Relativity 



that body, the Gaussian curvature within the strip will differ from zero* 

 while outside it will remain nil, the neighbourhood of the solar strip 

 being bent and possibly strained somehow but remaining developable 

 upon a plane, as is a piece of a cylinder, say, or of a cone*. The geodesic 

 lines of the surface, and therefore also the eqs. of motion in the vicinity 

 of the strip, will then be changed correspondingly. 



Now, what I propose is to emancipate the fundamental 

 tensor, at least outside ordinary matter, from the influence 

 of gravitation (as well as of any other agents), in spite of the 

 well-known exceptional properties of gravitational fields. 

 In other words, I propose to reject the gravitational 

 "equivalence hypothesis,'"' but to retain the postulate of 

 genera] covariance of physical laws. 



But here, at the very outset, a fundamental question pre- 

 sents itself. If the coefficients of the invariant line-element 

 ds 2 =gijdxidxj are not manufactured or moulded by gravi- 

 tating bodies, what does determine them physically ? What 

 determines the values of those tensor components, if in 

 different cases they were to be essentially different, i.e. not 

 reducible to one another by mere transformations of coordi- 

 nates ? A radical answer to this question easily suggests 

 itself, and is already announced bv having emphasized the 

 "if." It is this: 



Let the fundamental tensor gij be not different in different 

 physical circumstances hut always, under all circumstances 

 (at least in vacuo) essentially the same. In other words, let 

 us assume that ds 2 is, in vacuo, throughout the world essen- 

 tially the same quadratic form, or that it is always possible 

 to choose such coordinates x±, x 2 , x s , x± — ct in which ds 2 

 acquires a certain standard form, no matter whether suns or 

 galaxies are near at hand or very remote. This amounts to 

 postulating homogeneity of the four-manifold, which — in 

 view of the principle of causality, in its heuristic aspect — 

 seems to be a perfectly sound requirement. 



Now, our world, as any multi-dimensional manifold, has 

 a host of invariants, the differential invariants of various 

 orders of its line-element. Thus, if the world is to be 

 homogeneous (always in vacuo, at least), clearly all of its 

 invariants must each have throughout one and the same 

 numerical value ; and since one of tlipm (and even pro- 

 minent amongst them) is the differential invariant of 2nd 



* The idea of reducing physical phenomena to changes of curvature, 

 especially in connexion with particles of matter, is not altogether new. 

 It was suggested nearly fifty years ago by Clifford, with the only 

 difference that Clifford had no opportunity of associating the time- 

 coordinate with the remaining three. Of. Clifford's ' Math. Papers,' 

 p. 21, and his ' Common Sense of Exact Sciences,' 5th ed. p. 224 et seq. 



