SECTIONAL TRANSACTIONS.—A. 447 
were all about. But I think I still prefer the general relativity way of 
doing the sums. 
Two qualifications must be appended to this result. First, it holds, as 
stated, onlyifA =o. If A +0, we get a scattering of galaxies dependent 
on the value of \, and if we take the simplest case and neglect the gravita- 
tional interaction of the matter, we get de Sitter’s universe. If further we 
make A > 0, we find that the velocity of scattering > o also. So this kind 
of scattering is physically different from Milne’s. 
Second, I suspect Prof. Milne might object to my saying that his calcula- 
tion neglects gravitation. For he would contend, if I understand him 
aright, that large-scale gravitation is just such as to reproduce this particular 
kind of scattering found by him, and no other. I would answer that 
I believe it possible to show that his theory certainly does not take account 
of the detailed interaction of pairs of particles. As regards the large-scale 
effect it is, as a matter of fact, contrary to the strict spirit of general relativity 
to’ distinguish kinematical and gravitational effects, for the method of 
general relativity is to turn all dynamics into kinematics. Consequently, 
his objection would be quite consonant with a proper relativistic outlook. 
Nevertheless, I think that the language we have been employing does help 
in the physical understanding of the phenomena. 
This latter point seems to me largely one of epistemology. The point of 
real importance is, what happens to Milne’s theory when the detailed 
gravitational interaction of the particles is allowed for. Actually general 
relativity provides what is, within its own field of postulates, a complete 
solution of the problem which Milne sets out to solve, taking full account of 
gravitation. For, in starting with the invariance of the element of interval, 
general relativity ensures the constancy of the velocity of light. Thus the 
first of Milne’s requirements ts satisfied. 
Then, further, the expanding universes of general relativity are based on 
the requirements of isotropy and homogeneity. These ensure that an 
observer at rest in the spatial co-ordinates sees the world as isotropic, i.e. 
as spherically symmetrical about himself, and that all such observers have 
the same view of the world at the same cosmic instant.? If, then, these 
observers be taken as Milne’s fundamental class, they satisfy precisely the 
requirement he makes that each should have exactly the same sequence of world 
views. All the general relativity solutions satisfying these requirements are 
known.4 
It is important to recall, as I was recently reminded by a conversation 
with M. l’Abbé Lemaitre and Dr. McVittie, that these solutions are 
independent, in the first place, of Einstein’s law of gravitation, and are 
derived only from the invariance of the element of interval. The law of 
gravitation merely gives the value of the density-momentum tensor asso- 
ciated with each solution. A different law of gravitation might lead to 
a different value of the tensor associated with any particular universe, but 
it could not alter the whole class of universes satisfying the fundamental 
requirements. 
If, then, Prof. Milne ultimately arrives at conclusions different from 
those of general relativity, it must be either by denying the invariance of 
the element of interval, or else by something equivalent to using the same 
8 Milne in his original paper, Zeit. fiiy A strophysik, 6 (1933), I-99, criticised the 
concept of cosmic time, but it is pointed out by Kermack and McCrea, and by 
Robertson (/.c.), that cosmic time is implicit in his own theory. There it appears 
as the proper time of any one of his fundamental particlessince its passage through 
the space-time origin. 
4 H. P. Robertson, Proc. Nat. Acad. 
