248
DOC. 10 RESEARCH NOTES
Hieraus
Gleichungen
A?u =
^11
+
2-^^kk
^g
12 ~
^12
*
Ag
14
=
7^14
[eq.
155]
-1/2dg
KK
dxo
=
1/2
U
Egmvdxvdxm
du
[107]
dx
a
d2U
a?
+
.
+
.
+
du
dx"
Darstellbar
in
der verl. Form.
[104]Einstein
rewrites the
decomposed
harmonic condition
[eq.
143]
as [eq.
150],
where the
second condition
is
now
expressed
in
terms
of the deviations
gxik
from the zeroth-order
met-
ric.
Throughout
this
page
Einstein
uses an
imaginary
time
coordinate,
as
already
indicated
in
[eq.
146],
so
that his zeroth-order metric
is
diag(1,1,1,1).
[105]The
second coordinate condition of
[eq.
150]
is
problematic.
In
combination with the
weak field
equation
it
leads
to
Tkk = 0.
The
addition of the
trace term to
the
right-hand
side
of
[eq.
151]
may
be
an
attempt to
avoid this
problem.
[106]Einstein
returns to
the
original
harmonic coordinate
condition,
whose weak
field
form
is
[eq.
152]. Compatibility
between the coordinate condition
and
the weak
field
equations
is
restored
by
the addition of
a
trace term to
[eq.
146],
which
now
becomes
[eq.
153],
with
U
the
trace
of
gik,
and
the
equivalent
form
[eq. 155].
The factor
2
in
[eq. 154]
should
be
a
minus
sign.
[107]Einstein
starts
a
calculation
designed to
show
that the modified
field
equations
admit
a
stress-energy tensor
for the
gravitational
field.
The calculation
proceeds
on [p.
41].
[p. 40]
=
d%2
[108]
T1
=
dx
dt
-
4g\\X•2
+' +' +
2gj2xy
-
+2gHi
••
+g44
Extr.
8r|
-=
Impuls
ox
r|
~
O-L
2(g1ii +
Si2)5
+
-
+*i4)
2rj
=
~(g
11
dx
dx
dy
dt
+
g12dx
'
+?14^)dx
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