Examples of this type of vectors will be found on page 36, Φ = ωF, the electrical-rest-force, and ψ = 2ωf^*, the magnetic-rest-force. The rest-ray Ω = iω[Φψ]^* also belong to the same type (page 39). It is easy to show that
Ω = -i | ω_1 ω_2 ω_3 ω_4 |
| Φ_1 Φ_2 Φ_3 Φ_4 |
| ψ_1 ψ_2 ψ_3 ψ_4 |
When (ω_1, ω_2, ω_3) = 0, ω_4 = i, Ω reduces to the three-dimensional vector
| Ω_1, Ω_2, Ω_3 | = | Φ_1 Φ_2 Φ_3 |
| | | |
| ψ_1 ψ_2 ψ_3 |
Since in this case, Φ_1 = ω_4 F_{1 4} = e_n (the electric force)
ψ_1 = -iω_4 f_{2 3} = m_x (the magnetic force)
we have (Ω) = | e_x e_y e_z | , i.e., analogous to the
| | Poynting-vector.
| m_x m_y m_z |
[M. N. S.]
Note 16. The electric-rest force. (Page 37.)
The four-vector φ = ωF 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
ρ_0φ_1 = ρ_0[_1 F_{1 1} ω]_2 F_{1 2} + ω_3 F_{1 3} + ω_4 F_{1 4}]
= ρ_0/([sqrt](1 - V^2/c^2)) [d_x + 1/c (v_2 h_3 - v_3 h_2)]]
Now since ρ_0 = ρ[sqrt](1 - V^2/c^2)
We have ρ_0φ_1 = ρ[d_x + 1/c (v_2 h_3 - v_3 h_2)]]
N. B.—We have put the components of e equivalent to (d_x, d_y, d_z), and the components of m equivalent to
