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
energydensity
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
=yg(^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
generallycovariant 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
fourdimensional 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.