164
DOC.
13
GENERALIZED THEORY OF RELATIVITY
The
identity
that
is
being sought
is
thereby uniquely
determined;
if
one
constructs it
according
to
the
procedure indicated,15 one
obtains
2
9-?ußdxs-dXg
O8YTO
2
2dx"(y
9'
Y"?
dYt9dgtQ
dxl
dx
a
(it
(j
aßtq
dx-a(y
(12)
dy.
"V-..
"
d1^y
fiV afi
aßt'Q
+
2
2
y8xa
dfftoSY
8x
X
Zj
Y*
*
Y"
f
d9tdv
Jxa
}
•
a
[it
(j
et
ßtq
'
Thus,
the
expression
for
Iuv
that
is
enclosed between the
curly
brackets
on
the
right-hand
side
is
the
tensor
that
is
being sought
that
enters
into the
gravitational
equations
K0,xv
=
r.
To make these
equations
more
comprehensible,
we
introduce the
following
abbreviations:
/
dSrp
rp
1
, ,
dg
dY
(13)
-2k-0
e
Ya"Yßv
dx.
'
dxa
~ 2y,a
Yaß dx.
r p
/XV
dxß
aßrp
\
ß
OL
)
We
will
designate 0uv as
the "contravariant
stress-energy
tensor
of
the
gravitational
field"
The covariant
tensor
reciprocal
to
it will be denoted
by
tuv;
then
we
have
(14)
-2k^v
=
e dg
tp
dyTP
i
v
dgTP
dyT
aßrp
V
dx..
/x
dx..
-
v
-
2
MV
ß
dx..
~
'
a
dx-jS
Likewise,
for the sake
of
brevity,
we
introduce the
following
notations for
differential
operations
carried
out
on
the fundamental
tensors
y
and
g:
(15)
A^v(y) =
£
1
Ö
dy
»
/XV
e
yaßsrp
dy
[it
dyv
aß dXa aßrp
dxa
dxß
ß
/
and
(16)
Duv(y) =
e
y="j-
Y«ßJ-g
dg
/XV
-
e
YaßYr
dg^ 3gv
aß
J-g
ÖXa
\
dx
aßrp
P
dxa
dxß
/
15Cf.
Part
II,
§4,
No.
8.
[30]
[31]