DOC. 30 FOUNDATION
OF GENERAL RELATIVITY
153
We therefore
reach this
result:-In the
general theory
of
relativity,
space
and time cannot be defined
in
such
a
way
that
differences of
the
spatial
co-ordinates
can
be
directly
measured
by
the
unit
measuring-rod,
or
differences
in the
time
co-ordinate
by
a
standard
clock.
The method
hitherto
employed
for
laying
co-ordinates
into the
space-time
continuum
in
a
definite
manner
thus
breaks
down,
and
there
seems
to be
no
other
way
which
would allow
us
to
adapt systems
of
co-ordinates
to
the four-dimensional
universe
so
that
we
might expect
from
their
application
a
particularly simple
formulation
of
the
laws
of
nature.
So
there
is
nothing
for
it but
to
regard
all
imaginable systems
of co-ordinates,
on
principle,
as equally
suitable
for
the
description
of nature.
This
comes
to
requiring
that:-
The
general
laws
of
nature
are
to
be expressed
by
equations
which hold
good
for
all
systems
of co-ordinates,
that
is,
are
co-variant with
respect
to
any
substitutions
whatever
(generally
co-variant).
It
is clear
that
a
physical
theory
which
satisfies
this
postulate
will also be suitable
for the
general postulate
of
relativity.
For
the
sum
of
all
substitutions in
any
case
in-
cludes
those which
correspond
to all
relative motions
of
three-
dimensional
systems
of co-ordinates.
That
this
requirement
of
general
co-variance,
which
takes
away
from
space
and
time the last remnant
of
physical objectivity,
is
a
natural
one,
will be
seen
from
the
following
reflexion. All
our
space-time
verifications
invariably
amount to
a
determination
of
space-time
coincidences.
If,
for
example,
events consisted
[11]
merely
in the motion
of
material
points,
then
ultimately
nothing
would be observable
but
the
meetings
of two
or more
of
these
points.
Moreover,
the results
of
our
measurings
are
nothing
but
verifications of such
meetings
of the material
points
of
our
measuring
instruments with other material
points,
coincidences
between the hands
of
a
clock
and
points
on
the
clock
dial,
and
observed
point-events happening
at
the
same
place
at the
same
time.
The introduction
of
a
system
of
reference
serves
no
other
purpose
than
to facilitate the
description of
the
totality
of such
coincidences.
We
allot to
the universe
four
space-time
vari-
ables
x1, x2, x3,
x4
in such
a
way
that
for
every
point-event
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