286 DOC.
25
FOUNDATIONS OF GENERAL THEORY
matter
and the
gravitational
field taken
together.
Then there will have
to
be
a
system
of
equations
of the form
£
a(S°:+
tgv)
=
o
(5)
v
cMrv
where the tov
depend only
on
the
guv
and their derivatives. But there do
not
exist
generally
covariant
systems
of
equations
of
the
type
of
equations (5).
Instead,
closer
examination shows that such
systems are
covariant
only
with
respect
to
linear
transformations.
By demanding
that the field
equations
of
gravitation
be formulated
in such
a
manner
that the
validity
of the
conservation laws finds
expression
in
this
formulation,
we
therefore restrict the choice of the reference
system
in
such
a way
that
only
linear
transformations lead from
one justified
system to
another
one.
9.I
have
explained
several times how the
gravitational equations
with
respect
to
[15]
reference
systems specialized
in
this
way are
to
be found. One asks: What kinds
of
differential
expressions involving
the
guv are
the
$ov
to
be
equated
with in order for
equations
(4)
to
go
over
into
equations (5)
if
I replace
the
$ov
on
the
right-hand
sides of
(4)
with their
expressions
in
terms
of the
guv?
This
question
leads
to
the
differential
equations
[16]
a dy/XV
E
\Fsyaß8,
K(Sov + tov),
(6)
aß/x
°»dxß
where
we
have
set
A
/--{
\""V
^Tp
Tp
1
-2Ktov
£
Y -
£
SavY^--P-i^Yrp^Srp£
[ßrp
ÖXa
ÖXß
laß
rp
ÖXa
ÖXß
Here
Sov
= 1
or 0, depending
on
whether
o
=
v or o
#
v.
It
is
easy
to
show that
these
equations
are
covariant with
respect
to
linear transformations.
It
is
beyond
doubt that there
exists
a
number,
even
if
only
a
small
number,
of
generally
covariant
equations
that
correspond
to
the above
equations,
but their
derivation is
of
no special
interest either from
a
physical
or
from
a
logical point
of
view,
as
the
arguments
presented
in
point
8
clearly
show.
However,
the realization
that
generally
covariant
equations corresponding
to
equations
(6)
must
exist is
important
to
us
in
principle.
Because
only
in that
case was
it
justified
to
demand the
covariance of the
rest
of
the
equations
of the
theory
with
respect
to
arbitrary
substitutions. On the other
hand,
the
question
arises
whether
those other
equations
might
not
undergo specialization owing
to
the
specialization
of the reference
system.
In
general,
this does
not
seem
to
be the
case.
10.
One
can
see
from the
foregoing description
of the foundations of the
theory
that
no
special assumptions
need
to
be
used to
establish
it.
The
reason
why
this
is
otherwise
according
to
the
account
presented recently by
Mie in this
journal
is
[14]