Now since ρ₀ = ρ√(1 - V²/c²)
We have ρ₀φ₁ = ρ[dx + 1/c (v₂ h₃ - v₃ h₂)]
N. B.—We have put the components of e equivalent to (dx, dy, dz), and the components of m equivalent to hx hy hz), in accordance with the notation used in Lorentz’s Theory of Electrons.
We have therefore
ρ₀ (φ₁, φ₂, φ₃) = ρ (d + 1/c [v·h]),
i.e., ρ₀ (φ₁, φ₂, φ₃) represents the force acting on the electron. Compare Lorentz, Theory of Electrons, page 14.
The fourth component φ₄ when multiplied by ρ₀ represents i-times the rate at which work is done by the moving electron, for ρ₀ φ₄ = iρ [vxdx + vydy + vzdz] = vx ρ₀φ₁ + vy ρ₀φ₂ + vz ρ₀φ₃. -√(-1) times the power possessed by the electron therefore represents the fourth component, or the time component of the force-four-vector. This component was first introduced by Poincare in 1906.
The four-vector ψ = iωF* has a similar relation to the force acting on a moving magnetic pole.
[M. N. S.]