412
DOC.
42
SPECIAL AND GENERAL RELATIVITY
Relativity
and
the Problem
of
Space
173
Considered
four-dimensionally,
a
non-linear transformation
of the four co-ordinates
corresponds to
the transition
from
S1
to
S2.
The
question now
arises:
What kind of non-linear
trans-
formations
are
to
be
permitted,
or,
how
is
the Lorentz
trans-
formation
to
be
generalised?
In order
to
answer
this
question,
the
following
consideration
is
decisive.
We ascribe
to
the inertial
system
of the earlier
theory
this
property:
Differences
in co-ordinates
are
measured
by
station-
ary
"rigid" measuring rods,
and differences
in
time
by
clocks
at rest.
The
first
assumption is supplemented
by another,
namely,
that
for
the relative
laying
out
and
fitting
together
of
measuring
rods
at
rest,
the theorems
on
"lengths"
in Euclid-
ean
geometry
hold. From the results
of
the
special theory
of
relativity
it
is
then
concluded,
by elementary
considerations,
that this direct
physical interpretation
of the co-ordinates
is
lost
for
systems
of reference
(S2)
accelerated
relatively
to
in-
ertial
systems
(S1).
But if
this
is
the
case,
the co-ordinates
now
express only
the order
or
rank of the
"contiguity"
and
hence
also
the dimensional
grade
of the
space,
but
do
not
express
any
of
its
metrical
properties.
We
are
thus led
to
extend the
transformations
to arbitrary
continuous transformations.1
This
implies
the
general principle
of
relativity:
Natural
laws
must
be covariant with
respect to arbitrary
continuous transforma-
tions of the co-ordinates.
This
requirement
(combined
with
that
of the
greatest possible
logical
simplicity
of the
laws)
limits the natural laws concerned
incomparably more strongly
than the
special principle
of
relativity.
1
This inexact mode of
expression will
perhaps
suffice here.