104 Br. L. Silberstein on General Relativity 



Passing from a system u to any other, u', we shall have 



(ififcX)' = ™ (giKgtix'gik'gfiK). The left-hand members being 



properly developed, (5) are ultimately partial differential 

 eqs. for the gij. It is still to be proved that our above tensor 

 components (2) do actually satisfy all these differential 

 equations. This will be shown in the next section. 



To conclude the present one, notice that by (1) the velocity 

 of light, given by ds = 0, becomes, in natural coordinates, 



dX 



v =Tt^ 



that is, constant and independent of direction, throughout 

 the natural space (vacuum) of curvature -^- 2 , whatever the 



value or the sign of the latter. The " rays" of light will be 

 straight, shortest, lines in that particular space. Remember, 

 however, that it is only a space, among many others at your 

 disposal. In view of the above property we can call it visual 

 or optical space. If then, by convention, we desire to choose 

 ms our reference space that among an infinity of others which 

 has the above property, then there is certainly no objection 

 to calling it simply space as a short for " optical space."" 

 And since all more remote objects are explored by optical 

 means, such a choice will manifestly be by far the most 

 convenient one. If R 2 <0, so that 



R sin 75 = | R | sinh .-^ „ 

 Jx \±i\ 



then the optical space will be hyperbolic or Lobatchewskyan, 

 i. e. infinite but showing a defect in the angle sum of a 

 triangle proportional to its area, and so on ; if, in spite of 

 the negative value of the invariant R 2 , somebody would 

 prefer to use Euclidean geometry, there would be nothing to 

 present him doing so ; only in that case his optics will not be 

 so simple. Similarly, mutatis mutandis, for k = or &>0. 

 To put it in a few words : The world is so or so (to be 

 explored), while space — even with the requirement of homo- 

 geneity — is, in very wide limits, a matter of convention, 

 much as was predicted \ ears ago by Poincare. 



Positive constant world-curvature is a feature of Einstein's 1917- 

 theory ; of course, only in absence of gravitation, and with the unavoid- 

 able cooperation of a certain hypothetical u world -matter." An inter- 

 esting' modification of Einstein's newest theory due to de Sitter will be 



