210
DOC.
10 RESEARCH
NOTES
dg44
dx{
=
0
dg44
"
1^
+
1
=
0
=
0
=
0
dgu
dg24
+
dx2
3*!
=
0
2^H
+
!£n
=0
ax,
dx2
ndg
12
(
dg22
_
n
dx,
dx,
dg
12
dx
[23]
~P'
(*2) [eq. 20]
12
3A:
[eq.
22]
gn=P(*2)
£12 =
[eq.
23]
g22 =
V(*i)
=
-\|/'(x,)
[eq.
21]
[eq. 27]
C0
+
CjXJ
+
c2x2+
ax
j
x2
cp'
(x2)
=
-2
(cj +
ax2)
\|/'
(xj)
=
-2(c2
+ axj)
[eq.
28]
q(*2) =
gn
=
a
2
-2
(c1X2
+
- X2
+
K
)
[eq.
29] i|/(x,)
= g22 =
-2
(c2x,
+
«x,2
+
k'")
g
14
=
ßx2
+
K
[eq.
30]
Ä14
=
9
(*2)
g24 =
V(*l
[eq.
24]
[eq.
25]
g24 =
-ßx,
+
K'
[eq. 31]
p'
(x2)
+ \|/'
(x,)
=0
[eq.
26]
(p' (x2)
=
a
cp
=
ax2
+
k
vj/'(x,) =-a
\|/
=
-
ax,
+
k'
[21]On
this
page,
Einstein
computes
the
general
form of
a
metric
(x3
components sup-
pressed)
which satisfies
the
conditions
gik, 4
=
0
and
g(ik,
m)
=
0.
That
g(ik,
m)
transforms
as a
tensor
is
used
as a
coordinate condition
on [p.
45].
[22]This
list contains
all
nonredundant
triples
of indices for the condition
g(ik,
m)
=
0.
The
corresponding
equations
are
written
explicitly below,
where
the
terms
with
s/sx4 are
canceled
due
to
gik, 4
=
0.
[23]The
joint
general
solutions
are
represented
by [eqs.
20-26] (where
(p
and
\|f
of
[eqs.
20-
23]
need
not
represent
the
same
functions
as
in
[eqs.
24-26]) and
by
the
suppressed equation
g44
=
constant.
The
corresponding
metric
is
found
to
be
[eqs.
27-31].
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