D O C . 5 2 O N A F F I N E F I E L D T H E O R Y 4 9
(3a)
Let and be conceived of as tensor densities of the metric and electric
fields. If one plugs the expressions for and given by (1) and (2) into (3),
one obtains by variation the equations
(4)
where is the covariant extension of the tensor density in accordance
with the equation
(5)
and is the tensor density to be interpreted as the current density
. (6)
We introduce as metric tensor , or , that tensor which belongs to the sym
metrical tensor density , in accordance with the equations
(7)
This tensor is used as in Riemannian geometry in order to switch over from covar
iant tensor characters to contravariant ones and vice versa. In this sense, contravar
iant tensor and covariant tensor belong to the current density ; contravari
ant tensor and covariant tensor belong to the field density
.[4]
By means of such operations it is possible to resolve equations (4) for in
the familiar way, whereupon one obtains:
.
(8)[5]
Furthermore, we note that in accordance with (3a), and are functions of
the and in such a way that
is a complete differential. From this it follows that
∂γμν
∂H
gμν
=
∂φμν
∂H
fμν =
gμν fμν
γμν φμν
gμν;α
1
2
gμσ;σδα  ν –
1
2
gνσ;σδα  μ –
1
2
iμδα  ν –
1
2
iνδα  μ – 0 =
gμν;α gμν
[p. 138]
gμν;α
∂xα
∂gμν
gσνΓσα μ gμσΓσα ν gμνΓασ σ – + + =
iα
iμ
∂xσ
∂
fμσ =
gμν gμν
gμν
gμν gμν
g –= g gσσ = ( ).
gμσgνσ δμν =
iμ
iμ
iμ
fμν fμν fμν
Γμν α
Γμν
α
1
2
gαβ¨

∂xγ
∂gμβ
∂xμ
∂gνβ
∂xβ
∂gμν·
–+
© ¹
¸
§
1
2
gμνiα
 –
1
6
δμ 
αiγ
1
6
δγ 
αiμ
+ + =
gμν fμν
γμν φμν
gμνdγμν fμνdφμν +