196
DOC.
16
FOUNDATIONS OF GRAVITATION
(3).
We
are
therefore
looking
for
equations
that
express
the
equality
of
two tensors,
one
of which is the
given
tensor 3uv,
while the other
comes
from the fundamental
tensor
guv
through
differential
operations.
[9]
It has
now
turned
out
that the conservation laws
of
momentum and
energy
make
possible
the derivation of these
equations.
It has
already
been
emphasized
above that
the material
process
alone
cannot
satisfy
the conservation
laws;
but
we
must
demand
that the conservation laws be
satisfied for the material
process
and the
gravitational
field
together. According
to
the
arguments
presented above,
this
means
that there
must
exist four
equations
of the form
(4)
£
JL
($av
+ tov)
=
0. (0
=
1,2,3,4)
v
°XV
Here the tov characterize the
stress-energy
components
of the
gravitational
field in
a manner
analogous
to
the
way
in which the
quantities
$ov
characterize
those of the
material
process.
In
particular,
the
quantities
Sov
and
tov
must have the
same
invariant-theoretical character. It turned out
to
be
possible
to
show
by means
of
a
general
argument
that the
equations
that
completely
determine the
gravitational
field
[10] cannot
be covariant with
respect
to
arbitrary
substitutions. This fundamental
discovery
is
especially noteworthy
because all other
physical
equations,
such
as, e.g., equations
(2), possess general
covariance. In accordance with this
general result,
the
postulated
equations
(4) are
also covariant
only
with
respect
to
linear
substitutions,
but
are
not
[11]
so
with
respect
to
arbitrary
substitutions.
Hence,
we
will have
to
demand covariance
only
with
respect
to
linear transformations from the
gravitation equations
that
we
are
seeking.
It has turned
out that
one
is led
to
completely
determined
equations
if
one
adds
to
these considerations the demand that when
these
equations
are
applied
to
the
relevant
special
case
and
an approximate
solution is
sought, they
must
yield
Poisson's
[12] equation (3).
Using
the
way
indicated,
one
obtains the
following equations:
(5)
(6)
(V
9
7a
ß
ff
Oft Qßp ^
~
X
v
-f-
tn
v)
;
(O,
v =
1,2,3,4
Here
v=;
(.-fe),
k
is
a
universal constant that
corresponds
to the
gravitational constants;
6ov
is
1
or
0,
depending on
whether
o
and
v are
different
or equal.
One
can see
from the
system
of
equations (5),
which
corresponds
to
equation (3),
that
along
with the
stress-energy components
3ov
of the material
process,
those
of
the
gravitational
field
(namely, tov) appear
as an
equivalent field-inducing cause,
a
circumstance
that
obviously
must be
demanded;
for the
gravitational
effect
of
a
system
may
not
depend on
the
physical
nature
of
the
system's field-producing energy.