DOC. 25 FIELD
EQUATIONS
OF GRAVITATION
119
or
dT
x
Ejr--EW.
(7a)
X
axx
MV
When
one
multiplies (6) by dgim/dxa and sums
over
i
and
m,
one gets2
because
[p. 846]
of
(7)
and because of the relation
IVje
=
0
2^8im
dx"
~
dxa
that follows from
(3a),
the conservation theorem
of
matter
and
gravitational
field
combined
in the
form
E
-=A-
(Tax
+
f
*)
=
0,
(8)
A
where
tAa
(the
"energy
tensor" of the
gravitational field)
is
given by
E
g^r^va
-
(8a)
^
pvaß
fiva
The
reasons
that motivated
me
to introduce the second
term
on
the
right-hand
sides
of
(2a)
and
(6)
will
only
become
transparent
in
what
follows,
but
they
are
completely
analogous
to those
just quoted
(p.
785).
When
we multiply
(6)
by
gim
and
sum over i
and
m, we
obtain after
a
simple
calculation
E
~
"P1
*0-0,
aß
dxadxß
where, corresponding
to
(5), we
used the abbreviation
(9)
E
8
ptV
=
E
=
'•
(8b)
pa
a
It should
be
noted that
our
additional term is such that the
energy
tensor
of the
gravitational
field
occurs
in
(9) on
footing equal
with the
one
of
matter,
which
was
not the
case
in
equation (21)
l.c.
Furthermore,
one
derives in
place
of
equation
(22)
l.c. and in the
same manner
as
there,
with the
help
of
the
energy equation,
the
relations
2On
the
derivation
see
Sitzungsber.
44
(1915), pp.
784-785.
For the
following
I
ask
the reader also to
consult,
for
a comparison,
the deliberations
given
there
on
p.
785.
[3]