160
DOC.
13
GENERALIZED THEORY OF RELATIVITY
What differential
equations permit
us
to
determine the
quantities
gik,
i.e.,
the
gravitational
field? In other
words,
we
seek the
generalization
of Poisson's
equation
Acp
=
4rkp.
We have
not
found
a
method for the solution of this
problem
as
thoroughly
compelling as
that for the solution of the
problem
discussed
previously.
It
would be
necessary
to
introduce several
assumptions
whose
correctness
seems
plausible
but
not
evident.
The
generalization
that
we
seek would
likely
have the form
(11)
k-Ouv
= ruv,
where
K
is
a
constant
and
Tuv
a
second-rank contravariant
tensor
derived from the
fundamental
tensor
guv by
differentiatial
operations.
In line with the Newton-Poisson
law
one
would be inclined
to
require
that these
equations (11)
be second order. But
it must
be
stressed
that,
given
this
assumption,
it
proves impossible
to
find
a
differential
expression Fuv
that
is
a
generalization
of
Ap
and that
proves
to
be
a
tensor
with
respect
to
arbitrary transformations.10 To be
sure,
it
cannot
be
negated
a priori
that the
final,
exact
equations
of
gravitation
could be of
higher
than second
order.
Therefore there still exists the
possibility
that the
perfectly
exact
differential
equations
of
gravitation
could
be
covariant with
respect
to arbitrary
substitutions.
But
given
the
present
state
of
our
knowledge
of the
physical properties
of the
gravitational
field,
the
attempt
to
discuss such
possibilities
would
be
premature.
For
that
reason we
have
to
confine ourselves
to
the second
order,
and
we
must
therefore
forgo setting up gravitational equations
that
are
covariant with
respect
to
arbitrary
transformations.
Besides,
it
should be
emphasized
that
we
have
no
basis
whatsoever
for
assuming a
general
covariance of the
gravitational
equations.11
The
Laplacian
scalar
Ap
is obtained from the scalar
p
if
one
forms the
expansion (the gradient)
of the latter and then the inner
operator (the divergence)
of
this.
Both
operations
can
be
generalized
in such
a
way
that
one
can
carry
them
out
on
every
tensor
of
arbitrarily high rank, namely
while
permitting arbitrary
substitu-
tions of the basic
variables.12
But these
operations degenerate
if
they
are
carried
out
on
the fundamental
tensor
guv.13
From this
it
seems
to
follow
that the
equations
sought
will be covariant
only
with
respect
to
a
particular
group
of
transformations,
which
group,
however, is
as
yet
unknown
to
us.
10Cf.
Part
II,
§4,
No.
2.
11Cf.
also the
arguments given
at
the
beginning
of
§6.
12Part II,
§2.
13Cf.
the remark
on
p.
28
in
Part
II,
§2.
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