DOC. 30 FOUNDATION OF GENERAL RELATIVITY
183
It
is
to
be noticed
that
tao
is
not
a
tensor;
on
the other
hand
(49)
applies
to all
systems
of
co-ordinates for which
*/-
g
=
1.
This
equation expresses
the law
of
conservation
of momentum
and
of
energy
for
the
gravitational
field.
Actually
the
integration
of
this
equation over
a
three-
dimensional
volume
V
yields
the four
equations
A
j=
J(^
+
mtl
+
ȣ)iS.
.
(49a)
where
l,
m, n
denote the
direction-cosines of
direction
of
the
inward drawn normal at the element
dS of
the
bounding
sur-
face
(in
the
sense
of
Euclidean
geometry).
We
recognize
in
this the
expression
of
the
laws of
conservation in their usual
form.
The
quantities
tao
we
call
the
"energy components"
of
the
gravitational field.
I
will
now give
equations
(47)
in
a
third
form,
which
is
particularly
useful
for
a
vivid
grasp
of
our subject.
By
multiplication
of
the
field
equations
(47)
by
gvo
these
are
ob-
tained in the
"mixed"
form.
Note that
=,
jLf0»rj*
\
_
dy'p«,
9
öz. 7x\9
1
which
quantity, by
reason
of
(34),
is
equal
to
7XI
"Ö
{g"rI^y)
/
_-r*l \
~
ATtff TV*
~
9°ßTßa.r*y. -wmOL
a,
or
(with
different
symbols
for
the summation
indices)
~dx.
(g*ßKß)
-
-
9"KßTt.
The third term
of
this
expression
cancels
with the
one
aris-
ing
from
the
second
term
of
the
field
equations
(47); using
relation
(50),
the
second term
may
be written
*(£
-
$K0,
where
t
=
taa.
Thus instead
of
equations
(47) we
obtain
(g^Kß)
-
-
*(C
-
~
9
..
(51)