without the Equivalence Hypothesis. 101 



order repeatedly called world-curvature, we shall claim for 

 our world a constant curvature. This will henceforth be 

 denoted by 



k = 1/E 2 . 



Not pretending to know, or to be able to decide a priori 

 what its sign or value might be, we shall leave them 

 undetermined. 



If k is positive, R is a real length, and if negative, then iR is a real 

 length. If |_R|:=oo, the world is homaloidal. Notice that, whatever 

 the results of future observations, they can not lead to the conclusion 

 that the world is strictly homaloidal but can give only a lower limit 

 of | R |, say 10 9 astr. units or more ; this under the assumption that the 

 results of observation will be nil-effects, as in the case of Einstein's shift 

 and of all the aether-drift observations and experiments. It may, how- 

 ever, happen that some observations will point to a lower and an upper 

 limit of | R | together with a definite sign of R 2 . Then, whatever the 

 actual sign of R 2 , the result will be a very positive and an interesting 

 one. It may run thus, for instance : R 2 < 0, and 10 8 < | R | < 10 10 , 

 stating that the world has a negative curvature and fixing its amount 

 between two, more or less narrow limits. I must warn the reader, how- 

 ever, that if he lives long enough to hear of such a result, he must not 

 say that " the three-space " is negatively curved or hyperbolic of cur- 

 vature k== — 10~ 18 astr. un.~ 2 , but only that the four-manifold or the 

 world is so. In fact, if such be the world, he can choose in it a space * 

 just of the curvature k=— 10~ 18 (but not below it), as well as a linear 

 infinity of hyperbolic spaces, the homaloidal and all positively curved 

 spaces without upper limit. This freedom of conventional or opportunist 

 choice, limited only at the lower end by the invariant k, is based upon 

 a remarkable and very general theorem on manifolds of any number of 

 dimensions proved 33 years ago by Killing t and in part before him by 

 Beltrami, which may shortly be rendered thus : — 



Every w-dimensional space of constant curvature contains in itself 

 spherical space forms (Kugelgebilde) of less dimensions (v) whose 

 Riemannian curvatures form a continuous manifold having no maximum 

 but a minimum, viz. equal to the curvature of the w-space, this minimum 

 curvature belonging to the i/-dimensional plane. 



For our case it is enough to put in this admirable theorem w = 4 and 

 p=3. After this lengthy but (in view of certain recent misunderstand- 

 ings) not altogether needless digression, let us return to our subject. 



Having assumed a homogeneous world we have eo ipso 



accepted one of constant curvature, k= ™* (This being at 



any rate a necessary condition, it will be still incumbent to 

 show that the line-element to be written down presently 

 leads also to all other constant invariants, — which task may 

 be postponed to another opportunity.) Now, to obtain the 



* And a homogeneous one, or hypersphere of three dimensions. 



f W. Killing, Die Nicht-Euklidischen Raumformen, Leipzig, 1885, 

 pp. 79-83. This excellent old book will be helpful to every student of 

 general relativity. 



