DOC.
23
PROPAGATION OF LIGHT
383
(1b)
M'
-
M
=
c2
The
increase
in
gravitational
mass
is
thus
equal
to
E/c2,
thus
equaling
the
increase
in
inertial
mass
obtained
from
the
theory
of
relativity.
This
result
follows
even
more
directly
from
the
equivalence
of the
systems
K and
K',
according
to which
the
gravitational mass
with
respect
to
K
is perfectly equal
to
the
inertial
mass
with
respect
to
K';
hence, energy
must
possess
a gravitational mass
that
is
equal
to its inertial
mass.
If
a mass
M0
is suspended
from
a
spring
balance
in
the
system
K',
the balance
will
indicate
the
apparent
weight
M0y
because of the inertia
of
M0.
If the
energy
quantity
E
is
transferred
to
M0,
the
spring
balance
will indicate
M
+
-
° 2
Y,
in
accordance
with
the
principle
of
the
inertia of
energy.
According
to
our
basic
assumption,
exactly
the
same
thing
must
happen
if the
experiment is repeated
in
the
system
K,
i.e.,
in
the
gravitational
field.
§ 3.
Time
and
the
Velocity
of
Light
in the Gravitational Field
If the radiation emitted
in
S2
toward
S1
in
the
uniformly
accelerated
system
K'
had
the
frequency
v2
with
respect
to
a
clock
located
at
S2,
then
upon
its arrival at
S1,
its
frequency
with
respect
to
an identically
constituted
clock
located
at
S1
will
no
longer
be
v2,
but
a
larger frequency
v1,
such
that,
to
a
first
approximation,
(2)
v, =
v
2
Y
h
1
+
c2
For
if
we again
introduce the nonaccelerated reference
system
K0,
relative to which
K'
has
no velocity
at
the time the
light
is emitted,
then the
velocity
of
S1
with
respect
to
K0
will
be
y(h/c)
at
the time the radiation
arrives at
S1,
and from
this
we
immediately
obtain the relation
given
above with
the
help
of
Doppler's
principle.
According to
our
assumption
of the
equivalence
of the
systems
K and
K', this
equation
also
holds
for the
coordinate
system K,
which is
at rest
and
is
endowed
with
a
uniform
gravitational
field,
if
the radiation transfer described
above
takes
place
in
it.
Thus,
it follows
that
a ray
of
light
emitted
at
a given gravitational
potential
in
S2,
and
possessing
at
its emission
the
frequency
v2-compared with
a
clock
located
at
S2-will
possess
a
different
frequency
v1
at its arrival at
S1,
if this
frequency
is
measured
by
an
identically
constituted
clock
located
at
S1.
We substitute for
yh
the
gravitational
potential
G
of
S2,
with
respect
to
S1
as
the
zero
point,
and
assume
that
our
relation,
which
was
derived
for
the
homogeneous gravitational
field,
holds for
otherwise constituted
fields
as well; we
have
then