140 GRAVITATION— THE EXPLANATION 



We must try to reach the vivid significance which 

 lies behind the obscure phraseology of the law. Suppose 

 that you are ordering a concave mirror for a telescope. 

 In order to obtain what you want you will have to 

 specify two lengths (i) the aperture, and (2) the radius 

 of curvature. These lengths both belong to the mirror — 

 both are necessary to describe the kind of mirror you 

 want to purchase — but they belong to it in different 

 ways. You may order a mirror of 100 foot radius of 

 curvature and yet receive it by parcel post. In a certain 

 sense the 100 foot length travels with the mirror, but 

 it does so in a way outside the cognizance of the postal 

 authorities. The 100 foot length belongs especially to 

 the surface of the mirror, a two-dimensional continuum; 

 space-time is a four-dimensional continuum, and you will 

 see from this analogy that there can be lengths belonging 

 in this way to a chunk of space-time — lengths having 

 nothing to do with the largeness or smallness of the 

 chunk, but none the less part of the specification of the 

 particular sample. Owing to the two extra dimensions 

 there are many more such lengths associated with space- 

 time than with the mirror surface. In particular, there 

 is not only one general radius of spherical curvature, but 

 a radius corresponding to any direction you like to take. 

 For brevity I will call this the "directed radius" of the 

 world. Suppose now that you order a chunk of space- 

 time with a directed radius of 500 trillion miles in one 

 direction and 800 trillion miles in another. Nature 

 replies "No. We do not stock that. We keep a wide 

 range of choice as regards other details of specification; 

 but as regards directed radius we have nothing different 

 in different directions, and in fact all our goods have the 

 one standard radius, x trillion miles." I cannot tell you 

 what number to put for x because that is still a secret 

 of the firm. 



