106
DOC.
21
GENERAL RELATIVITY
[p. 786] §4.
Some
Remarks
on
the
Physical Qualities
of
the
Theory
A first
approximation
of
the
equations (22a)
is
iß
dxadxß
=
0.
This does not
yet
fix the coordinate
system,
because this would
require
4
equations.
We
are
therefore entitled to
put
for
a
first
approximation arbitrarily
{3}
EB
sgaB/B
sXB
=
0.
For further
simplification we
want
to
introduce the
imaginary
time
as a
fourth
variable. The
field
equations
(16a)
then
take,
as a
first
approximation,
the form
1/2Ea s2gmv/sXa2
= kTmv,
from which
one sees immediately
that it contains
Newton’s
law
as an approxima
tion.
That the
new
theory complies
with the
relativity
of
motion follows from the fact
that
among
the
permissible
transformations
are
those that
correspond
to
a
rotation
of
[6]
the
new
relative to the old
system
(with
arbitrarily
variable
angular velocity),
and also
those
where the
origin
of the
new system performs an
arbitrarily prescribed
motion
relative to that
of
the old
one.
Indeed,
the substitutions
(22)
(16b)
x
=
x
cos
t +
y sin
r
y'
=
x
sin
t +
y cos
r
z'
=
z
t'
=
t
and
X'
=
X

T1
y
=
y

t2
z'=
z

T3
t'
=
t,
where
r
and
T1, T2, T3
respectively are arbitrary
functions
of
t
and substitutions with
the determinant
1.