The equations (60), (62) and (63) give thus a generalisation of Maxwell's field-equations in vacuum, which remains true in our chosen system of co-ordinates.
The energy-components of the electro-magnetic field.
Let us form the inner-product
(65) K_{σ} = F_{σμ} J^μ.
According to (61) its components can he written down in the three-dimensional notation.
{ K_{1} = ρE_{x} + [i, H]_{x}?]
(65a) { — — —
{ K_{4} = — (i, E).
K_{σ} is a covariant four-vector whose components are equal to the negative impulse and energy which are transferred to the electro-magnetic field per unit of time, and per unit of volume, by the electrical masses. If the electrical masses be free, that is, under the influence of the electro-magnetic field only, then the covariant four-vector K_{σ} will vanish.
In order to get the energy components T_{σ}^ν of the electro-magnetic field, we require only to give to the equation K_{σ} = 0, the form of the equation (57).
From (63) and (65) we get first,
K_{σ} = F_{σμ} [part]F_{μν}/[part]x_{ν}
= [part]/[part]x_{ν} (F_{σμ} F^{μν}) - F^{μν} [part]F_{σμ}/[part]x_{ν}.
