354 DOC. 71 PRINCETON LECTURES
THE
GENERAL THEORY
This
is
the
place to
speak
of
the third
consequence
of
the
theory
which
can
be tested
by
observation,
namely,
that which
concerns
the motion
of
the
perihelion
of the
planet Mercury.
The secular
changes
in the
planetary
orbits
are
known with such
accuracy
that the
approxima-
tion
we
have been
using
is
no
longer
sufficient for
a com-
parison
of
theory
and observation. It
is
necessary
to
go
back
to
the
general
field
equations
(96).
To
solve this
problem
I
made
use
of
the method of
successive
approxi-
mations.
Since
then,
however,
the
problem
of the central
symmetrical
statical
gravitational
field has been
completely
[115]
solved
by
Schwarzschild and
others;
the
derivation
given
[116]
by
H.
Weyl
in his
book,
“Raum-Zeit-Materie,” is
particu-
larly elegant.
The calculation
can
be
simplified
somewhat
if
we
do
not
go
back
directly to
the
equation
(96),
but
base
it
upon
a
principle
of variation that
is
equivalent to
this
equation.
I shall
indicate the
procedure only
in
so
far
as
is
necessary
for
understanding
the method.
[117]
In the
case
of
a
statical
field, ds2
must
have the
form
(
ds2
=
-
da2 +f2dx42
I
da2
=
2) 7aßdxadxB
\
1-3
where the summation
on
the
right-hand
side
of the last
equation is
to
be
extended
over
the
space
variables
only,
The central
symmetry
of
the
field
requires
the
yuv
to
be
of the
form,
(110)
y
aß
=
usaß
+
\xaxB
[118] ƒ2,
U
and
X are
functions
of
r
=
\x12 +
x22
+
x32
only.
One of
these
three functions
can
be chosen
arbitrarily,
because
our system
of co-ordinates
is,
a
priori,
completely
arbitrary;
for
by
a
substitution
(109)
[114]
[94]