DOC. 42 SPECIAL AND GENERAL
RELATIVITY 349
Space-Time
Continuum
of
the
General
Theory
105
ature,
and with which
we
made
acquaintance
as an
example
of
a
two-dimensional continuum.
Just
as
it
was
there
impossible
to construct
a
Cartesian co-ordinate
system
from
equal rods,
so
here it
is
impossible to
build
up
a
system
(reference-body)
from
rigid
bodies and
clocks,
which
shall
be
of
such
a
nature
that
measuring-rods
and
clocks,
arranged rigidly
with
respect
to
one
another,
shall indicate
position
and time
directly.
Such
was
the
essence
of
the
difficulty
with which
we were con-
fronted
in
Section
23.
But the considerations
of
Sections
25
and
26
show
us
the
way
to
surmount
this
difficulty.
We
refer the four-dimensional
space-time
continuum in
an
arbitrary
manner
to
Gauss
co-
ordinates. We
assign to every point
of the continuum (event)
four
numbers,
x1,
x2,
x3, x4
(co-ordinates),
which have
not
the
least direct
physical significance,
but
only
serve
the
purpose
of
numbering
the
points
of
the continuum in
a
definite
but
ar-
bitrary
manner.
This
arrangement
does
not
even
need
to
be of
such
a
kind that
we
must regard
x1, x2, x3
as
"space"
co-
ordinates and
x4
as a
"time"
co-ordinate.
The
reader
may
think that
such
a
description
of the world
would be
quite
inadequate.
What does it
mean
to assign to
an
event
the
particular
co-ordinates
x1,
x2, x3, x4,
if
in
themselves
these
co-ordinates have
no
significance?
More careful consid-
eration
shows,
however,
that this
anxiety
is
unfounded.
Let
us
consider, for instance,
a
material
point
with
any
kind of
mo-
tion.
If
this
point
had
only
a
momentary
existence without
duration,
then
it would be described in
space-time by
a
single
system
of
values
x1, x2, x3, x4.
Thus
its
permanent
existence
must
be characterised
by
an
infinitely
large
number
of
such
[56]