DOC.
21
GENERAL RELATIVITY
105
character
only
under
linear
transformations. After
a simple rearrangement, one gets
from
(20a)
and
(19a)
ta =
Ul £
-
£
SMVCrl-.
(20b)
^
pvaß
pva
Finally,
it
is
of
interest to derive two scalar
equations
that result from the field
equations.
After
multiplying
(16a)
by gßv
with
summation
over p
and
v,
we get
after
simple rearranging
£
aß
&8“ß
_
V
çorycc Tß
^
d
(
qßdlgfH
dx"dxa
“Ó
018
Tfí
dxA(
dx,
=
-k£
ra.
(21)
'a^xß
oraß aß 'JAa
'JAß
On the other
hand, multiplying (16a) by
gvA
and
summing
over v,
we get
E
x-fe’X)
-
E
s’"!:/».
-
dx
av
aßv
or,
also
considering
(20b),
d~K8
Lßv) ~
7°,
av
'-"'a
^
fivaß
Taking
(20)
into
account,
and
after
simple rearranging,
this
yields
£
¿-(svirMv)
-
Is;
£
=
-k(t;
+1¿).
dx,
£
aß
a2g«*
dx"dxa
£
8'
maß
rr«
yß
*
aß
^
ra
=
0.
However,
we
demand somewhat
beyond
that:
iß
dxßxß
whereupon
(21)
becomes
EaB
s/sX
E
«"i*!*
=
0,
8
aß
dig\f~g
dxß
/
-k£C
(22)
(22a)
(21a)
Equation (21a)
shows the
impossibility
to
choose the coordinate
system
such
that
\f-g equals
1,
because
the scalar
of
the
energy
tensor cannot be set to
zero.
Equation (22a)
is
a
relation of the
gmv
alone;
it would not be valid in
a
new
coordinate
system
which would result from the
original one by a
forbidden
transformation. The
equation
therefore shows how the coordinate
system
has to be
adapted
to the manifold.
[5]