364 DOC.
71
PRINCETON
LECTURES
THE GENERAL THEORY
[137]
assumption
of
the
constancy
of
a, is
of
constant
curvature,
being
either
spherical
or
elliptical;
for
then
the
boundary
conditions
at infinity
which
are so
inconvenient
from
the
standpoint
of the
general
theory
of
relativity,
may
be
[138]
replaced
by
the much
more
natural conditions
for
a
closed
space.
According to
what
has
been
said,
we are
to
put
(119) ds2
=
dx
2
-
7
^dx^dx,
in
which the
indices
u
and
v run
from
1
to 3 only.
The
yuv
will be such functions of
x1, x2,
x3 as
correspond to
a
three-dimensional continuum
of
constant
positive
curva-
ture.
We
must
now
investigate
whether such
an assump-
tion
can satisfy
the
field
equations
of
gravitation.
In order
to
be
able
to
investigate this,
we
must
first
find
what differential conditions the three-dimensional
manifold
of
constant curvature
satisfies. A
spherical
manifold
of three
dimensions,
embedded in
a
Euclidean
continuum
of
four dimensions,* is
given by
the
equations
x,2
+
x22
+
X32
+
X42 =
a2
dxi2
+
dx*2
+
dx32
+
dx
42
=
ds2.
By
eliminating
x4, we
get
ds2
=
dXl2
+
dx
22
+
dx
32
+
(x\dxx
+
x2dx*
+
x3dx3)2
a2
~
x32
~
Xi2
-
x32
Neglecting terms
of
the third and
higher
degrees
in the
xv, we can
put,
in
the
neighbourhood
of
the
origin
of
co-ordinates,
ds2
=
dX'dx,.
*The
aid of
a
fourth
space
dimension has
naturally
no
significance except
that of
a
mathematical
artifice.
[104]