156
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
-1
0
0
0
0
-1
0 0
0
0 -1
0
. . .
(4)
0 0
0
+1
We
shall find
hereafter that the
choice of such
co-ordinates
is,
in
general,
not
possible
for
a
finite
region.
From the considerations
of
§
2
and
§
3
it
follows
that
the
quantities
gro
are
to
be
regarded
from
the
physical
stand-
point
as
the
quantities
which
describe
the
gravitational
field
in relation
to
the
chosen
system of
reference. For,
if
we now assume
the
special
theory
of
relativity
to
apply
to
a
certain four-dimensional
region
with the
co-ordinates
properly
chosen,
then the
gor
have the
values
given
in
(4).
A free
material
point
then
moves, relatively
to
this
system,
with
uniform
motion in
a
straight
line.
Then if
we
introduce
new
space-time
co-ordinates
x1, x2,
x3, x4,
by
means
of
any
substi-
tution
we
choose,
the
gor
in
this
new
system
will
no
longer
be
constants,
but functions
of
space
and time. At the
same
time
the
motion
of the
free
material
point
will
present
itself
in the
new
co-ordinates
as a
curvilinear non-uniform
motion,
and
the
law of
this motion
will be
independent
of
the
nature
of
the
moving particle.
We
shall
therefore
interpret
this
motion
as a
motion under the
influence
of
a
gravitational
field.
We thus
find
the
occurrence
of
a
gravitational
field
connected
with
a
space-time
variability
of
the
go
.
So,
too,
in the
general
case,
when
we are no longer
able
by a
suitable
choice
of
co-ordinates to
apply
the
special
theory
of
relativity
to
a
finite
region,
we
shall
hold
fast
to
the
view
that
the
gor
describe the
gravitational
field.
Thus,
according
to
the
general
theory
of
relativity,
gravi-
tation
occupies an exceptional position
with
regard
to
other
forces, particularly
the
electromagnetic forces,
since
the
ten
functions
representing
the
gravitational field
at the
same
time
define
the
metrical
properties
of
the
space
measured.
B. Mathematical Aids
to the
Formulation
of
Generally
Covariant
Equations
Having
seen
in
the
foregoing
that the
general
postulate
of relativity
leads to
the
requirement
that the
equations
of