50 DOC.
9
FORMAL FOUNDATION OF
RELATIVITY
dg,MV
dxa
_ =
E
or
&
dx
gang
aii&rv
(35)
dg
fiv
dxa
=
-E dg
dx
g
a/1g
TV
(36)
or
a
Extension
and
divergence
of
a
four-vector.
The extension of the covariant four-
vector is
given by (28a). By exchanging
the
indices
u
and
v
and
subtracting, one
obtains the
anti-symmetric
tensor
^
/xv
A
V/x
dx
dV
(28b)
The extension
Auv
of
the contravariant
four-vector
Au follows
from
(30)
as
am
_
dA»
v~äT
E
VT
AT.
The
divergence
of
this is
4
=
es;-E
/XV /X
dA*
dx..
ILT
/1T
or,
due to
(33),
*
=
^E
a
\[g
M-
").
Ö(/
(37)
Now
replacing
Au with the contravariant
vector
Eguvdo/dxv, where o denotes
dxv
a
scalar,
one
obtains the well-known
generalization
of
the
Laplacian
Acf:
[p.
1052]
*
-
EJ
a
/IV
\/i
dxi
Jgg
/xv
50
(38)
Extension
and
divergence
of
a
tensor
of
rank
two.
Applied
to covariant and
contravariant tensors of rank
two, (29)
and
(30)
resp. produce
tensors
of
rank three.
/xvs
_
'
t
\
us\ \vs 7K
-
T
/XT
(29a)
dA ^
\ST Ar
-
V
?
I
TV
+
ST
V
^4/XT
(30a)