DOC.
13
GENERALIZED THEORY OF RELATIVITY
153
set
(1a)
where
Sijfídt) =
0,
H
ds
--m
dt
is
posited,
if
m
designates
the
rest
mass
of
the material
point.
From this
we
obtain,
in
the familiar
way,
the
momentum
Jx,
Jy,
Jz,
and the
energy
E of the
moving point:
dH
x
(2)
m
dx
m
etc.
dH
.
dH
.
dH
. -x + -v + -z
dx
dy
dz
H
=
m-
This mode of
representation
differs from the
customary one only by
the fact that
in the latter
Jx,
Jy,
Jz,
and E contain also
a
factor
c.
But since
c
is constant
in the
customary theory
of
relativity,
the
system given
here
is
equivalent
to
the
ordinary
one.
The
only
difference
is
that J and E
possess
dimensions other than those
in
the
customary
mode of
representation.
I
have shown in
previous
papers
that the
equivalence hypothesis
leads
to
the
consequence
that in
a
static
gravitational
field the
velocity
of
light
c
depends
on
the
gravitational potential.
This
led
me
to
the view that the
customary theory
of
relativity
provides
only
an
approximation
to
reality;
it
should
apply only
in the limit
case
where differences
in
the
gravitational potential
in the
space-time region
under
consideration
are
not too
great.
In
addition,
I
found
again equations (1)
or
(1a)
as
the
equations
of motion of
a mass
point
in
a
static
gravitational
field; however,
c
is not
to
be conceived of here
as a
constant
but rather
as a
function of the
spatial
coordinates that
represents a measure
for the
gravitational potential.
From
(1a) we
obtain in the familiar fashion the
equations
of motion
dc
(3)
mc
dx
íe
It
is
easy
to
see
that the
momentum
is
represented
by
the
same
expression
as
above.
In
general, equations (2)
hold for the material
point moving
in the static
gravitational
field. The
right-hand
side of
(3) represents
the force
Rx
exerted
on
the
mass
point by
the
gravitational
field. For the
special case
of
rest
(q
=
0) we
have
dc
St
=
-m
dx
[7]
[8]
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