DOC. 30
FOUNDATION OF GENERAL RELATIVITY 189
with the
given
equation
between
p
and
p,
and
the
equation
dxa
dx&
-«
-ar
-
are
sufficient,
gaB
being given,
to
define
the
six
unknowns
dx1
dx2 dx3
dx4
p'
p,
ds,
ds, ds, ds
If
the
guv
are
also unknown,
the
equations
(53) are
brought
in.
These
are
eleven
equations
for
defining
the ten
functions
guv,
so
that these functions
appear
over-defined.
We must
remember,
however,
that the
equations
(57a) are
already
contained in the
equations (53), so
that the latter
represent
only
seven independent equations.
There
is
good
reason
for
this
lack of
definition,
in
that
the
wide
freedom
of
the
choice of
co-ordinates
causes
the
problem
to remain
mathematically
undefined
to such
a degree
that three
of
the
functions
of
space
may
be chosen at will.*
§
20.
Maxwell's
Electromagnetic
Field
Equations
for Free
[28]
Space
Let
Qv
be
the
components
of
a
covariant vector-the
electromagnetic potential
vector.
From them
we
form,
in
accordance
with
(36),
the
components
Fpo
of
the covariant
six-vector of
the
electromagnetic field,
in
accordance with
the
system
of
equations
Fp'=
tot
" to^
. . .
(59)
It
follows from
(59)
that
the
system
of
equations
=
o
. .
.
(60) [29]
is
satisfied,
its
left
side
being, by
(37),
an
antisymmetrical
tensor of
the third rank.
System
(60)
thus contains essenti-
ally
four
equations
which
are
written
out
as
follows:-
*
On the
abandonment
of
the
choice of co-ordinates with
g
=
-
1,
there
remain
four
functions
of
space
with
liberty
of
choice, corresponding
to
the
four
arbitrary
functions at
our disposal
in the
choice
of
co-ordinates.