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} ∂F_{μν}/∂_x__{ν}
= ∂/∂_x__{ν} (F_{σμ} F^{μν}) - F^{μν} ∂F_{σμ}/∂_x__{ν}.
On account of (60) the second member on the right-hand side admits of
the transformation—
F^{μν} ∂F_{σμ}/∂_x__{ν} = -½ F^{μν} ∂F_{μν}/∂_x__{σ}
= -½ _g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}.
Owing to symmetry, this expression can also be written in the form
= -1/4 [_g_^{μα} _g_^{νβ} F_{αβ} ∂F_{μν}/∂_x__{σ}
+ _g_^{μα} _g_^{νβ} ∂F_{αβ}/∂_x__{σ} F_{μν}],
which can also be put in the form
- 1/4 ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ} F_{αβ} F_{μν})
+ 1/4 F_{αβ} F_{μν} ∂/∂_x__{σ} (_g_^{μα} _g_^{νβ}).
The first of these terms can be written shortly as
- 1/4 ∂/∂_x__{σ} (F^{μν} F_{μν}),
and the second after differentiation can be transformed in the form
- ½ F^{μτ} F_{μν} _g_^{νρ} ∂_g__{στ}/∂_x__{σ}.
If we take all the three terms together, we get the relation
(66) K_{σ} = ∂τ_{σ}^ν/∂_x__{ν} - ½ _g_^{τμ} ∂_g__{μν}/∂_x__{σ}
τ_{τ}^ν
where
(66a) τ_{σ}^ν = -F_{σα} F^{να} + 1/4 δ_{σ}^ν F_{αβ} F^{αβ}.
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.
Public-domain text, read in full here on John Shaqi.
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