192
DOC. 30 FOUNDATION OF GENERAL RELATIVITY
which latter
expression
may,
for
reasons
of symmetry,
also
be
written in
the form
-
1/4 [gua
gusd+
eg
seg+]
But
for
this
we
may
set
-
1/4d/dx
(fq
we+gwe
eg
+
gw)
The first
of
these
terms
is
written
more
briefly
-
1/4
d/dx
(FurFuv);
the
second,
after the differentiation is
carried
out,
and
after
some
reduction,
results in
-
1/2
Fur Fvr
gvp
Taking
all three terms
together
we
obtain the
relation
'•
-
w.
-
•
•
(66)
where
Tvo =
-
FoaFva
+
1/4SvoFaßFaB.
Equation (66),
if
ko
vanishes,
is, on
account
of
(30),
equivalent
to
(57)
or (57a) respectively.
Therefore the
Tvo
are
the
energy-components
of
the
electromagnetic
field.
With
the
help
of
(61)
and
(64),
it
is
easy
to
show
that these
energy-components
of
the
electromagnetic
field
in the
case
of the
special
theory of
relativity give
the
well-known Maxwell-
Poynting expressions.
We
have
now
deduced
the
general
laws
which
are
satisfied
by
the
gravitational
field
and
matter,
by
consistently using
a
system
of
co-ordinates
for
which
-
g
=
1.
We have
thereby
achieved
a
considerable
simplification
of formulae
and
calculations,
without
failing
to
comply
with the
require-
ment
of general
covariance; for
we
have
drawn
our equations
from
generally
covariant
equations
by specializing
the
system
of co-ordinates.
[31]