2 7 4 D O C . 2 8 2 O N E D D I N G T O N ’ S T H E O R Y

(11)

or according to (4a)

(11a)

These equations, together with (2), (3), and the equations (1c), determine the field

equations. is the electromagnetic energy tensor of Maxwell’s theory. Let it be

noted that a term of the form could be appended additively to the func-

tion H that would correspond to the “cosmological” term of the general theory of

relativity.

By the calculation indicated one obtains field equations of the form

(12)

where means, as in (37, 2), the contracted Riemann tensor. As concerns the

physical interpretation of these equations, one does in any event have to interpret

as the tensor of the electromagnetic field. The first of equations (12) exactly

corresponds to the familiar field equations of the general theory of relativity in the

case being considered here, that apart from the metric field only an electromagnetic

one exists, but that to which another energetic term is added that is determined by

the current density. The second of equations (12) seems to directly contradict ex-

perience, for it requires that the electromagnetic field vanish wherever the charge

density vanishes.

This objection however does not hold, since in fact we do not know whether very

small densities of the electrically charged masses aren’t associated with electro-

magnetic fields. In order to be able to assess the permissibility of equations (12),

we must furthermore take into account that the unit of the electromagnetic field has

to be changed if length is supposed to be measured in centimeters, and mass, i.e.,

energy, in grams. We then have to write instead of (12)

dH* β

1

4

--gαβfστfστ - fασfβ¹

σ·

–

©

§

δgαβ fαβδfαβ + –=

γμν

1

4

--gαβfστfστ - fασfβ¹

σ·

–

©

–β§

–βEμν = =

ϕμν βfμν. –=

Eμν

const –g ⋅

Gμν βEμν –

1

6

--iμiν - –=

βfμν

1§

6©

--¨ -

∂xν

∂iμ

∂xμ¹

∂iν

–

¸

·

, =

Gμν

[p. 371]

fμν