DOC.
13
GENERALIZED THEORY OF RELATIVITY
159
J
i- ^
dxv
dxA
V
E
-
^a
-E«.
fa"
v
ruv Z^*4v ^ ^
•E djTuv
dx^
dxv
\xv
dxx
ds ds
K.K
©uv
ßV
= Po
ru
ds ds
We
note
that
[21]
is
a
second-rank contravariant
tensor
with
respect
to
arbitrary
substitutions. From the
foregoing
one
surmises that the
momentum-energy
law will have the form
(10)
\
E
f~S'^©"v
=
0
•
(a- 1234)
[22]
jiv
Z
The first three of these
equations (o
=
1,2,3)
express
the
momentum
law,
and the
last
one
(a
=
4)
the
energy
law.
It
turns out
that these
equations
are
in fact covariant
with
respect
to
arbitrary
substitutions.9
Also,
the
equations
of motion of the material
point
from which
we
started
out
can
be rederived from these
equations
by integrating
over
the thread of flow.
We
call
the
tensor
0uv
the
(contravariant) stress-energy
tensor of the
material
flow.
We ascribe
to
equation (10)
a
validity range
that
goes
far
beyond
the
special
case
of the
flow
of incoherent
masses.
The
equation represents
in
general
the
energy
[23]
balance between the
gravitational
field and
an
arbitrary
material
process;
one
has
only
to
substitute for
0uv
the
stress-energy
tensor
corresponding
to
the material
system
under consideration. The first
sum
in
the
equation
contains the
space
derivatives of
the
stresses
or
of the
density
of the
energy
flow,
and the time derivatives of the
momentum
density
or
of the
energy
density;
the second
sum
is
an expression
for the
effects exerted
by
the
gravitational
field
on
the material
process.
§5.
The
Differential
Equations
of the Gravitational
Field
Having
established the
momentum-energy
equation
for
material
processes
(mechani-
cal, electrical,
and other
processes)
in
relation
to
the
gravitational
field,
there remains
for
us
only
the
following
task. Let the
tensor
0uv
for the material
process
be
given.
9Cf.
Part
II, §4,
No.
1.