DOC. 42 SPECIAL AND GENERAL RELATIVITY
415
176
Relativity
cial
theory
of
relativity, space (space-time)
has
an
existence
independent
of
matter
or
field.
In order
to
be able
to
describe
at
all
that which
fills
up space
and
is dependent
on
the
co-
ordinates, space-time
or
the
inertial
system
with its metrical
properties must
be
thought
of
at
once as
existing,
for other-
wise the
description
of
"that
which
fills
up space"
would have
no
meaning.1
On the
basis
of the
general theory
of
relativity,
on
the
other
hand,
space as opposed
to
"what
fills
space,"
which
is
dependent
on
the
co-ordinates,
has
no
separate
ex-
istence.
Thus
a
pure gravitational
field
might
have
been
de-
scribed in
terms
of
the
gik
(as
functions of the
co-ordinates),
by
solution
of
the
gravitational equations.
If
we
imagine
the
grav-
itational
field,
i.e.
the functions
gik,
to
be
removed,
there does
not
remain
a
space
of the
type
(1),
but
absolutely
nothing,
and
also
no
"topological space."
For the functions
gik describe
not
only
the
field,
but
at
the
same
time
also
the
topological
and
metrical structural
properties
of the manifold.
A
space
of the
type
(1),
judged
from
the
standpoint
of
the
general theory
of
relativity, is not
a
space
without
field,
but
a special case
of the
gik
field,
for
which-for
the co-ordinate
system
used,
which in
itself
has
no
objective
significance-the
functions
gik
have
val-
ues
that
do
not
depend
on
the
co-ordinates.
There
is
no
such
thing
as an
empty space, i.e.
a
space
without
field.
Space-time
does
not
claim existence
on
its
own,
but
only
as a
structural
quality
of
the
field.
1
If
we
consider that which
fills space (e.g.
the
field)
to
be removed, there
still
remains the
metric
space
in accordance with
(1),
which would also determine the inertial behaviour of
a
test
body
introduced into
it.