DOC.
4
THEORY OF
STATIC GRAVITATIONAL
FIELD
115
/adr
over a
space
for which
c
is constant at
infinity,
then the
principle
of
equality
of
action
and reaction
requires
that this
integral
vanish. Otherwise the
totality
of the
masses
in the
space
under
consideration,
which
we
will
imagine
to
be
affixed to
a
rigid
massless
framework,
would strive
to start
moving.
But
according
to
(4)
and
(3a)
JSdt=-|agradcdt=-1/k|Ac/cgradcdt,
and
it is
easy
to
prove
for the last of these
integrals
that it in
general
does
not
vanish.
Thus,
we
have arrived
at
a
rather
questionable result,
which
is
likely
to
raise
doubts
as
to
the
admissibility
of the entire
theory
that
we
have
developed
here. It
is
certain that this result
points
to
a deep
lacuna in the
very
basis of
our
two
investiga-
tions;
for it would
scarcely
be
possible
to
infer from the
expression
c0
+ ax,
which
we
found for
c
of the
uniformly
accelerated
system,
an
equation worthy
of
consideration other than
equation (3)
that,
for its
part,
entails of
necessity equation
(3a).
To resolve this
difficulty,
at
first
one
would feel
inclined,
in view
of
the results
of the old
theory
of
relativity,
to
ascribe
a
gravitational
mass
to
the framework
subjected
to
stresses,
so
that forces that the
gravitational
field
exerts
on
the
parts
of
the
framework
subjected
to stresses
would be added
to
the forces that
it exerts
on
the
masses
of
density
o.
However,
the
following argument
leads
to
the
rejection
of such
a hypothesis.
Let
a
box with
reflecting
walls be situated in
a
static
gravitational field,
and let
radiation,
the
energy
of
which,
measured with
a
"pocket
instrument,"
is
E,
be
enclosed in the
box;
i.e.,
let
E=1/f(S2+S2)dr
2
If
the dimensions of the box
are
small
enough,
then
it
follows from
equation (4)
of
this
paper
that the
sum
of the forces exerted
by
the radiation
on
the walls of the box
possesses
the
value
-E
grad c.
[22]
This
sum
of
forces
must
be
equal
to
the resultant of the forces that the
gravitational
field
exerts
on
the whole
system (box together
with
radiation)
if the box
is
massless,
and if
the circumstance that the walls of the box
are
subjected
to stresses
due
to
radiation
pressure
does
not
result in the
gravitational
field
acting on
the
walls of
the
box. If the latter
were
the
case,
the resultant of the forces exerted
by
the
gravitational
field
on
the
box
(together
with
its content)
would differ from the value -E
grad c,
i.e.,
the
gravitational mass
of the
system
would be different from
E.
On the other
hand,
if
our
radiation box
is
situated in
a
space
of
constant
c,
then
the
results
of
the old
theory
of
relativity
will hold for
it.
In
particular,
it follows then