DOC.
13
GENERALIZED THEORY OF RELATIVITY
183
(40)
e"-E4'«,.)•3Xß
We derive the dual contravariant
tensor
of second rank
©*
from
a
contravariant
tensor
of the second rank
©uv
in the
following
way.
First
we
form the
complement11
(41)
^ik
e
ik
/xv
'
®/xv'
^
/XV
or,
thus,
Tn
=
fg-®
34'
"^13
Jfi
'
®
42'
^14
"
'
@23'
(41a)
^23
'
®14'
"^24
'
®31'
-^34
' ®12*
The dual
tensor
that
is
being sought
is
the
reciprocal
of this
complement,
and
thus has the form
(42)
=
E
Tik
=
4E
yirykse,kßv
V
ils
^
^ i'fc/xv
ilr•«%«
Because of the
reciprocity
of the
two
discriminant
tensors,
the
sequence
of the
two
operations-the
construction of
the
complement
and of
the
reciprocal
tensor-can
be reversed.
-
§4.
Mathematical
Supplements
to the
Physical
Part
1.
Proof
of the
Covariance
of the
Momentum-Energy Equations
It has
to
be
proved
that the
equations (10)
of Part
I, page 10,
which,
neglecting
the
factor
yj-1,
read
E
37-(v£
-gar •
@Mv)
-
•
E
= ° a
= 123'4)
/XV
^
Atv
are
covariant with
respect
to
arbitrary
transformations.
According
to
formula
(35),
the
divergence
of the contravariant
tensor
©uv
is
__ ___
fvkl
E
vk
vk
H
The covariant
vector
To reciprocal
to
this contravariant
vector
©u
is thus
11The factor
1/2
serves
to
simplify
the result but
is
inconsequential
from the
point
of
view of the
theory
of invariants.
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