DOC.
9
FORMAL FOUNDATION OF RELATIVITY
71
lA(8.J)
=
/(Ax^S,
+ SjF.
(65a)
Z
M
Since the
d1-variations
of
the
guv
and all their derivatives vanish at the
boundary
of
£,
the
quantity d1F
(which can
be transformed into
a
surface
integral)
also vanishes
according
to
(65b).
After this and with
(70), our equation (65a)
turns into the relation
A(81J)
=
0.
(68a)
Proof for the variation of
S2:
The variation
S2J
is
equivalent
to
an
infinitesimal
coordinate transformation under fixed coordinates of the
boundary.
Since the
coordinate
system
is
an
adapted
one
relative
to
the unvaried
gravitational
field,
it
follows from
the definition of
adapted
coordinate
systems
that
S2J
=
0.
Next,
we
assume
the variation of the
gravitational
field relative
to
the coordinate
system
K1
to
be chosen
as a d2-variation;
then
we
have
d2(J1) =
0.
If
this variation
is
a
d2-variation
also relative to
K2-as
we
shall
prove
later-then
we
have
an analogous equation
relative to
K2,
i.e.,
d2(J2) =
0.
The
equation
to be
proven
follows then
by
subtraction
S2(AJ)
=
A(82J)
=
0.
(68b)
We still have to show that the variation under consideration is
a S2-variation
also
[p.
1073]
relative to
K2.
The unvaried tensors guv relative to
K1
and
K2
be denoted
symbolically by
G1
and
G2,
respectively; similarly,
the varied tensors
guv
relative
to
K1
and
K2
are
G1*
and
G2*,
respectively.
The coordinate transformation
T
brings
us
from
G1
to
G2,
also from
G1*
to
G2*
resp.;
and the inverse transformation will be
T-1.
Furthermore,
the coordinate transformation
t
brings
G1
to
G1*.
Consequently,
G2*
is obtained from
G2
by
the
sequence
of
transformations
T-1
-
t
-
T,
which is
again a
coordinate transformation. In this
manner
it
is shown that the
variation
of
the
guv
which
we
have under consideration here
is
also
a
d2-variation
relative to
K2.
The
equation
(68),
which
is
to
be
proven,
follows
finally
from
(68a)
and
(68b).
From
the
proven
theorem
we
deduce the existence of
a complex
of
10
components,
which has tensorial character if
we
limit ourselves to
adapted
coordinate
systems. According
to
(61)
one
has