70 DOC.
9
FORMAL
FOUNDATION OF RELATIVITY
[39] §14.
The
H-Tensor
Equation
(65)
leads
us
to
a
theorem that is of fundamental
importance
to
the entire
theory.
If
we
vary
the
gravitational
field of
the
guv
by an
infinitely
small
amount, i.e.,
replace
the
guv
by
guv + dguv,
where the
8guv
shall vanish in
a zone
of finite width
adjacent
to
the
boundary
of E, then H becomes H
+
dH and
J
becomes
J
+
dJ.
We
now
claim that the
equation
A{dJ}
=
0
(68)
always
holds whichever
way
the
8gßV
might
be
chosen,
provided
the coordinate
systems
(K1
and
K2)
are adapted
coordinate
systems
relative
to the unvaried
gravitational
field. This
means
that under the restriction
to
adapted
coordinate
systems,
dJ
is
an
invariant.
In order
to
prove
this,
we
imagine
the variations
dguV
to
be
composed
of
two
parts,
and
we
therefore
write
SgMV =
8lg^
+
82g»\
(69)
where the
two
parts
of
the
variation
are
chosen such that
a.
The
d1guv
are
taken
in
a manner
that the coordinate
system
K1
is
not
only
adapted to the
(true)
gravitational
field
of
the
guv
but also
to
the
(varied)
gravita-
tional field of the
guv +
dgßV.
This
means
that not
only
the
equations
Bu
=
0
but also the
equations
81B
`a
=
0
(70)
are
valid. In other
words,
the
d1guv
are
not
independent
of each
other;
there
are
rather 4 differential
equations
between them.
b.
The
d2guv
are
taken
just
as
one
would
get
them without
changing
the
gravitational
field,
by
mere
variation
of
the coordinate
system, specifically, by
variation in that subdomain of
£
in which the
8guv
differ from
zero.
Such
a
[p.
1072]
variation is
determined
by
four
mutually
independent
functions
(variations
of
coordinates).
Obviously,
in
general,
d2Bu
*
0.
The
superposition
of
the two variations is therefore determined
by
(10
-
4) +
4
=
10
mutually independent
functions,
and thus
will
be
equivalent
to
an arbitrary
variation
of
the
dguv.
Hence,
the
proof
of
our
theorem will be
completed
when
equation (68)
is
proven
for the two
partial
variations.
Proof
for
the variation
of
d1:
By
d1-variation
of
(65) one
obtains in
a
straightfor-
ward
manner