258
DOC.
10
RESEARCH
NOTES
[p.
48]
Divergenz
eines
Ebenentensors 0ik
Der Ausdruck
^aß^%xß'\
1
d
(
dgaß^Ya
dX;
(^'£
dx£
dxa
j
2
dxa
l^'e
dxz
dXj
[122]
[eq.
189]
verschwindet für das
System
[eq.
190]
g
-1 0
wy
0
-1
-wx
-y -wx
1
-
w2
(x2
+
y2)
obiger
Ausdruck liefert:
[123]
[eq.
192]
^^aß
d
l'y.f£
^^aß'j
^
^
dg
*
v.
*
-
1
+
w2y2
wxy
wy
y
wxy
-
1
+ w2x2
-wx
[eq. 191]
wy
-wx
1
9x"
dx!;
V
dxj
+
Y;e
dxc
dx;dx"__
d28aßj^
l^/e^aß^Y«aß
^i
J^^dx0
dx¡
2
dxa dxe
dx,
Hierdurch nahe
gelegt
d
(
^aß^]
l^£¡e*^íf
3x,
,fe
dx,£
2
dxa
dxß
[eq. 193]
Probiert
am
Fall
des
rot.
Körpers
a
= 1
ß
=
1
liefert -w2.
^ ?aß^%xß
^Yaß^aß
5xö dxj dxa
dx¡
d
8aß
dYaß
dgaß
d
Yaß
*aß
dx£dxö
dx¡
dx"
dxcdx-
d\ß
dgaß
|
Byttß
d2gaß
c)x£c)xa
dx,
dxa
dx£dxj
[122][Eq.
189]
is
the
(coordinate)
divergence
of
the stress-energy
tensor
of the
gravitational
field
of Einstein and Grossmann
1913 (Doc.
13), part
1, pp.
15-16.
[Eq.
190]
and
[eq.
191]
are
the covariant
and
contravariant forms of the Minkowski metric
in
a
Cartesian coordinate
system rotating uniformly at angular speed w.
See
[p.
50]
for their
correct
forms. Since
/-G
= 1
for
[eq. 190]
and
[eq.
191],
the
vanishing
of
[eq.
189]
would
correspond
to
the
ob-
taining
of
the
energy
conservation
law
of Einstein and Grossmann
1913 (Doc.
13), part
1, p.
17.
[123]Einstein
expands
[eq. 189]
to
[eq.
192].
If the cancelations
in
[eq. 192] are
unrescind-
ed,
the reduced
[eq. 192]
is
equivalent to
[eq. 193]
contracted with
yap,
g.
This
suggests
that
the
vanishing
of
[eq. 189]
derives from the
stronger
condition
that
[eq. 193]
vanish, but this
supposition
is
refuted
by
the
computation
of
[eq. 193]
for indices
a=1,ß=1
and for
metric
[eq.
190],
[eq.
191],
which
yields
the
non-zero
result
-w2.
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