190
DOC. 30 FOUNDATION OF
GENERAL RELATIVITY
dF
dF34
dF42
23
=
0
+
dx2
dx3
dF41
dx1
+
dx3 dx4
=
0
dF41
dF12
dx2
+
dX4
+
dx1
=
0
dF12
+
dF23
+
dF31
dx2
=
0
.
(60a)
[30]
This
system corresponds
to the
second
of
Maxwell's
systems of
equations.
We
recognize
this
at
once by
setting
F23
F31
F12
=
Hx,
F14
=
Hy,
F
Hz,
F
Ex
Ey
Ez
(61)
Then in
place
of
(60a) we
may set,
in the usual notation of
three-dimensional vector
analysis,
aH
It
div
H
=
0
=
curl E
(60b)
We
obtain
Maxwell's first
system by
generalizing
the
form
given by
Minkowski.
We introduce the
contravariant
six-vector associated
with
Faß
Fuv
=
guagvBFaß
(62)
and also
the
contravariant vector Ju of
the
density
of
the
electric current. Then,
taking
(40)
into
consideration,
the
following
equations
will be
invariant
for
any
substitution
whose
invariant
is
unity (in
agreement
with
the
chosen
co-
ordinates):-
*
F*"
=
J1*
. .
. .
(63)
Let
7xl
F23
=
H'x,
F14
=
-
E'x
F31
=
H'y, F24
= -
E'y
F12
=
H'z, F34
=
-
E'z
(64)
which
quantities are
equal
to the
quantities
Hx...Ez
in