DOC. 30 FOUNDATION OF GENERAL RELATIVITY
179
culty
from the
following
consideration. With
respect
to
K0
the law
of motion
corresponds
to
a
four-dimensional
straight
line, i.e.
to
a
geodetic
line. Now since
the
geodetic
line
is
defined
independently
of
the
system
of reference,
its
equations
will also be
the
equation
of
motion
of
the material
point
with
respect
to
K1.
If
we
set
Truv
=
-
{uv,
T}
. . .
(45)
the
equation
of
the motion
of
the
point
with
respect
to
K1,
becomes
d^Xf
-p
T
dXfj.
dXy
"3?"
~
1
ds
ds
.
.
.
(46)
We
now
make the
assumption,
which
readily
suggests itself,
that this covariant
system
of
equations
also defines
the motion
of
the
point
in the
gravitational
field
in
the
case
when there
is
no
system
of
reference
K0,
with
respect
to
which the
special
theory
of
relativity
holds
good
in
a
finite
region.
We have
all
the
more justification
for
this
assumption
as
(46)
contains
only
first
derivatives
of
the
guv,
between
which
even
in the
special case
of
the existence
of
K0, no
relations
sub-
sist.*
If the
Truv
vanish,
then the
point
moves
uniformly
in
a
straight
line.
These
quantities
therefore condition the devi-
ation
of
the motion
from
uniformity.
They
are
the
com-
ponents
of the
gravitational field.
§
14.
The Field Equations
of
Gravitation in the Absence
of
Matter
We make
a
distinction hereafter between
"gravitational
field" and
"matter"
in this
way,
that
we
denote
everything
but the
gravitational
field
as
"matter."
Our
use
of
the
word
therefore
includes
not
only
matter
in the
ordinary sense,
but
the
electromagnetic
field
as
well.
Our
next task
is to find
the
field
equations
of
gravitation
in the
absence of
matter. Here
we
again
apply
the method
*It
is
only
between
the
second
(and first)
derivatives that,
by
§
12,
the
relations
BpmsT
=
0 subsist.