DOC. 30 FOUNDATION OF GENERAL RELATIVITY
155
in the
sense
of
the
special
theory
of
relativity. By
the
special theory
of
relativity
the
expression
ds2
=
- dX21 - dX22 -
dX23
+
dX24
. .
(1)
then has
a
value which
is
independent
of
the orientation
of
the
local
system
of
co-ordinates,
and
is
ascertainable
by
measurements
of
space
and time.
The
magnitude
of
the
linear element
pertaining
to
points
of
the
four-dimensional
continuum
in infinite
proximity,
we
call ds.
If the
ds
belong-
ing
to
the
element
dX1
...
dX4
is
positive,
we
follow
Minkowski in
calling
it
time-like;
if
it is
negative,
we
call
it
space-like.
To
the
"linear
element" in
question,
or
to the two
infin-
itely proximate point-events,
there will also
correspond
definite differentials
dx1
...
dx4
of
the
four-dimensional
co-ordinates
of
any
chosen
system
of reference.
If this
system, as
well
as
the
"local"
system,
is
given
for
the
region
under
consideration,
the
dXv will allow
themselves
to be
represented
here
by
definite
linear
homogeneous expressions
of
the dXo:-
dXv
=
XavadXa
....
(2)
Inserting
these
expressions
in
(1),
we
obtain
ds2
=
2g
ardxadxr,.
.
.
.
(3)
where the
gar
will be
functions
of
the
xo.
These
can no
longer
be
dependent
on
the orientation and
the
state of
motion
of
the
"local"
system
of co-ordinates,
for
ds2
is
a
quantity
ascertainable
by
rod-clock
measurement
of
point-
events
infinitely proximate
in
space-time,
and defined inde-
pendently
of
any particular
choice
of
co-ordinates.
The
gor
are
to
be chosen
here
so
that
gor
=
gra;
the summation
is
to
extend
over
all values of
o
and
t, so
that the
sum
consists
of
4
x
4 terms,
of which twelve
are
equal
in
pairs.
The
case
of
the
ordinary
theory
of
relativity
arises out of
the
case
here
considered,
if
it
is
possible,
by
reason
of
the
particular
relations
of
the
gor
in
a
finite
region,
to
choose
the
system
of reference
in the finite
region
in such
a
way
that
the
gor assume
the
constant values
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