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 zerothorder
met
ric.
Throughout
this
page
Einstein
uses an
imaginary
time
coordinate,
as
already
indicated
in
[eq.
146],
so
that his zerothorder 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
righthand
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
stressenergy 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
~
OL
2(g1ii +
Si2)5
+

+*i4)
2rj
=
~(g
11
dx
dx
dy
dt
+
g12dx
'
+?14^)dx