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.
E. §21. Newton’s theory as a first approximation.
We have already mentioned several times that the special relativity theory is to be looked upon as a special case of the general, in which gμν’s have constant values (4). This signifies, according to what has been said before, a total neglect of the influence of gravitation. We get one important approximation if we consider the case when gμν’s differ from (4) only by small magnitudes (compared to 1) where we can neglect small quantities of the second and higher orders (first aspect of the approximation.)