DOC. 9 FORMAL FOUNDATION OF RELATIVITY 69
a2
va
aav
dxa
dxa
8
(65a)
\
F
-/*£
=
a
8 va
dH^g
_
a
8
vo-
a//^
Ax (65b) {17}
OCTV/l
a^v
a*_ a*.
a&r
\
F
can
be transformed
into
a
surface
integral.
It vanishes when
AxU
and
5Ax
dAxu
dxo
vanish at the
boundary.
Adapted
coordinate
systems.
We consider
again
our
continuum and its domain
[37]
£,
which
we assume
finite in all of its
coordinates,
and referred to the coordinate
system
K.
Starting
from K
we
imagine successively
introduced coordinate
systems
K', K"
etc.,
all
infinitely
close to each
other such that the
Axu
and
the
dAxu/dAxa
vanish
at the boundaries.
We call all these
systems
"coordinate
systems
with
coinciding
boundary
coordinates." For
every
infinitesimal coordinate transformation between
neighboring
coordinate
systems
of
the
totality
K, K',
K"...
we
have
F=
0,
so
that instead
of
(65) we
have the
equation
JAJ=EfdrA*A- (66)
[38]
L
M
Among
all
systems
with
coinciding boundary
coordinates
will be
some
for which
J
attains
an
extremum
as compared
with the J-values
of
neighboring
systems
with
likewise
coinciding boundary
values.
We call these coordinate
systems
"coordinate
systems adapted
to the
gravitational
field."
The
equations
Bu =
0
(67)
hold
for these
adapted systems according
to
(66)
because the
Axu
can
be chosen
freely
inside
of
E.
Inversely, (67)
is
a
sufficient condition
that the coordinate
system
is
adapted
to
[p. 1071]
the
gravitational
field.
We avoid the
difficulty
mentioned in
§13
by
henceforth
writing only
differential
equations
of
the
gravitational
field that claim
validity merely
for
adapted
coordinate
systems.
Under the limitation to
adapted
coordinate
systems,
it is indeed
no longer
allowed to extend
a
coordinate
system
that is
given
for the exterior
of
E
into the
interior
of
E
in
an arbitrary manner.