13
289
Doc. 26
on
"Outline of
a
Generalized
Theory
of
Relativity
and of
a
Theory
of Gravitation"
by
A. Einstein
[Zeitschrift
für
Mathematik und
Physik
62
(1914):
260-261]
Regarding §5
and
§6.
When
we
were writing
this
paper
we
felt it
a
deficiency
of the
theory
that
we
did
not succeed
in
constructing equations
for the
gravitational
field
that
are
generally covariant, i.e.,
covariant with
respect
to
arbitrary
substitutions.
Subsequently
I found
out, however,
that
equations
that determine the
yuv univocally
from the
0uv
and
are generally
covariant
cannot
exist
at
all;
the
proof
of
this
is
obtained
as
follows.
Suppose
that
our
four-dimensional manifold contains
a
part
L
in which
no
"material
process"
takes
place,
in
which, therefore,
the
0uv
vanish.
According
to
our
assumption,
the
yuv
are
completely
determined
everywhere,
and thus also in the
interior of
L,
by
the
0uv
given
outside of
L.
Let
us
now imagine
that
new
coordinates
x'v
of
the
following
kind
are
of
the
original
coordinates
xv.
Outside
of
L,
let
us
have
everywhere
xv
=
xv';
but inside L let
us
have
xv
=
xv'
at
least for
a
part
of
L
and
at
least for
one
index
v.
It is
clear that
by
means
of
such
a
substitution
one can
obtain
y'uv
#
yuv
for
at
least
a part
of
L.
On the other
hand,
one
will
have
O'uv
=
0uv
everywhere, namely,
outside
L
because for this
region
x'v
=
xv,
but inside
L because for this
region
Ouv =
0
= 0'uv.
From this it follows
that in the
case
under
consideration,
where
all
substitution
are
as
justified,
more
than
one
system
of the
yuv
is associated with the
same
system
of
the
0uv.
Thus,
if
one
sticks
to
the
demand-as
has been done in
our
paper-that the
yuv
should be
completely
determined
by
the
0uv,
then
one
is forced to restrict the choice
of the reference
system.
In
our paper we
realized this restriction
by postulating
the
validity
of
the conservation
laws, i.e.,
the
validity
of
four
equations
of
the form
of
equations (19),
for the material
process
and the
gravitational
field taken
together.
This
is
in fact the
postulate
from which
we
derived
equations
(18)
for the
gravitational
field in
§5.
Equations (19)
are
covariant
only
with
respect
to
linear
transformations,
so
that
in the
theory developed
in
our
paper only
linear
transformations
are
to
be considered
justified. Hence,
we can
designate
the
axes
of such
systems
as
"straight
lines,"
and
the coordinate surfaces
as "planes."
It is
very noteworthy
that the conservation laws
[1]
[2]
[3]
[4]
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