154
DOC.
13
GENERALIZED THEORY OF RELATIVITY
From this
one sees
that
c
plays
the role of the
gravitational
potential.
From
(2)
it
follows that for
a
slowly moving point
J
mx
x
C
,
(4)
1
2
-mq
E
-
mc
=
-
At
a given
velocity,
the
momentum
and the kinetic
energy
are
thus
inversely
proportional
to
the
quantity c;
in other
words:
the inertial
mass,
as
it
enters
into the
momentum
and
energy,
is
m/c,
where
m
denotes
a
constant
that is characteristic
of
the
mass
point
and
independent
of the
gravitational potential.
This
is consonant
with
Mach's
daring
idea that inertia has
its
origin
in
an
interaction between the
mass
point
under consideration and all of the other
mass
points;
for if
we
accumulate
masses
in
the
vicinity
of the
mass
point
under
consideration,
we
thereby
decrease the
gravitational potential c,
thus
increasing
the
quantity
m/c
that
is
determinative of
inertia.
§2.
Equations
of Motion
of the Material Point
in
an
Arbitrary
Gravitational
Field.
Characterization
of the
Latter
By introducing
a
spatial variability
of the
quantity
c,
we
have breached the frame of
the
theory presently designated
as
the
"relativity theory";
for
now
the
expression
designated by
ds
no
longer
behaves
as an
invariant with
respect
to
orthogonal
linear
transformations of the coordinates.
Thus,
if the
relativity
principle
is
to
be
maintained-which
is not to
be
doubted-then
we
must
generalize
the
relativity
theory
in
such
a
way
that the
theory
of the static
gravitational
field whose elements
have
been indicated
above will be
contained in
it
as a
special
case.
If
we
introduce
a new
space-time system K'(x', y',
z',
t') by
means
of
an arbitrary
substitution
[10]