166
DOC.
13
GENERALIZED THEORY OF RELATIVITY
(19)
£
+
ümv)}
=
0
=
1,2,3,4)
fJiV V
This
shows
that
the
conservation laws hold
for
the matter
and
the
gravitational
field
taken
together.
In the
foregoing
we
have
given preference
to
the contravariant
tensors,
because
the contravariant
stress-energy
tensor
of the flow of incoherent
masses can
be
expressed
in
an
especially simple
manner.
However,
we can
express
the fundamental
relations that
we
have obtained
just
as
simply
by
using
covariant
tensors.
Instead of
0uv,
we
must
then take
Tuv
=
^
g^gvß®aß
as
the
stress-energy
tensor
of the
aß
material
process.
Instead of
equation (10), we
obtain
through term-by-term
reformulation
(20)
Es-(^V,)
*
-
o-
It follows from this
equation
and
equation
(16)
that the
equations
of the
gravitational
field
can
also be written
in
the form
(21)D
~D^(g)
-
K(t)iv
+
7v);
these
equations
can
also
be
derived
directly
from
(18).
The
equation
that
corresponds
to
(19)
reads
(22)
*'"))
=0-
V
UXV
§6.
Influence of the
Gravitational
Field
on
Physical Processes,
Especially on
the
Electromagnetic
Processes
Since
momentum
and
energy play a
role in
every physical process
and,
for their
part,
also determine the
gravitational
field and
are
influenced
by
it,
the
quantities
guv
that
determine the
gravitational
field
must
appear
in
all
systems
of
physical equations.
Thus,
we
have
seen
that the motion of the material
point
is
determined
by
the
equation
S|/fc}
-
0,
[34]
[35]