134
DOC. 27 MAXWELL'S
EQUATIONS
(1)
+
curl
c
=
0
dt
div
f) =
0
(2b)
These last
equations can
be retained in the
case
of the
general theory
of
relativity
if
one stays
with the
defining equations (3), i.e.,
when the six-vector
(e,
f))
is treated
as
a
covariant six-vector.
Regarding
the first
system
of
Maxwell's
equations,
we
stay
with the
generaliza-
tion of
Minkowski's
scheme,
which has been elaborated
upon
in
§11
of the
repeatedly
quoted paper.
We introduce the covariant V-six-vector
tf
=
fg
Tg^S^F,aß
aß
?
*
(4)
and demand that the divergence of this contravariant six-vector equals the
contravariant V-four-vector
ZJ"
of the electric current density in vacuum, viz.,
(5)
This
system
of
equations
is
truly equivalent
to the first
system
of Maxwell,
as can
be
seen by calculating
from
(4)
for the
case
of
special relativity
where the
have the values
-1
0
0
-1
0 0
0 0
(3)
and
(4) yield
for this
special
case
C
=
S
=
fc
0 0
0 0
-1
0
0
+1
f:
tr~-
-e
-e
-e
(6)
By setting
in addition
31-«»
32
= iv,
S3
=
i,
34=p
(7)
(5)
takes the familiar
form
curl
ff
-
=
i
dt
div
e
=
p
(5b)
[p. 187] Equations
of the form
(5b)
also
apply
in the
general theory
of
relativity,
but the
(three-dimensional)
vectors e
and
f)
are
no
longer
the
same
as
in
(2b).
One rather