D_{_x_} = P_{_y_}(U_{_z_}^* V_{_l_}^* - U_{_l_}^* V_{_z_}^*) +
P_{_z_}(U_{_l_}^* V_{_y_}^* - U_{_y_}^* V_{_l_}^*)
+ P_{_l_}(U_{_y_}^* V_{_z_}^* - U_{_z_}^* V_{_y_}^*)
= P_{_y_} φ_{_z_ _y_}^* + P_{_z_}^* φ_{_l_ _y_} + P_{_l_} φ^*_{_y_
_z_}.
= P_{_y_} φ_{_x_ _y_} + P_{_z_} φ_{_x_ _z_} + _P__{_l_} φ_{_x_ _l_},
which coincides with (Pφ_{_x_}) according to our definition.
Examples of this type of vectors will be found on page 36, Φ = wF, the
electrical-rest-force, and ψ = 2wf^*, the magnetic-rest-force. The
rest-ray Ω = iw[Φψ]^* also belong to the same type (page 39). It is easy
to show that
Ω = -_i_ | w₁ w₂ w₃ w₄ |
| Φ₁ Φ₂ Φ₃ Φ₄ |
| ψ₁ ψ₂ ψ₃ ψ₄ |
When (Ω₁, Ω₂, Ω₃) = 0, w₄ = _i_, Ω reduces to the three-dimensional
vector
| Ω₁, Ω₂, Ω₃ | = | Φ₁ Φ₂ Φ₃ |
| |
| ψ₁ ψ₂ ψ₃ |
Since in this case, Φ₁ = w₄ F₁₄ = _e__{_n_} (the electric force)
ψ₁ = -_i_w₄ f₂₃ = _m__{_x_} (the magnetic force)
we have (Ω) = | _e__{_x_} _e__{_y_} _e__{_z_} |
| _m__{_x_} _m__{_y_} _m__{_z_} |
[M. N. S.]
Note 16.
The electric-rest force. (Page 37.)
The four-vector φ = wF which is called by Minkowski the
electric-rest-force (elektrische Ruh-Kraft) is very closely connected to
Lorentz’s Ponderomotive force, or the force acting on a moving charge.
If ρ is the density of charge, we have, when ε = 1, μ = 1, _i.e._, for
free space
ρ₀φ₁ = ρ₀[w₁ F₁₁ w₂ F₁₂ + w₃ F₁₃ + w₄ F₁₄]
= ρ₀/(√(1 - V²/_c²_)) [_d__{_x_} + 1/_c_ (_v₂_ _h₃_ -
_v₃_ _h₂_)]