72 DOC. 9 FORMAL FOUNDATION OF RELATIVITY
dJ
=
8[jHRdT^
d(HR) d(HR)
/XV
/XV
dguv
Sg
dgaMV
Sg,
a
or,
since
dg0uv
=
d/dxo(dguv),
after
partial integration
and
considering
the
vanishing
of
the
d(guv)
at the
boundary
a
MR
o
dxa
a*r
(71)
We have
now
proven
that under limitation
to
adapted
coordinate
systems
dJ
is
an
invariant. Since the
dguv
need differ from
zero only
in
an infinitely
small
domain,
and since
gdr
is
a scalar,
the
integral
divided
by
/-g
is also
an invariant,
i.e.,
the
quantity
R
(72)
where
I.E
`lv
agMv
a
ax0
(73)
Now, however,
dguv
is
a
contravariant tensor
just
as guv is,
and the ratios of the
dguv
can
be chosen
freely.
From this follows that under limitation
to
adapted
[p. 1074]
coordinate
systems
and substitutions between
them,
vc~
is
a
covariant tensor and
Suv
itself
is the
corresponding
covariant
V-tensor
and
according
to
(73)
a
symmetric
tensor.
§15.
Derivation
of the
Field
Equations
One
may expect
the tensor
Suv
to have
a
fundamental
role in the field
equations
of
gravitation
that
we
want to
find,
and that those
equations
have
to take
the
place
that
Poisson's
equation
has
in the Newtonian
theory.
After the
deliberations
of
§§13
and
14
we
have to demand that the desired
equations-as
well
as
the tensor Suv-are
only
covariant with
respect
to
adapted
coordinate
systems.
The
equations we are