DOC. 9 FORMAL FOUNDATION OF RELATIVITY 79
E. Some
Remarks
on
the
Physical
Content of
the
General Laws
Developed
In the derivation
of
the
laws I let
myself
be guided-inasfar
as
this
was
possible-by
solely
formal
aspects.
In order to leave the
elaboration
of
the
subject
matter
not
too
incomplete, we
shall
now
also
highlight
the obtained results from their
physical
side.
Not to be smothered
by
mathematical
complications, we
limit ourselves to the
consideration
of
approximations.
§17.
The
Establishment of
Approximative
Equations,
Seen
from
Various
Aspects
One
sees
from the
far-reaching
usefulness
of
the
equations
of
the
original theory
of
relativity
that in the
space-time
domain that is
perceptible
to
us,
the
guv
can
almost
be
treated like constants.
Consequently, we put
[p. 1081]
811V
8 *v
=
*r
(82)
where the
guv0
and the
g0uv
take the values
-1
0
0 0
0
-1
0 0
0
0
-1
0
0
0
0 -1
(82a)
The
huv
and the
huv
are
treated
as infinitesimally
small
quantities
of
first
order;
and
neglecting
infinitesimals
of
second order
they obey
the relations
huv
= -huv.
The time coordinate is chosen
as
purely imaginary (as
Minkowski
also
does)
and
by
this
we accomplish
that
(g44)0 =
g044
=
-1 and
also the covariance of
the
systems
of
equations
under linear
orthogonal
transformations. An
imaginary
choice
for
the
time coordinate makes
g14, g24, g34
and also
-g
imaginary.
But the
validity
of the
equations
we
have
developed
is
retained,
because
one can go
from
a
real-valued to
an imaginary-valued
time variable
by means
of
a
linear transformation.
Fixing
the
values
as was
done in
(82a)
achieves the
agreement
of
naturally
measured
lengths
with coordinate
length
in the domain
of
consideration,
up
to infinitesimals
of
first
order.
We
now replace equations
(81)
and
(81a)
with others
in which
infinitesimally
small
quantities
of
second and
higher
order
are neglected.
tvo
will then vanish and
we
obtain