168
DOC.
13
GENERALIZED THEORY OF RELATIVITY
with
respect
to
a
comoving body
(rest volume).
We define
e/dv0
=
p0
as
the
true
density
of the
electricity;
this
is
a
scalar
by
definition. Hence
[38]
dx
(V =
1,2,3,4)
ds
is
a
contravariant
four-vector,
which
we
reformulate
by defining
the
density
p
of the
electricity,
referred
to
a
coordinate
system,
by
the
equation
p0dv0
=
p
dV.
Using
the
equation
dV0ds
=
\f^g'dV'dt
from
§4, we
obtain
dxv
=
i
dxv
Po
ds
,/T7P
dt
'
i.e.,
the contravariant
vector
of the electric
current.
We reduce the
electromagnetic
field
to
a
special,
contravariant
tensor
of second
[39]
rank
puv
(a
six-vector),
and form the "dual" contravariant
tensor
of second rank
pUv
by
the method
explained
in
Part
II,
§3
(formula 42). According
to
formula 40 in
§3
of Part
II,
the
divergence
of
a special
contravariant
tensor
of second rank
is
4=E
V
UA^
v
[40]
As
a
generalization
of the Maxwell-Lorentz field
equations,
we
set
up
the
equations
"
7)
, ,
dx
(23) 52
=
(dt
=
dx4)
V
~-vv
(24)
-p;,)
=
0,
V
UJiv
the covariance of which is self-evident. If
we
set
V~~9
•
923
=
§x,
.
Psi
=
§y,
V-
yzrj.
9u-
@x,
-
e
y^~g
-pu=-
and
dx^
_
P
dt
~
U
then the
system
of
equations
(23),
written
out
in
a more
detailed
manner,
takes
the
form