DOC.
71
PRINCETON LECTURES 367
THE GENERAL THEORY
From
this follows
(123)
er
a
=
2/ka
If
the universe
is quasi-Euclidean,
and
its
radius of
curvature
therefore
infinite,
then
a
would
vanish. But it
is improbable
that the
mean
density
of
matter
in
the
universe
is
actually zero;
this
is
our
third
argument against
the
assumption
that the universe
is quasi-Euclidean.
Nor
does
it
seem
possible
that
our
hypothetical pressure
can
vanish;
the
physical
nature
of
this
pressure
can
be
appreci-
ated
only
after
we
have
a
better theoretical
knowledge
of
the
electromagnetic
field.
According to
the
second
of
equations (123)
the
radius,
a,
of
the
universe
is
determined
[143]
in
terms
of the total
mass,
M,
of
matter,
by
the
equation
(124)
MK
a = -.
4~.2
The
complete dependence
of
the
geometrical upon
the
physical properties
becomes
clearly apparent by
means
of
this
equation.
Thus
we
may present
the
following
arguments against
[144]
the
conception
of
a
space-infinite,
and
for
the
conception
of
a
space-bounded,
or
closed,
universe:-
1.
From the
standpoint
of the
theory
of
relativity, to postu-
late
a
closed
universe
is
very
much
simpler
than
to postulate
the
corresponding boundary
condition
at infinity
of
the
quasi-Euclidean
structure
of
the
universe.
2.
The idea that Mach
expressed,
that inertia
depends
upon
the mutual action
of
bodies,
is
contained,
to
a
first
approximation,
in the
equations
of the
theory
of
relativity;
[107]