DOC.
9
FORMAL FOUNDATION OF RELATIVITY
83
For
light rays
(ds
=
0),
one
has
dx2
+ dy2 +
dz2
=
1
+
o
dt
The
speed
of
light
is, therefore,
independent
in
its direction but varies with the
gravitational potential,
and
consequently one
has
a
curved
progression
of
light rays
in
a gravitational
field.
Finally,
we
calculate momentum and
energy
of
a
material
point
in
a
Newtonian
field for which
we
do not
use
the
equations
(86a)
but rather the
rigorous equations
(51).
If
we
substitute
into these for the
gou
the
values
-1
0
0
0
0 -1
0 0
0
0 -1
0
0 0
0 -1
+
h44
and for the
dxv
the
quantities
dx
dy dz
idt,
furthermore
limiting
ourselves for
h44
to
quantities
of
first
order,
and
replac-
ing
(h44/2)
with
o,
and neglecting terms of higher than second order in the velocities,
one
obtains
"Ii
=
m(
1
- ß)q;
[p. 1085]
il4
=
m
(W)
|(1
- f)q2
Since
(-I1)
is the
X-component
of the
momentum
and
(iI4)
is
the
energy
of
the
material
point,
one comes
to
the conclusion
that
the inertial
mass
increases
with
diminishing gravitational potential.
This is
very
well in
agreement
with the
spirit
of
the
interpretation
taken here. As there
are
no
independent physical qualities
of
space
in
our
theory,
the
inertia
of
mass
is
a consequence
of
the
mutual action between each
mass
and all the other
masses.
This interaction
must,
therefore,
increase when other
masses
are brought
closer
to
the
mass
that is under
consideration, i.e.,
if 0 is
decreased.
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