446 SECTIONAL TRANSACTIONS—A. 
to fathom the general relativity theory!’ It is, however, my purpose to-day 
to show that it is fortunate that things turned out as they did. For I believe 
that general relativity provides the best mathematical method for dealing 
with Milne’s phenomenon, and I am thankful that he had patience to wait 
till this method had been discovered before giving us his own theory, which 
otherwise might have been taken to be the last word. In short, general 
relativity was all the time studying the simple phenomenon contemplated 
by Milne, but somehow never realised the fact! Only general relativity 
goes somewhat further, and takes account of the influence of gravitation, 
and also of the influence of the cosmical constant—that is, if Mrs. Harris 
really does interfere. 
The results I am about to summarise were arrived at in collaboration 
with Dr. Kermack of Edinburgh.1.We discovered afterwards that 
Prof. H. P. Robertson? of Princeton had simultaneously arrived at 
identical conclusions. 
Milne has stated his postulates. He wants a universe in which every 
observer, or every one of a particular class of observers, sees the same 
sequence of world-pictures. To fix the ideas we take the simplest solution 
of his problem, which is provided by his ‘ hydrodynamic’ solution. This 
represents a set of particles all setting out from a single point with uniform 
velocities relative to each other. If the velocities are distributed from zero 
up to the velocity of light, in the particular manner calculated by Milne, then 
each observer will see himself for all time as the centre of the whole bunch. 
Any observer at any instant will describe the universe by associating with 
each point the velocity and density of particles found by Milne. 
Now, in arriving at these results, Milne chooses to map his events in 
a four-dimensional Minkowski space. He is quite entitled to do this. 
For his space-time is introduced from a different standpoint from space- 
time in general relativity. He chooses his space-time first, and then seeks 
the law of gravitation for the actual universe which will reproduce the 
observed state of affairs. General relativity, on the other hand, puts the 
law of gravitation first, and then seeks the form of space-time for the actual 
universe, again so that the observed state of affairs is reproduced. ‘These 
two distinct methods of approach must always be possible. 
Now there is one case in which the space-time of general relativity reduces 
to the space of Milne’s map. That is when we neglect the effect of gravity. 
In general relativity, however, the result of neglecting gravity is equivalent 
to flattening out curved space-time. But the general relativity theory of 
the expanding universe admits a whole class of curved spaces. What we 
can show then is that, 7f we choose the right one of these expanding universes, 
with \ =o, and flatten it out, i.e. neglect the gravitational interaction of the 
particles, then we get exactly Milne’s universe described above. We get 
identically the same velocity and identically the same density associated with 
a given point at a given time. So this particular general relativity solution 
enjoys all the virtues of Milne’s solution, in particular its simplicity and its 
explanation of why the universe appears to be expanding and not contracting. 
This is very important, for, as I have already mentioned, it means that the 
mysterious expansion of the universe is nothing more or less than the simple 
kinematical scattering of Milne’s theory, allowing, of course, in the general 
case for the influence on the scattering of the gravitational interaction of the 
particles. General relativity has, so to speak, invented a way of doing sums, 
but had to wait for Prof. Milne to come along and tell us what the sums 
1 W. O. Kermack and W. H. McCrea, M.N., R.A.S., 98, 519-529 (1933). 7 
2 H. P. Robertson, Zeit. fiir A strophysik, 6, in press (1933). 
