68
DOC.
9 FORMAL
FOUNDATION OF RELATIVITY
(f
at
some
point
in the
continuum,
then
one has, according
to
(17),
first
A (födr)
=
0
(62)
and
furthermore
A#
=
£ |^-Ag^
+
-^-A
/XV(7
dg
/XV
dga/XV
(62a)
The
AgßV
can
be
expressed by means
of
(8)
in
terms
of the
Axß
by taking
the
relations
Ag
/XV
_
-y/XV
_
s
gafl\
A*M
=
x'
~
into account. One
obtains
AgßV
=
E
a
8
8AxV
dxa
dAx
+
g
V« M
3x
(63)
[36]
Agf
=
E
a
a
Ät
/ia
5Axv
9xa
+
gva
3Ax
/a
5x«
_
dg
/XV
3Axa
(63a)
The
equations
(62a), (63), (63a),
present
AH
as
linear
homogeneous
functions
of
the first
and second derivatives of the
Ax
in the coordinates.
We
have,
so
far,
not made
any
assumptions
of
how H shall
depend upon
the
gMV
and the
g"v.
Now
we
shall
assume
H
to
be invariant under linear
transformations,
i.e.,
AH shall vanish when
--=-^-does.
We obtain under
this
assumption
oxadxa
-AH
=
E
Gvg.
2
.£i
dxadxa
(64)
/xv ra
[p. 1070]
By means
of
(64)
and
(62)
one gets
and
from
this,
by partial
integration,
2 J
dg? dx*dxa
IA
J=
f
drj;
(Ax^BJ
+
F,
(65)
where
we
used the abbreviations