DOC.
30
FOUNDATION OF GENERAL RELATIVITY
185
required general
field
equations of gravitation
in
mixed form.
Working
back from
these,
we
have in
place
of
(47)
TL
+
T^aTya
=
-
tf(TM"
-
ig^T!),.
J
"9
=
1
(53)
It
must be
admitted that this introduction of
the
energy-
tensor of
matter is not
justified by
the
relativity postulate
alone.
For this
reason
we
have
here deduced
it from the
requirement
that the
energy
of
the
gravitational
field
shall
act
gravitatively
in
the
same way as any
other kind
of
energy.
But the
strongest
reason
for
the
choice of
these
equations
lies in
their
consequence,
that
the
equations
of
conservation
of
momentum and
energy,
corresponding
exactly
to
equations
(49)
and
(49a),
hold
good
for
the
components
of
the total
energy.
This
will
be shown in
§
17.
§
17.
The Laws
of
Conservation in the General Case
Equation
(52)
may
readily
be transformed
so
that the
second
term
on
the
right-hand
side
vanishes. Contract
(52)
with
respect
to
the
indices
u
and
a,
and after
multiplying
the
resulting equation
by
1/2Sau,
subtract it
from
equation
(52).
This
gives
^irßKß
-
=
-
C
+
TZ).
(52a)
On
this
equation
we
perform
the
operation
d/dav.
We have
2
~bxj)x,
g'T
I
-
i
7)xJ)x,
9aA
)Xß
^Xft 7Xx.
The first
and
third
terms of
the
round brackets
yield
con-
tributions which
cancel
one
another,
as
may
be
seen
by
interchanging,
in
the
contribution
of the third term, the
summation
indices
a
and
a
on
the
one
hand,
and ß and
\
on
the
other. The
second term
may
be re-modelled
by
(31),
so
that
we
have
a2
(g'ßKß)
=
i
gaß
'bXn'dx,
fi..
(54)
The
second term
on
the left-hand
side of
(52a) yields
in
the