5 0 D O C . 5 2 O N A F F I N E F I E L D T H E O R Y

must also be a complete differential of a quantity H* (scalar density), which we

want to consider as a function of the and .

Thus

(9)

must be set, whereby equations (9) may completely be substituted for equations

(3a).[6]

We now only still have to choose the scalar density H* as a function of the

and . The most general possible form is

, (10)

whereby Φ stands for an arbitrary function of th e two known invariants of the elec-

tromagnetic field. The most natural approach according to our present knowledge

reads

. (10a)

If one substitutes the left-hand sides of equations (1) and (2) on the basis of

equations (9) and (10a) and the right-hand sides on the basis of equations (8) by

expressions in the field quantities, one obtains the equations

(11)

(12)[7]

Equations (6), (11), and (12) are the field equations of the theory developed here.[8]

Up to now we have been using units of unknown magnitude for the field inten-

sities of the metric and electric fields. Upon conversion into the gram-centimeter

system, the field equations assume the form

(11a)

(12a)

where α and β mean other constants; κ is the gravitational constant.

γμνdgμν φμνdfμν

+

gμν fμν

γμν

∂H*

∂gμν

---------- -=

φμν

∂H*

∂fμν

---------- =

fμν gμν

H* –gΦ(J1, J2) =

[p. 139]

H* 2α –g

β

2

--fμνfμν - –+=

Rμν αgμν – β© fμσfν

σ

–

1

4

--gμνfστfστ¹ -

+

§ ·

1

6

--iμiν -

+ –=

–βfμν

1§

6©

--¨ -

∂xν

∂iμ

∂xμ¹

∂iν

–

¸

·

. =

Rμν αgμν – κ fμσfν

σ

–

1

4

--gμνfστfστ¹

- +

©

§ ·

1

β

--iμiν - + =

–fμν

1

β©

-- -

∂xν

∂iμ

∂xμ¹

∂iν

–

¨ ¸

§ ·

, =