414
DOC.
42
SPECIAL
AND GENERAL RELATIVITY
Relativity
and
the Problem
of
Space
175
nificant
quantity
ds is
expressed
in
the
new system
of
co-
ordinates
by
the relation
ds2
=
gikdxidxk
(1a)
which
has
to
be summed
up
over
the indices
i
and
k for all
combinations
11,
12,...
up to
44.
The
term
gik
now are
not
constants,
but
functions of the
co-ordinates,
which
are
deter-
mined
by
the
arbitrarily
chosen
transformation.
Nevertheless,
the
terms
gik
are
not arbitrary
functions of the
new co-
ordinates,
but
just
functions of such
a
kind
that
the form
(1a)
can
be transformed back into the
form (1)
by
a
continuous
transformation of the four co-ordinates. In order
that
this
may
be
possible,
the function
gik
must satisfy
certain
general
cova-
riant
equations
of
condition,
which
were
derived
by B.
Rie-
mann more
than half
a century
before the formulation of the
general theory
of
relativity
("Riemann
condition").
According
to
the
principle
of
equivalence,
(1a)
describes in
general co-
variant
form
a
gravitational
field of
a
special kind,
when
the
functions
gik
satisfy
the Riemann condition.
It
follows
that
the
law for
the
pure gravitational
field
of
a
general
kind
must
be satisfied when the Riemann condition
is
satisfied;
but
it
must
be weaker
or
less
restricting
than the
Riemann condition. In this
way
the field
law
of
pure gravita-
tion
is
practically completely
determined,
a
result which
will
not
be
justified
in
greater
detail here.
We
are now
in
a
position to
see
how
far
the transition
to
the
general theory
of
relativity
modifies the
concept
of
space.
In
accordance with
classical
mechanics and
according to
the
spe-