112
DOC.
4
THEORY OF STATIC GRAVITATIONAL FIELD
density
of the
electromagnetic energy.
Thus,
we
will obtain the
equation
that
corresponds
to
the
energy principle through
scalar
multiplication
of
the first
of
equations (1a)
by
cS
and the third
by
c§,
adding
the
two,
and then
integrating
over
an
arbitrary
closed
space.
We obtain then in the familiar fashion:
[14]
(3)
ft
c
@
pdr
+ (@2 +
tf)dr\
=
f
[cS,
c£]ndo,
dt
if
dx denotes the element of the
space,
do
the element of the
boundary surface,
and
n
its inward-directed normal.
Thus,
the
energy principle
is
satisfied,
with the
vector
c2
[©,$]
equal
to
the
energy
flux.
We
now
derive the
momentum
law
by
vector
multiplication
of the first of
equations
(1a)
by
S
and the third of these
by
-S,
and then
adding.
If
we
take
as
the
expressions
for the Maxwell
stresses
A-, =
«?
0!
-
\
e2
-
^02i,
x,
-
c(«,e,
+
0,0,),
x,
-*
«,)
etc.,
we
get
(4)
p{cGx +
[v,0L
+
4
[S&]r
=
fc
+ +
dX
Z
/
1
(@2
+
^2)|£ox
2
dt
v
dx
dy
dz
[15]
[16]
as
well
as
the
equations
obtained from this
by cyclic permutation.
The first term
of
this
equation expresses
the
X-component
of the
quantity
of
momentum
that the
electric
masses
impart
to
the
system's ponderable masses per
unit time and unit
volume.
Thus,
up
to
the factor
c,
the
expression
for the
ponderomotive
force
is
the
one given by
H. A. Lorentz. The second
term
of the left-hand side
expresses
the
increase in the
momentum
of
the volume unit.
If
the
spatial
derivatives
of
c
vanish,
i.e.,
if
no
gravitational
field
is
present,
then the increase in the
electromagnetic
momentum
of the volume unit that
corresponds
to
the left-hand side is caused
by
electromagnetic
stresses,
as
is
the
case
in
electrodynamics
without
taking
into
account
the
gravitational
field.
However,
in the
case
where
a gravitational
field is
present,
then it follows from the last
term
of the
right-hand
side that this
term must
be
regarded
as
the
source
of the
momentum
for the
electromagnetic
field. The
electromagnetic
field
energy
receives
a
momentum
from the
gravitational
field
exactly
like
a
ponderable
mass
at
rest;
for
it
has been
found
in the
previous paper
that
the
gravitational
field transfers
per
unit time the
momentum
-m
grad
c
to
the
mass
at
rest
m.
Thus,
it
turns out,
for
example,
that
cavity
radiation
possesses
a
gravitational
mass
that
corresponds exactly
to
its inertial
mass;
this result is
already
contained in
equations
(1a)
and
in
the
expression
for the
ponderomotive
forces
acting on
electrical
quantities,
since the last
of
the above-written
equations
for the
momentum
is
a