158
DOC.
13
GENERALIZED THEORY OF RELATIVITY
point.
We consider
an
infinitely
small
(four-dimensional)
piece
of the
space-time
thread of
our
material
point.
Its volume is
Iff
/dx1dx2dx3dx4
=
Vdt.
If
we
introduce the natural differentials
ds
in
place
of the
dx, assuming
that the
measuring body
is at rest
with
respect
to
the material
point,
we
have
to set
fffdsdsds3
=
VQ,
i.e., equal
to
the "rest volume" of the material
point.
Further,
we
have
fds4
=
ds,
where ds has the
same
meaning
as
above.
If the
dx
are
related
to
the
ds, by
the substitution
dxu =
£
aM°d^
a
then
we
have
d(dx1dx2,dx3,dx4)
or
Vdt
=
V0ds-apa
But since
=
E
=
+
^2
+
-
dt4,
/xv /xv p a
there obtains the
following
relation between the determinant
&
= IguvI,
i.e.,
the discriminant of the
quadratic
differential form
ds2,
and the substitution
determinant
|otpa|:
8
'
(Kai)2
=
^
pa
1
V
Thus,
one
obtains
the
following
relation for
V:
Vdt
= V0ds
From this
one
obtains with the
help
of
(7), (8),
and
(9),
if
one
substitutes
p0
for
m/
vo