DOC.
71
PRINCETON LECTURES 355
THE GENERAL
THEORY
x'4
=
X4
X'a
=
F(r)Xa
we can always
insure
that
one
of these
three functions
shall be
an
assigned
function
of
r'. In
place
of
(110)
we
can
therefore
put,
without
limiting
the
generality,
(110a)
yaB
=
5aB
+
\xaxB.
In
this
way
the
guv
are
expressed
in
terms
of
the
two
quantities
X
and
ƒ.
These
are
to
be
determined
as
func-
tions of
r,
by introducing
them into
equation
(96),
after
first
calculating
the
Tauv
from
(109)
and
(110a).
We have
[119]
roaB
-
X.
X'x0-*3
+
2\rôag
r
'
1
+
Xr2
(for
a, ß,
r =
1,
2,
3)
(110b)
{
r444 =
Tig
=Ttg=0
(for
a,i3
=
1,
2,
3)
¿J
oxa
4
a
rA44 =
dxB
[120]
{5}
With the
help
of these
results,
the
field
equations
furnish
[121]
Schwarzschild’s
solution:
(109a)
ds2=(1+a/r)dl2
f dr2
r
+
r2(sin2
Qdÿ1
+
dO2)
in which
we
have
put
x4 =
l
X1 =
r
sin 0 sin
(o
X2
=
r
sin
6 cos
6
x3
=
r
cos
e
_
kM
A
47T
(109b)
[95]
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