84 DOC. 17 GRAVITY AND MATTER
194
ATOMIC
STRUCTURE
term of
this
equation
lies
in its
causing
the
scalar
of
the
left-
hand
side,
T(G",
-
iflwG),
to vanish
identically,
as
the
scalar
guvTuv
of
the
right-hand
side
does
by
reason
of
(3).
If
we
had
reasoned
on
the
basis
of
equations
(1)
instead of
(1a), we
should,
on
the
contrary,
have obtained the
condition G
=
0,
which would have
to
hold
good everywhere
for
the
guv,
independently
of
the
electric
field.
It
is clear that
the
system of
equations [(1a), (3)]
is
a consequence
of
the
system [(1), (3)],
but
not
conversely.
We
might
at first
sight feel
doubtful
whether
(1a)
together
with
(6)
sufficiently
define the
entire
field.
In
a generally
relativistic
theory
we
need
n
-
4
differential
equations,
in-
dependent
of
one
another, for
the definition
of
n
independent
variables,
since in the
solution,
on
account
of
the
liberty of
choice
of
the
co-ordinates,
four
quite arbitrary
functions
of all co-ordinates must
naturally
occur.
Thus
to
define
the sixteen
independent quantities
guv
and duv
we
require
twelve
equations,
all
independent
of
one
another. But
as
it
happens,
nine
of
the
equations (1a),
and
three
of
the
equations
(6)
are
independent of
one
another.
Forming
the
divergence
of
(1a),
and
taking
into
account
that the
divergence
of Guv
-
1/2guvG
vanishes,
we
obtain
1~G
4~T~~J
+
-
=
0
4,c
•
(4a)
From
this
we
recognize
first of
all that
the
scalar
of
curvature
G
in the four-dimensional domains in which the
density of
electricity
vanishes, is constant.
If
we
assume
that
all
these
parts
of
space are
connected,
and
therefore that the
density
of
electricity
differs from
zero
only
in
separate “world-
threads,”
then the scalar
of curvature, everywhere
outside
these
world-threads,
possesses a
constant
value
G0.
But
equation
(4a)
also allows
an
important
conclusion
as
to the
behaviour
of G
within the domains
having
a
density
of
elec-
tricity
other than
zero.
If,
as
is
customary,
we
regard
elec-
tricity
as a
moving
density
of
charge,
by
setting
[15]
y
dxa
•
(7)
jGTg-Pds