Page:The principle of relativity (1920).djvu/228

 If we take all the three terms together, we get the relation

(66) K_{σ} = [part]Τ_{σ}^ν/[part]x_{ν} - 1/2 g^{τμ} [part]g_{μν}/[part]x_{σ} Τ_{τ}^ν

where

(66a) Τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{[alpha beta]}.

On account of (30) the equation (66) becomes equivalent to (57) and (57a) when K_{σ} vanishes. Thus Τ_{σ}^ν'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 [sqrt](-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