DOC.
13
GENERALIZED THEORY OF RELATIVITY
157
physical meaning (measurability
in
principle)
of the coordinates
x1,
x2,
x3, x4.
We
note in this
connection that ds
is to
be conceived
as
the invariant
measure
of the distance between
two
infinitely
close
space-time
points.
For that
reason,
ds
must
also
possess
a
physical meaning
that
is
independent
of the chosen reference
system.
We will
assume
that ds
is
the
"naturally
measured" distance between the
two
space-time points,
and
by
this
we
will understand the
following.
The immediate
vicinity
of the
point
(x1,
x2,
x3,
x4)
with
respect
to
the coordinate
system
is
determined
by
the infinitesimal variables
dx1, dx2, dx3, dx4.
We
assume
that,
in their
place,
new
variables
ds1
ds2, ds3,
ds4 are
introduced
by
means
of
a
linear transformation in such
a
way
that
ds2
=
ds21
+
ss22
+
ds23
-
ds24.
In this transformation the
guv are
to
viewed
as
constants;
the real
cone
ds2
=
0
appears
referred
to its
principal
axes.
Then the
ordinary theory
of
relativity
holds
in
this
elementary
ds,
system,
and the
physical meaning
of
lengths
and times shall be
the
same
in this
sytem
as
in
the
ordinary theory
of
relativity, i.e.,
ds
is
the
square
of
the four-dimensional distance between
two
infinitely
close
space-time points,
measured
by
means
of
a rigid body
that
is not
accelerated
in
the
ds
-system,
and
by
means
of unit
measuring
rods and clocks
at rest
relative
to it.
From this
one sees
that,
for
given
dx1, dx2, dx3,
dx4,
the natural distance that
corresponds
to
these differentials
can
be determined
only
if
one
knows the
quantities
guv
that determine the
gravitational
field. This
can
also be
expressed
in the
following way:
the
gravitational
field influences the
measuring
bodies and clocks
in
a
determinate
manner.
From the fundamental
equation
dxu,dxv
one sees
that,
in
order
to
fix the
physical
dimensions of
the
quantities
guv
and
xv, yet
another
stipulation
is
required.
The
quantity
ds has the
dimension
of
a
length.
Likewise,
we
wish
to
view the
xv
(x4
too)
as lengths,
and
thus
we
do not
ascribe
any
physical
dimension
to
the
quantities guv.
§4.
The Motion of
Continuously
Distributed Incoherent
Masses
in
an
Arbitrary
Gravitational
Field
In
order
to
derive the law of motion of
continuously
distributed incoherent
masses,
we
calculate the
momentum
and the
ponderomotive
force
per
unit volume and
apply
the law of the conservation of
momentum.
To this
end,
we
must
first calculate the three-dimensional volume
V
of
our mass
[17]
[18]
[19]
[20]