8
DOC.
2
COVARIANCE PROPERTIES
[8]
(1)
Er
dIav/dxv
=
1/2
Euv
dguv/dxa
Yup
Ipv.
If
one
introduces in
place
of the
energy
tensor of the
gravitational
field the
"complex
of
the
energy-density
of the
gravitational field,"
that
is,
the
quantities
(2)
tor
j/
~g
•
yafltuv
~9
'
then the
"Outline"
equations
(14)
and
(13) respectively yield
(2a)
-
2xtay
=y-g(^Yßyd-^dJll
-
i^darya^^iyPat(jpa
pqt a P
where
8av
=
0
or
1
depending
on
a
f
v or a
=
v.
In
place
of
the
gravitational equations
(21)
and
(18), respectively,
of the
"Outline"
we now
get
the
equations
`c.,0
_
(II)
~,axa(Y7apYoiz
x(t~,
+
t0;).
In
a manner
analogous
to the
one
used in
§5
of the "Outline"
one can now get
from
(I)
and
(II)
the
general
conservation
theorems,
which
can
take the form
(III)
-
°-
§2.
Remarks
on
the Choice
of the
Coordinate
System
We
want to show
now
that,
completely independent
of
the
gravitational
equations
we
established,
a
complete
determination
of
the
fundamental
tensor
Yuv
of
a
gravitational
field with
given
©uv
by a
generally-covariant system
of
equations
is
impossible.
[p.
218]
We
can
prove
that
if
a
solution for the
yuv
for
given
©uv
is
already
known,
then
[9]
the
general
covariance
of
the
equations
allows
for the existence
of
further solutions.
Assume
a
domain L within
our
four-dimensional manifold such that
no
"material
process"
shall exist within
L, i.e.,
where the
©uv
therefore vanish.
By
virtue
of
the
given
©uv
the
yuv
are
assumed
determined
everywhere
outside
of L
and, therefore,
also inside L
(assumption
a).
Instead of the
original
coordinates
xv
we now imagine new
coordinates
Xv'
introduced in the
following
manner.
Everywhere
outside
of
L
we
have
xv' =
xv,
but inside L at least for
part
of it and
at
least for
one
index let there be
Xv'
=
xy.
[10]
Obviously,
at
least for
part
of
L,
this substitution achieves
y'uv
=
yuv
.
On
the
other hand
we
have
©uv'
=
©uv
everywhere,
that
is,
outside
of
L,
because for this
[11]
domain
xv' =
xv,
and inside of
L
because for this domain
©'uv
=
0
=
©uv.