DOC.
10
RESEARCH NOTES
245
1
d8ipd8ma
1
dg
-j
dgmo
=
~
~ynnyr,^-^- +
oYnrJ
2'P°'KldxI
dxK
2'P°'Kldxp
dxK
Der mit
2 multiplizierte
Ebenentensor
erhält also die Form
d28im
1
a?K/
^/
+
3YK/
dgn
+
dy«
d8mi_
[eq.
141]
k/3xk3X/
2
3xm
3x,
3xm
öxK
3*.
5S,p
d8mo
dgu
d8mO
Y^Yw-r-
-r-
+Y""Y
P5'Kldxl
dxK
'P°'K/3xp dxK
[100]
Resultat sicher. Gilt für
Koordinaten,
die der
Gl.
A(p =
0
genügen.
[96]After contraction with
y [eq.
135]
is
the
Ricci
tensor,
which
is to be expanded
in
a
coordinate
system satisfying
the harmonic condition
A2xi
=
0
(see
also
the
discussion in
the
editorial
note,
"Einstein's Research Notes
on a
Generalized
Theory
of
Relativity").
A
term
corresponding to
the
first
term
in
the summation of
[eq.
91] (the
full
form of the Riemann
ten-
sor)
has been
suppressed
since it vanishes
on
application
of
the
coordinate condition.
[97]Of
the second-derivative
terms,
only
this
one,
[eq.
136],
will
remain
to
first
order for
weak fields, in
accord with
requirement
(1)
discussed
in
the editorial
note,
"Einstein's
Re-
search
Notes
on a
Generalized
Theory
of
Relativity."
[98]The
condition
A2xi =
0
entails
[eq. 137]
(see
[eq. 37] on [p.
11]
for the definition of
A2;
see
also the
reworking
of this
operator
in
Einstein and Grossmann
1913
[Doc. 13], p.
29).
Differentiation of
[eq.
137]
and
[eq. 138]
yields
the
expression
[eq. 139]
for the three
second–
derivative
terms
of the Ricci
tensor
indicated, which vanishes
to
first order.
[99]At
[eq.
140],
Einstein
expands
the first-derivative
terms
of the Ricci
tensor
and
com-
bines with
[eq.
135]
and
[eq.
139] to produce
the reduced Ricci
tensor
[eq.
141],
which
now
satisfies condition
(1)
discussed
in
the editorial
note,
"Einstein's Research Notes
on a
Gener-
alized
Theory
of
Relativity."
[100]This
condition,
written
as
A2cp
=
0
for coordinate
(p,
was
commonly
used
as
the
"iso-
thermal" condition in
the
theory
of two-dimensional Gaussian surfaces
(see
Bianchi
1896,
§§36-37).
Für die
erste
Annäherung
lautet
unsere
Nebenbedingng.
[p. 38]
Xykk
k
fj8i
K
dgKK\ [101]
2^
5-
|
=
0
[eq. 142]
v k
dxi
Zerfällt vielleicht
in
=
0 & ^AKSkk
=
konst.
[eq. 143]
a*K