DOC. 9 FORMAL FOUNDATION OF RELATIVITY
77
-1
0
0 0
0 -1
0
0
0
0 -1
0
0
0
0
1
at
an arbitrarily given point.
This is the
case
when the surface
of
second
degree
E
s".
5"5V
=
1
/XV
has
always
(for every system
of the
guv
which
occurs
in
our
continuum)
three
imaginary-valued axes
and
one
real-valued axis.
If
X1,
X2,
X3,
X4
are
the
squares
of
the
reciprocals
of the semi-axes of the
surface,
the
equation
of
fourth
degree
Iv
-
K\
=0
=
-
*)(*
-
*)
is satisfied.
Consequently,
X1 X2 X3 X4
=
g.
In order to
prevent
the
guv
from
reaching
infinite
values, one
has to demand that
g
vanishes
nowhere,
since the
guv
are equal
to the minors
of
the
guv
-determinant
divided
by g.
No
X
can
then
ever
become
zero. Therefore,
whenever
X10,
X20, X30,
X40
for
one point
of
the
continuum,
it is true for
every point.
Thus,
the
space-time
character
of
our
continuum in the
neighborhood
of
all
points
remains the
same as
it
was
in the
original theory
of
relativity. Mathematically
this
can
be
expressed by
saying:
among
four
mutually "orthogonal"
line elements
originating
from
one
point, one
element is
always
"time-like,"
the other three
are "space-like."
However,
this does
not
yet
establish
space-like or
time-like relations with the
coordinate
system
of
the
xv.
In the
original
theory
of
relativity, every
line element
that deviates from
zero
only
in
dx4
is time-like. The
same
statement
cannot be
claimed for
our adapted
coordinate
systems. Considering sufficiently large parts
of
the
universe,
it
is
very
well
imaginable
that
no
coordinate
axis
can
be
denoted
as
"time-axis,"
but
that rather the line elements
of
one
axis
are
in
parts time-like,
in
[p. 1079]
other
parts space-like.
The
equivalence
of the four dimensions
of
the world would
then be not
only
a
formal
one
but
a complete one.
For the time
being one
has to
leave this
important question an open one. [47]
An
even deeper-reaching
question
of
fundamental
significance
shall
now
be
brought
up-and I
am
not able to
answer
it.
In the
ordinary theory
of
relativity, every
line that describes the movement of
a
material
point,
i.e.,
every
line
consisting only
of time-like
elements,
is
necessarily
nonclosed,
the
reason being
that such
a
line
never
contains elements for which
dx4
vanishes.
An
analogous
statement cannot be
claimed for the
theory developed
here. It is therefore
a priori possible
to
imagine a
point
movement where the four-dimensional
curve
of
the
point
is
almost
a
closed
one.
In this
case
one
and
the
same
material
point
could exist in
an
arbitrarily
small
space–