104 DOC. 21 GENERAL RELATIVITY
s{f(2
-
kEmv gmv
Tmv)}dr,
Tap
Tta
8
=
E
maß
(17)
where the
guv
have
to
be varied while the
Tmv
are
to
be
treated
as
constants. The
reason
is that
(17)
is
equivalent
to
the
equations
Es/sx
deßV
\uSa
/
88
dg”
=
-
K
T
fiv’
(18)
where
8
has
to
be
thought
of
as
a
function
of
the
gMv
and the
dg fiv
8x"
(= guvo).
On
the
other
hand, a lengthy
but
uncomplicated
calculation
yields
the relations
88
dgßV
Er“
pP
l/jß1
vor
aß
88
a¡r
(19)
(19a)
These
together
with
(18)
provide
the field
equations
(16a).
It
can now
also be
easily
shown that the
principle
of
the conservation of
energy
and momentum is satisfied.
Multiplying
(18) by
g£mv
with summation
over
the
indices
p
and
v, one
obtains after
customary rearrangement
Emv
s/sxa
88)
fcrj
8
8
dx"
=
-
k
y T
iiv
iiv
«T-.
According
to
(14), on
the other
hand,
for the
total
energy
tensor
of
matter
one
has
8gMV j
Y
dxx
~
2£
dxa
^
From the last
two
equations
follows
E
4-(To
+
Ó
=
0- (20)
a.
dxx
where
taA
-
1/2k(2sta)
E*r
88
'
(20a)
[p.
785]
denotes the
"energy
tensor" of the
gravitational
field
which, by
the
way,
has tensorial