244
DOC.
41
HAMILTON'S PRINCIPLE
the value
of
the
integral
in
equation
(2)
extended
over
the
boundary
does
not
change
during
the transformation. We therefore have
and
thus6
A
(F) =
0
A
{|
OJrfr}
=
A
{|
©rfr}.
But the left-hand side
of
the
equation
must
vanish since both R/-g and
/-gdr
are
[10]
{6}
invariants.
Consequently,
the
right-hand
side
vanishes
also.
Due
to
(13), (14),
and
(15) we
next
get
the
equation
d2Ax
/
8
/XV
dr
=
0.
(16)
pa
dxv
dxa
Rearranging
after twofold
partial integration,
and
considering
the free choice of
the
Axa, one
has the
identity
a2
dR
uv
dxv
dxa
8
=
0.
(17)
[11]
{7}
We
now
have
to
draw conclusions from the two identities
(15)
and
(17),
which
follow from the invariance of
R/-g
,
i.e.,
from the
postulate
of
general relativity.
The field
equations (7)
of
gravitation
can
be transformed first
by
mixed
multiplication
with guv. One obtains then
(also exchanging
the indices
o
and
v) as
an
equivalent
of
the field
equations
(7)
the
equations
3
(dR*
guv
=
-(«;
+
o,
dx
pa
(18)
where
we put
SI
=
-
dm
8 /IV
dg
(19)
t:
=
(d®
8a
Mv
+
d®
8
/XV
I
_
1
ll®8;
J-
*
d®
8a
^
{d8r
a§
po
2
^8
pa
v
(20)
The
latter
expression
for
tva
is
justified by (14)
and
(15).
After differentiation
of
(18)
with
respect
to
xn
and summation
over v,
with consideration
of
(17),
follows
6Introducing
R
and
R
instead
of
b
and
b*.