286 RELATIVITY, THE GENERAL THEORY(a proposal which again has nineteenth century origins), where p is a uniform
density, then the solution0==_l
is dynamically acceptable.
Is it also physically acceptable? Constant p means an isotropic, homogeneous
universe. In 1917 the universe was supposed to consist of our galaxy and presum-
ably a void beyond. The Andromeda nebula had not yet been certified to lie
beyond the Milky Way. Today an individual galaxy is considered as a local dis-
turbance of a distribution which is indeed isotropic and homogeneous, to a degree
which itself demands explanation [SI3]. Einstein had no such physical grounds
for assuming these two properties—except for the fact that, he believed, they led
to the first realization of the relativity of inertia in the model he was about to
unveil. That this model is of the static variety is natural for its time. In 1917 no
large-scale galactic motions were yet known to exist.
Let us return to the transition from Eq. 15.17 to Eq. 15.18. There are three
main points in Einstein's paper. First, he performs the very same transition in
general relativity, that is, he replaces
by
(15.21)
Second, he constructs a solution of Eq. 15.21 that resolves the conundrum of the
Newtonian infinite. Third, he proposes a dynamic realization of the relativity of
inertia. His solution, the Einsteinian universe, had to be abolished in later years.
It will nevertheless be remembered as the first serious proposal for a novel topol-
ogy of the world at large. Let us see how he came to it.
Einstein had applied Eq. 15.20 with great success to the motion of planets,
assuming that far away from their orbits the metric is flat. Now he argued that
there are two reasons why this boundary condition is unsatisfactory for the uni-
verse at large. First, the old problem of the Newtonian infinite remains. Second—
and here Mach enters—the flatness condition implies that 'the inertia [of a body]
is influenced by matter (at finite distances) but not determined by it [his italics]
If only a single mass point existed it would have inertia ... [but] in a consistent
relativity theory there cannot be inertia relative to "space" but only inertia of
masses relative to each other.' Thus Einstein began to give concrete form to
Mach's ideas: since the g^ determine the inertial action, they should, in turn, be
completely determined by the mass distribution in the universe. He saw no way
of using Eq. 15.20 and meeting this desideratum. Equation 15.21, on the other
hand, did provide the answer, it seemed to him,* in terms of the following solution
(i,k =1,2, 3):
*He also noted that this equation preserves the conservation laws, since gme = 0.
(15.20)