NON-EMPTY SPACE 153 



Non-Empty Space. The law that the directed radius is 

 constant does not apply to space which is not completely 

 empty. There is no longer any reason to expect it to 

 hold. The statement that the region is not empty means 

 that it has other characteristics besides metric, and the 

 metre rod can then find other lengths besides curvatures 

 to measure itself against. Referring to the earlier (suf- 

 ficiently approximate) expression of the law, the ten 

 principal coefficients of curvature are zero in empty 

 space but have non-zero values in non-empty space. It 

 is therefore natural to use these coefficients as a measure 

 of the fullness of space. 



One of the coefficients corresponds to mass (or 

 energy) and in most practical cases it outweighs the 

 others in importance. The old definition of mass as 

 "quantity of matter" associates it with a fullness of 

 space. Three other coefficients make up the momentum 

 — a directed quantity with three independent com- 

 ponents. The remaining six coefficients of principal 

 curvature make up the stress or pressure-system. Mass, 

 momentum and stress accordingly represent the non- 

 emptiness of a region in so far as it is able to disturb 

 the usual surveying apparatus with which we explore 

 space — clocks, scales, light-rays, etc. It should be 

 added, however, that this is a summary description and 

 not a full account of the non-emptiness, because we 

 have other exploring apparatus — magnets, electroscopes, 

 etc. — which provide further details. It is usually con- 

 sidered that when we use these we are exploring not 

 space, but a field in space. The distinction thus created 

 is a rather artificial one which is unlikely to be accepted 

 permanently. It would seem that the results of ex- 

 ploring the world with a measuring scale and a magnetic 

 compass respectively ought to be welded together into 



