234 DOC. 10
RESEARCH
NOTES
[67][Eq.
91]
is
the
fully
covariant form of the Riemann
curvature
tensor,
almost
exactly
as
given
in
Einstein and Grossmann
1913 (Doc.
13),
p.
35.
For
a
discussion of Einstein's collab-
oration with Marcel
Grossmann,
see
the editorial
notes,
"Einstein's Research Notes
on a
Gen-
eralized
Theory
of
Relativity"
and
"Einstein
on
Gravitation
and
Relativity:
The Collaboration
with Marcel Grossmann."
[68]Einstein
forms the
fully
covariant Ricci
tensor
by
contracting
once
and
proceeds to
compute
a
reduced form
([eq.
92])
for the
terms containing
first
derivatives
given
by summa-
tion
over
the Christoffel
symbols.
[69]These
three second-derivative
terms
from
the Ricci
tensor
"should vanish"
("sollte[n]
verschwinden")
to
leave
yklgim,
Kl
as
the
only
second-derivative
term in
the Ricci
tensor,
if
the Ricci
tensor
is
to satisfy
the condition
(1)
for
a
gravitation
tensor
discussed in
the editori-
al note,
"Einstein's Research Notes
on a
Generalized
Theory
of
Relativity."
Einstein
consid-
ers a
weak
field
metric with
components diag(1,1,1,1) to
zeroth order
so
that the
contracting
metric need
not appear explicitly
in
[eq.
93].
[p. 28]
/
d2Sil
\
p
=
I
yim\i
ö
im
imKl
V
dxKdx, dxKdxm
/
\
[70]
+
X
V-Jk/
i
m K
/
i i
K m
[eq. 94]
pGimK/ V
O
p
a
P /
fdSia
dg
mo
dgjm^\(dg
Kp
dg[p
[71]
+ +
p
oimKl
\
dx,n
dx dx
o /
V
dxl
dxK 3xp
dSKi
dsK,
~8
lC
dxm
°ma
dX;
im
dxa
dxK
^
,
^im ^im
dSim d1K,
Ki
d8Kt
+8ma
dx.
+y'm
dxa
)(g*P
dx./
S,pdxK
P
dx^
+
Jk
^'dx
}
171
/
O K
iv
p1*0
dy^ dy.
dig
G
aigc
'pm
'/p
dxm
dx,
PG
dx
O
dxp
V
_
J
Y
dypa
dx
a