154
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
there
is
a
corresponding
system
of
values of
the
variables
x1
...
x4.
To two
coincident
point-events
there
corre-
sponds one
system
of
values of
the
variables
x1
...
x4,
i.e.
coincidence is
characterized
by
the
identity of
the co-ordinates.
If,
in
place
of
the
variables
x1
...
x4,
we
introduce
functions
of them,
x'1,
x'2, x'3,
x'4,
as a new
system
of
co-ordinates,
so
that the
systems
of
values
are
made to
correspond
to
one
another without
ambiguity,
the
equality
of all
four
co-ordin-
ates in the
new
system
will also
serve as an
expression
for
the
space-time
coincidence of
the
two
point-events.
As all
our physical experience
can
be
ultimately
reduced to such
coincidences,
there
is
no
immediate
reason
for
preferring
certain
systems
of
co-ordinates to
others,
that
is
to
say,
we
arrive
at
the
requirement
of
general
co-variance.
§
4.
The
Relation of
the
Four
Co-ordinates
to
Measure-
ment in
Space
and Time
It
is
not
my purpose
in this
discussion to
represent
the
general theory
of
relativity
as a
system
that
is
as simple
and
logical as possible,
and
with the minimum number
of axioms;
but
my
main
object
is to
develop
this
theory
in such
a way
that the reader
will feel
that the
path
we
have
entered
upon
is
psychologically
the natural
one,
and that
the
underlying
assumptions
will
seem
to have
the
highest
possible degree
of
security.
With this
aim in
view
let it
now
be
granted
that:-
For
infinitely
small four-dimensional
regions
the
theory
of
relativity
in the restricted
sense
is
appropriate,
if
the
co-
ordinates
are
suitably
chosen.
For this
purpose
we
must choose
the
acceleration
of
the
infinitely
small ("local")
system
of
co-ordinates
so
that
no
gravitational
field
occurs;
this
is
possible
for
an
infinitely
small
region.
Let
X1,
X2,
X3,
be
the
co-ordinates of
space,
and
X4
the
appertaining
co-ordinate
of
time
measured
in the
appropriate
unit.* If
a
rigid
rod is
imagined
to be
given
as
the unit
measure,
the
co-ordinates,
with
a
given
orientation
of
the
system of co-ordinates,
have
a
direct
physical meaning
*
The unit
of
time
is
to be chosen
so
that the
velocity
of light
in
vacuo as
measured in
the
"local"
system
of
co-ordinates is to
be
equal
to
unity.