180
DOC. 30
FOUNDATION OF GENERAL RELATIVITY
employed
in the
preceding
paragraph
in
formulating
the
equations
of
motion
of
the material
point.
A
special
case
in
which
the
required equations
must
in
any case
be
satisfied
is
that
of
the
special
theory
of
relativity,
in which the
guv
have
certain
constant values.
Let
this
be
the
case
in
a
certain
finite
space
in relation
to
a
definite
system
of co-ordinates
K0.
Relatively
to this
system
all
the
components
of
the Riemann
tensor
BpuoT,
defined
in
(43),
vanish.
For the
space
under
consideration
they
then
vanish, also
in
any
other
system
of
co-ordinates.
Thus the
required equations
of
the matter-free
gravita-
tional
field
must in
any
case
be satisfied if all
BpuoT
vanish.
But this
condition
goes
too far.
For
it
is
clear that,
e.g.,
the
gravitational
field
generated
by a
material
point
in its environ-
ment
certainly
cannot be
"transformed away"
by
any
choice
of
the
system
of co-ordinates, i.e.
it
cannot
be transformed
to
the
case
of
constant
guv. This
prompts
us
to
require
for
the matter-free
gravitational
field
that the
symmetrical
tensor
Guv,
derived from
the
tensor
BpuvT,
shall vanish.
Thus
we
obtain ten
equations
for
the
ten
quantities
guv,
which
are
satisfied
in
the
special
case
of
the
vanishing
of all
BpuvT.
With the
choice which
we
have
made
of
a
system of
co-ordinates,
and
taking
(44)
into considera-
tion,
the
equations
for the
matter-free field
are
If
-
-
0
J
-
9
=
1...
(47)
It
must be
pointed
out
that there
is
only a
minimum
of
arbitrariness
in the
choice of
these
equations.
For
besides
Guv
there
is
no
tensor
of second
rank which
is
formed from
the
guv
and its
derivatives,
contains
no
derivations
higher
than
second,
and
is
linear in these
derivatives.*
These
equations,
which
proceed,
by
the method
of
pure
*Properly
speaking,
this
can
be affirmed
only
of
the tensor
+
*-9ßV9a$Gaß,
where
A
is
a
constant. If,
however,
we
set
this
tensor
=
0,
we come
back
again
to
the
equations
Guv
=
0.