The first of these terms can be written shortly as
- 1/4 ∂/∂xσ (Fμν Fμν),
and the second after differentiation can be transformed in the form
- ½ Fμτ Fμν gνρ ∂gστ/∂xσ.
If we take all the three terms together, we get the relation
(66) Kσ = ∂τσν/∂xν - ½ gτμ ∂gμν/∂xσ ττν
where
(66a) τσν = -Fσα Fνα + 1/4 δσν Fαβ Fαβ.
On account of (30) the equation (66) becomes equivalent to (57) and (57a) when Kσ vanishes. Thus τσν’s are the energy-components of the electro-magnetic field. With the help of (61) and (64) we can easily show that the energy-components of the electro-magnetic field, in the case of the special relativity theory, give rise to the well-known Maxwell-Poynting expressions.
We have now deduced the most general laws which the gravitation-field and matter satisfy when we use a co-ordinate system for which √(-g) = 1. Thereby we achieve an important simplification in all our formulas and calculations, without renouncing the conditions of general covariance, as we have obtained the equations through a specialisation of the co-ordinate system from the general covariant-equations. Still the question is not without formal interest, whether, when the energy-components of the gravitation-field and matter is defined in a generalised manner without any specialisation of co-ordinates, the laws of conservation have the form of the equation (56), and the field-equations of gravitation hold in the form (52) or (52a); such that on the left-hand side, we have a divergence in the usual sense, and on the right-hand side, the sum of the energy-components of matter and gravitation. I have found out that this is indeed the case. But I am of opinion that the communication of my rather comprehensive work on this subject will not pay, for nothing essentially new comes out of it.