DOC.
71
PRINCETON LECTURES 359
THE GENERAL THEORY
universe,
however
large they
may
be,
are
quasi-Euclidean,
is
a
wholly
different
question.
We
can
make
this
clear
by
using
an
example
from the
theory
of
surfaces
which
we
have
employed many
times.
If
a
certain
portion
of
a
surface
is practically plane,
it
does
not at
all follow
that
the
whole surface has
the form
of
a
plane;
the surface
might
just
as
well be
a
sphere
of
sufficiently
large
radius.
The
question
as
to
whether the universe
as a
whole
is
non-
Euclidean
was
much
discussed
from the
geometrical point
of view before
the
development
of
the
theory
of
relativity.
But with the
theory
of
relativity,
this
problem
has
entered
upon
a new
stage,
for
according to
this
theory
the
geo-
metrical
properties
of bodies
are
not
independent,
but
depend upon
the distribution
of
masses.
If the universe
were
quasi-Euclidean;
then Mach
was
wholly wrong
in his
thought
that
inertia,
as
well
as
gravita-
tion,
depends upon
a
kind
of
mutual
action between
bodies.
For in
this
case,
for
a
suitably
selected
system
of
co-
ordinates,
the
g",
would be
constant at
infinity, as
they
are
in the
special
theory
of
relativity,
while
within
finite
regions
the
gß,
would differ
from these
constant
values
by
small
amounts
only,
for
a
suitable
choice
co-ordinates,
as a
result
of
the influence
of the
masses
in finite
regions.
The
physical
properties
of
space
would
not
then
be
wholly
independent,
that
is,
uninfluenced
by
matter,
but
in
the
main
they
would
be,
and
only
in
small
measure
condi-
tioned
by matter.
Such
a
dualistic
conception is
even
in
[128]
itself
not satisfactory;
there
are, however,
some
important
physical arguments against it,
which
we
shall
consider.
The
hypothesis
that the universe
is
infinite and Euclidean
at
infinity, is,
from
the relativistic
point
of
view,
a com-
plicated
hypothesis.
In the
language
of the
general
theory
of
relativity
it demands that the Riemann
tensor
of
[99]