76 DOC.
1
MANUSCRIPT
ON SPECIAL RELATIVITY
system
of
reference. We need
only
one more
equation,
which
expresses
(3*k)
in
terms
of the field
quantities.
If
we
multiply
the first
of
equations
(Ic)
by
(Gm)we
obtain,
in
light
of
(60), (62),
and
49,
[
or
]
(Gy)
d
(SjiV
+Puv)Hv
dx.
j=
(3J)
(GJ
= p0(G|1)(G(l)
=-p0
or
[p. 61]
Po
=
"«V
(
mv
+
mv/Xv).
...(65)
In
conjunction
with
(50), (64),
and
(54)
and
(62),
this
equation
determines
(3^k)
in
terms
of
(mv); this
equation
is
merely
a
consequence
of the first
of
the
vector
equations
(Ic)
and
therefore does
not
need
to
be added
to
the
system
of the
fundamental
equations.
Considering
that
by
virtue of
(64)
and the definitions
given
at
the
beginning
of
the
§,
the
vectors
(^Pnv),
(9W[iv),
and
(3^l)
can
be
expressed
in
terms
of
(Suv)*
and
(G|x), equations (Ic)
contain all of the conditions that the
electromagnetic
field has
to
satisfy
if
H,
e, i,
o are
given
as
functions of
x,
y, z,
t.
For the
system
(Ic)
contains
8
condition
equations
for
p0
and the
6
components f^nv.
The
surplus
condition
expresses
the absence
of
magnetic charges.
It should be noted
that
our
equations
differ
only formally
from those
of
Minkowski.[106]
We
prefer
this
system
because the
quantities appearing
in it have
a
much
more
direct
physical meaning,
and because this
system
lends itself better
to
an
extension
to
cases
in which the connections between field
strength
and
polarization
are
less
simple.
Finally,
it also
turns
out
that the derivation
of
the
conservation laws
(and
the
ponderomotive forces)
becomes
simpler.
We
now
write the
equations
of the
theory
in the form
of
the three-dimensional
vector calculus
by splitting
both of the six-vectors of the
polarizations
(*miv)
and
(9W^v)
into two
ordinary
vectors
p*,
p**
and
m*, m**,
respectively.[107]
Instead of
the
field
equations
(Ic),
we
then obtain
**
»
,
•
lö(c
+
P )
1
•
q
rot
(t)
+
p
)--
^
=
-» +
P-
**
1
/•
div
(C
+
p ) =
-
(U
-)
+
P
c c
.
iö(t)
+
m
)
rot
(e
-
m
)
h
=r
=
0
c
ot
y (1d)
div
(h +
m
)
=
0
J
The
vectors
of
polarization are
connected with those of the
electromagnetic
field
by
the
following
chain of
relations