144 GRAVITATION— THE EXPLANATION 



This explanation introduces no new hypothesis. In 

 saying that a material system of standard specification 

 always occupies a constant fraction of the directed radius 

 of the region where it is, we are simply reiterating 

 Einstein's law of gravitation — stating it in the inverse 

 form. Leaving aside for the moment the question 

 whether this behaviour of the rod is to be expected or 

 not, the law of gravitation assures us that that is the 

 behaviour. To see the force of the explanation we 

 must, however, realise the relativity of extension. Exten- 

 sion which is not relative to something in the surround- 

 ings has no meaning. Imagine yourself alone in the 

 midst of nothingness, and then try to tell me how large 

 you are. The definiteness of extension of the standard 

 rod can only be a definiteness of its ratio to some other 

 extension. But we are speaking now of the extension 

 of a rod placed in empty space, so that every standard 

 of reference has been removed except extensions be- 

 longing to and implied by the metric of the region. It 

 follows that one such extension must appear from our 

 measurements to be constant everywhere (homogeneous 

 and isotropic) on account of its constant relation to what 

 we have accepted as the unit of length. 



We approached the problem from the point of view 

 that the actual world with its ten vanishing coefficients 

 of curvature (or its isotropic directed curvature) has a 

 specialisation which requires explanation; we were then 

 comparing it in our minds with a world suggested by 

 the pure mathematician which has entirely arbitrary 

 curvature. But the fact is that a world of arbitrary 

 curvature is a sheer impossibility. If not the directed 

 radius, then some other directed length derivable from 

 the metric, is bound to be homogeneous and isotropic. 

 In applying the ideas of the pure mathematician we 



