DOC. 30
FOUNDATION OF GENERAL
RELATIVITY
178
where
G"
=
= Rm"
+
SMK
Rmf
= - ^-{/w,
a}
+
{/mx,
£}
{f/3, a}
_
a-
.
|in
ai'°e^
*
7)Xu.^xv
xa
j
(44)
Note
on
the
Choice
of
Co-ordinates.-It
has
already
been
observed
in
§
8,
in
connexion
with
equation
(18a),
that the
choice of
co-ordinates
may
with
advantage
be made
so
that
s/-g
=
1.
A
glance
at the
equations
obtained in the
last
two
sections shows
that
by
such
a
choice
the
laws of
forma-
tion
of
tensors
undergo an important
simplification.
This
applies
particularly
to
Guv,
the
tensor
just
developed,
which
plays
a
fundamental
part
in the
theory
to be set forth. For
this
specialization
of
the
choice of
co-ordinates
brings
about
the
vanishing
of
Suv,
so
that the tensor
Guv
reduces to Ruv.
On
this account I
shall hereafter
give
all relations
in
the
simplified
form
which this
specialization
of
the
choice
of
co-
ordinates
brings
with it.
It
will
then
be
an easy
matter to
revert to the
generally
covariant
equations,
if this
seems
desirable
in
a special
case.
C.
Theory
of the
Gravitational Field
§
13.
Equations
of Motion of
a
Material
Point in the
Gravitational
Field. Expression for the Field-com-
ponents
of
Gravitation
A
freely
movable
body
not
subjected
to
external
forces
moves, according
to
the
special
theory
of
relativity,
in
a
straight
line
and
uniformly.
This
is also
the
case, according
to
the
general
theory
of
relativity,
for
a
part
of four-di-
mensional
space
in
which
the
system
of
co-ordinates
K0,
may
be,
and
is,
so
chosen
that
they
have the
special
constant
values
given
in
(4).
If
we
consider
precisely
this
movement from
any
chosen
system
of co-ordinates
K1,
the
body,
observed
from
K1,
moves,
according
to the considerations
in
§
2,
in
a
gravitational field.
The
law of
motion with
respect
to
K1
results without
diffi-