DOC. 10 RESEARCH
NOTES
247
servation for such
a
choice
of
gravitation tensor.
The left-hand side
is
proportional to
the
gravitational
force
density acting
on
the
source masses.
The
right-hand
side
can
be
expressed
as
the
divergence
of
a
second-rank tensor,
the
stress-energy tensor
of the
gravitational
field.
The role of identities such
as [eq.
145]
in
the formation of
field
equations
is
stressed
in Ein-
stein and Grossmann
1913 (Doc.
13), part
1,
§5.
[103]Einstein
confirms the
compatibility
of
two
conditions that
must apply to pressureless
dust: the
continuity equation
and
the
law
of conservation of
energy momentum.
The
continu-
ity
equation
for dust
with
rest density
p0
in
the weak
field
case
is
[eq.
147]
or
its
equivalent
[eq. 148]
(with
vi
=
dxi/dx
and
x
the
proper
time).
The
continuity equation
is
rewritten
as
[eq. 149].
The
vanishing
of the
first
line
is
equivalent to
the
energy-momentum
conservation
law in
the weak
field
case,
Tik, k
=
0,
with
Tik = p0vivk
the
stress-energy tensor.
The
two
conditions
are
compatible
if
Dvi/Dx
=
0.
Presumably
these conditions also
test
the coordi-
nate
condition
yklgiK l
=
0,
for this condition
together
with the
field
equation
ykigim,
kl = KTim
gives
the
energy-momentum
conservation
law
yklTik, l
=
0.
[p.
39]
dxK dxK
0
dl
dx
[eq.
151]
für
gleiche
i &
K.
I
/
d8iK
1
dg
kk
\
V
dxK
2
dxi
j
=
0
[eq. 152]
A^.
=
°im
[106]
XSkk
=
U
Gravitationsgleichungen
A^n-^)=7'n
A
gn
=
Tn
4
=
T
14
[eq.
153]
2^U
=
Xtkk
[eq.
154]
[104]
g~
= 0
[eq. 150]
[105]