284 DOC. 25 FOUNDATIONS OF GENERAL THEORY
[8]
to
arbitrary
transformations. It is
owing
to
this fact that the construction of the
theory
can go on
without
arbitrariness
despite
the
occurrence
of the
guv.
6.
But the
new
theory
of
relativity
also has
to solve
a
problem
to
which
none
corresponds
in the
original theory
of
relativity.
That is
to
say,
it also has
to
yield
equations
that the
gravitational
field itself
satisfies,
i.e., equations
from which the
quantities
guv
are
to
be calculated when the
quantities referring
to
the material
processes
are
known. The
energetic
behavior of
a
system
is
characterized
by
the
$
energy
tensor
Iov/v-g
(mixed tensor).
Since
energy
and inertia
on
the
one
hand,
and
f-g
inertia and
gravitation on
the other
hand,
determine
one
another,
we
must
demand
that the
gravitational
field
be determined
by
the
quantities
$ov.
Thus,
we are
to
seek
differential
equations
that
are
to
be considered
a generalization
of
Poisson's
equation
and
which, therefore,
permit
the
calculation
of
the guv
from the $ov; these
equations
must
be
generally
covariant.
7.
We have
not
succeeded in
setting up
this relation between the
guv
and Xov
[9]
in
a generally
covariant
form;
and it is this circumstance that
my colleagues
believe
[10] they
can use
to set
up a
lethal
trap
for
our theory.
In what
follows,
I shall
explain
why they
are,
in
my opinion,
wrong.-
If
we are
given equations connecting any quantities
whatsoever2 that
are
valid
only
for
a special
choice of
the coordinate
system,
then
one
has
to
distinguish
between
two
cases:
1.
To these
equations
there
correspond generally
covariant
equations, i.e.,
equations
valid with
respect
to
arbitrary
reference
systems.
2.
There
are no
generally
covariant
equations
that
can
be
deduced
from the
equations given
for
the
specially
chosen reference
system.
In
case 2,
the
equations say
nothing
about the
things
described
by
the
quantities
in
question; they only
restrict the choice of the reference
system.
If the
equations say
anything
at
all about the
things represented by
the
quantities,
then
we are
dealing
with
case
1,
i.e.,
in that
case,
there
always
exist
generally
covariant
equations
between the
quantities.
Thus,
if,
without
knowing
the
generally
covariant
equations
of the
gravitational
field,
we
specialize
the reference
system
and
set
up
the field
equations
of
gravitation
for the
special
reference
system only,
then the sole
objection
that
can
be raised
against
the
theory
is that the
equations
we
have
set
up might, perhaps,
be void
of
any
physical
content.
But
no
one
is
likely
to
think
in earnest
that this
objection
is
justified
in
the
present
case.
2Of
course,
the
transformation
properties
of
the
quantities
themselves must be
considered
here
as being
given
for
arbitrary
transformations.