DOC. 30 FOUNDATION OF GENERAL RELATIVITY
181
mathematics,
from the
requirement
of
the
general theory
of
relativity,
give
us,
in combination with the
equations
of
motion
(46),
to
a
first
approximation
Newton's law of
at-
traction,
and
to
a
second
approximation
the
explanation
of
the
motion
of
the
perihelion of
the
planet Mercury
discovered
by
Leverrier
(as
it remains after corrections
for
perturbation
have
been
made).
These
facts must,
in
my
opinion,
be
taken
as a convincing proof
of
the correctness
of
the
theory.
§
15.
The Hamiltonian
Function
for the Gravitational
Field. Laws
of
Momentum
and
Energy
To show
that the field
equations
correspond
to
the
laws of
momentum and
energy,
it
is most
convenient
to
write them
in the
following
Hamiltonian
form
:-
sffiMr
=
0
iEL-jrr^it}.
.
.
(47a)
J
-
9
=
1
where,
on
the
boundary
of the finite four-dimensional
region
of
integration
which
we
have
in
view,
the variations
vanish.
We
first
have
to
show
that the
form
(47a)
is
equivalent
to
the
equations (47).
For this
purpose
we
regard
H
as
a
function
of
the
guv
and
the guvo
(=aguv/axo).
Then in the
first
place
SH
=
+
2rr;ß
sit.
=
-
r;,rl
sr
+
21^ s(ri£).
But
«(Wt.)
-
-
+
lr
-
g^)}.
[22]
The
terms
arising
from the last
two terms
in round brackets
are
of different
sign,
and
result
from each
other
(since
the
de-
nomination
of
the
summation
indices is
immaterial) through
interchange
of
the indices
u and
ß. They
cancel each
other
in
the
expression
for
8H,
because
they
are
multiplied
by
the