64 DOC. 9
FORMAL
FOUNDATION
OF
RELATIVITY
system,
the
components
of the field
strengths are
proportional
to
these
polarizations.
We obtain the field
equations
for this
case
from
(56)
by adding
on
their
right-
hand sides the
expressions
for the V-four-vectors of the electric and
magnetic
polarization
current,
respectively.
We
represent
the
electric
polarization by a
contravariant V-four-vector
(S£})
whose
components
in the
co-moving
normal
system
are
determined
by
the
equations
Vi,
=
-*«(e)
V«
=
324;
Vi
=
-a«
3*;'
Vi
=
o.
(e)
ö
'
V(e) "(e) O V(e) "(e)
O T'(e)
These conditions
are
met
by
the
equations
pa
dx
= CT(e)E
gapIJ ds
(57)
From this V-four-vector
we
form the V-six-vector
-
Ki
"
»w
(58)
ds ds
and from this the contravariant V-four-vector of the electric convection
current
E
&
(59)
V
°xv
by forming
the
divergence according
to
(40).
We
note
that the
components
of this
vector
in the normal
system
are
a@ce)
c~)
a(Cfr)
e~)
d(t7(e)
c~)
/
+
a(açej
c~)
+
a(Cfr)
cJ)
EU
`
&
`
ay
Therefore,
if
(59)
is
added
on
the
right-hand
side of the first
equation (56),
one
obtains
equations
which,
in
the normal
system,
transform
into the
first
Maxwellian
equation system
for bodies
at rest.
This
justifies
statements
(57), (58), (59).
For the
magnetic polarization
we
set,
in
analogy,
{13}
Ȁ
-»"ESrfST^f-
(57a)
W
"
Ȁ
^
(58a)
ds ds
[p. 1066]
which
yields
the
components
of
the
V-four-vector
of
the
magnetic polarization
current
as
E
(59)
v
OXv
One obtains in this
manner
the
field
equations