DOC. 9 FORMAL FOUNDATION OF RELATIVITY 75
dg
=
gY,8mdg0r
=
-gY^gardg
OT
or
or
dgocß
=
Sa^gß.dg^
/XV
one
obtains from
(78)
[43]
(79)
{19}
\^sT,S
VT
dy
ög^'
PP
Wir
ö-*" d*T
dg** dg
VK
Z
pp'K
a*p
3-v
with
£
g^
PP
dgm
agvK
/JT
PP
K
dxp dxp.
dH\[^g
=
_1
^E
r
g
dgm.
dg*
and
VT
Z
/IT
/JT
/^T
dxo
dxT
V
From these relations follows the claim made
above.
Without
using
our physical
knowledge
of
gravitation, we
have arrived
in
a purely
formal
manner
at
quite
distinct field
equations.
In order to
get
them into
a more
explicit
notation, we
multiply (74) by
gVT
and
sum over
the
index
r.
Considering
(73),
we
thus
get
t
k
s;
=
£g
VT
dHs/^g
_
d
ra
dgm a*a dg7
(80) [44]
or
-
E
a
g VT
dHj^g
=
k$;+£
pVTdH£s
-
0"dH^£
(80a)
[45]
OT
dxa
dg?
o
dg
or^
oa
dg7
OCT
Since
our
coordinate
system
is
an
adapted
one,
the
equation
E
^
|gVT
dH^g
=
0
ar\
dx"
dx
a
d8:
[p. 1077]
also holds
due
to
(67)
and
(65a) and, therefore,
considering
(80),
also the
equation
E
a
vr
dHf-g
_
vr
dH\f-g
0.
(80b)
[46]
dx
K
ar
o
dg
or-v
oa
dlC
Utilizing (78), (79),
and
(46), we can replace equations (80a)
and
(80b) by
the