DOC.
13
GENERALIZED THEORY OF RELATIVITY
163
differential
quotients.
If the differential
equation
for
p
were
not
yet known,
the
problem
of
finding
it
would be reduced
to
that of
finding
this
identity.
What
is
essential for
us
to
realize
is
that this
identity
can
be derived
if
one
of the terms
occurring
in it
is
known.
All
one
has
to
do
is to
apply repeatedly
the
product
differentiation rule in the forms
and
a
du dv
dxv
(uv)
=
dxv
+
dxv
dv
u
d
(uv) /
x
- -
du
v,
dx..
dx
dx
V
and then
finally
to
put
the
terms
that
are
differential
quotients
on
the left side and the
rest
of the
terms
on
the
right
side. For
example,
if
one
starts
with the first
term
of
the above
identity,
one
obtains,
one
after
another,
\
E
a
5tp
9(p
=
E
3cp
.
32cp
+
E
9(p
_
52cp
a*v
fa
3*
dx
M
dxß dxvdx/j
/j- v
3cp
n
32cp
3
/
5(p
\2
E
+
dx
dx
dx
V
\ /V
from which
we
obtain
the
above
identity upon rearrangement.
Now
we
turn
again
to
our problem.
It follows from
equation (10)
that
|e^-
0
(a
=
1,2,3,4)
^
/IV
dx.
is
the
momentum
(or
energy) imparted by
the
gravitational
field
to
the
matter
per
unit
volume. For the
energy-momentum
law
to
be
satisfied,
the differential
expressions
Fuv of the fundamental
quantities yuv
that
enter
the
gravitational equations
K'0,,v
=
rMV
must be
chosen such that
dgMv
r
2K
dxa
/XV
can
be rewritten in such
a
way
that it
appears
as
the
sum
of differential
quotients.
On the other
hand,
we
know that the
term
(a) appears
in the
expression sought
for
Tuv.
Hence the
identity
that
is
being sought
has the
following
form:
Sum of differential
quotients
\
iE^
E
d
'
Yabl)xß)
/XV3y
^
/IV
dxo
aß
dxa
+
the other
terms,
which vanish with the first
approximation.}