24 SECTIONAL ADDRESSES. 



by the following concrete problem : if all matter were annihilated except 

 one particle which is to be used as a test-body, would this particle have 

 inertia or not ? The view of Mach and Einstein is that it would not ; and 

 in support of this view it may be urged that, according to the deductions 

 of general relativity, the inertia of a body is increased when it is in the 

 neighbourhood 'of other large masses ; it seems needless, therefore, to 

 postulate other sources of inertia, and simplest to suppose that all inertia 

 is due to the presence of other masses. When we confront this hypothesis 

 with the facts of observation, however, it seems clear that the masses of 

 whose existence we know — the solar systems, stars, and nebula? — are 

 insufficient to confer on terrestrial bodies the inertia which they actually 

 possess ; and therefore if Mach's principle were adopted, it would be 

 necessary to postulate the existence of enormous quantities of matter in 

 the universe which have not been detected by astronomical observation, 

 and which are called into being simply in order to account for inertia in 

 other bodies. This is, after all, no better than regarding some part of 

 inertia as intrinsic. 



Under the influence of Mach's doctrine, Einstein made an important 

 modification of the field-equations of gravitation. He now objected to 

 his original equations of 1915 on the ground that they possessed a solution 

 even when the universe was supposed void of matter, and he added a 

 term — the ' cosmological term ' as it is called — with the idea of making 

 such a solution impossible. After a time it was found that the new term 

 did not do what it had been intended to do, for the modified field-equations 

 still possessed a solution — the celebrated ' De Sitter World ' — even when 

 no matter was present ; but the De Sitter World was found to be so 

 excellent an addition to the theory that it was adopted permanently, and 

 with it of course the cosmological term in the field-equations ; so that 

 this term has been retained for exactly the opposite reason to that for 

 which it was originally introduced. 



The ' De Sitter World ' is simply the universe as it would be if all minor 

 irregularities were smoothed out : just as when we say that the earth is a 

 spheroid, we mean that the earth would be a spheroid if all mountains were 

 levelled and valleys filled up. In the case of the De Sitter universe the 

 levelling is a more formidable operation, since we have to smooth out the 

 earth, the sun, and all the heavenly bodies, and reduce the world to a 

 complete uniformity. But after all, only a very small fraction of the cosmos 

 is occupied by material bodies ; and it is interesting to inquire what 

 space-time as a whole is like when we simply ignore them. 



The answer is, as we should expect, that it is a manifold of constant 

 curvature. This means that it is isotropic (i.e. the Riemann curvature is 

 the same for all orientations at the same point), and is also homogeneous. 

 As a matter of fact, there is a well-known theorem that any manifold 

 which is isotropic in this sense is necessarily also homogeneous, so that 

 the two properties are connected. A manifold of constant curvature is 

 a projective manifold, i.e. ordinary projective geometry is valid in it 

 when we regard geodesies as straight lines ; and it is possible to move 

 about in it any system of points, discrete or continuous, rigidly, i.e. so 

 that the mutual distances are unaltered. 



The simplest example of a manifold of constant curvature is the 



