62
DOC. 9 FORMAL FOUNDATION OF RELATIVITY
when
we integrated
this
subject
matter
into the
original theory
of
relativity; we
can,
therefore,
be brief here.
[p.
1063]
Electromagnetic equations
in
vacuum.
Let
$ and
if
be
two
dual and
[33]
contravariant V-six-vectors
(see (24)).
It then follows from
(40)
that the
expressions
^
dx
'
^
dx
are
the
components
of
contravariant
V-four-vectors. One obtains the Maxwellian
equations
for
a vacuum
in
a generally-covariant
form if
one
sets
these
components
equal
to
zero.
It is indeed
easily
seen
that these
equations
go
into their Maxwellian
uy
*
form
if
one
denotes the
components
of
f$|
and $
according
to the scheme
If
If
if if if
If
r
if*
If* if* if* if*
% -*X
-*z
*
-«X
-e*y
-i
-o;
and
observes, furthermore,
that
according
to
(24)
t
=
t
whenever the
guv
take the
special
values
given
in
(18).
Charge density,
convection
current.
There is
obviously an
electrical
charge
density
in the
co-moving
normal
system.
This
density
is
a
scalar
by
definition.
Multiplying
it with
v/-g
produces
a
V-scalar
which
we
denote
by
p^.
Together
with
dx
the contravariant four-vector
-it
forms
the contravariant
V-four-vector
of
the
ds
convection
current
P(e)
dx
ds
The
Lorentz
equations in
vacuum.
When
all
interactions between matter
and
electromagnetic
fields
are
reduced
to
the movement
of
electrical charges-as Lorentz
does-one
has
to
base
everything
upon
the
equations
V
dxv
ds
E
_
V
dxv
=
0
(54)
They are
the
fundamental
equations
of
Lorentz's
theory
of
the
electrons
in
general-
[p. 1064]
covariant form.
They
explain by
which laws the
gravitational
field
acts
upon
the
electromagnetic
field.