The Principle of Relativity · Albert Einstein
[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_)] Now since ρ₀ = ρ√(1 - V²/_c²_) We have ρ₀φ₁ = ρ[_d__{_x_} + 1/_c_ (_v₂_ _h₃_ - _v₃_ _h₂_)] N. B.—We have put the components of _e_ equivalent to (_d__{_x_}, _d__{_y_}, _d__{_z_}), and the components of _m_ equivalent to _h__{_x_} _h__{_y_} _h__{_z_}), 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 = _i_ρ [_v__{_x_}_d__{_x_} + _v__{_y_}_d__{_y_} + _v__{_z_}_d__{_z_}] = _v__{_x_} ρ₀φ₁ + _v__{_y_} ρ₀φ₂ + _v__{_z_} 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 = _i_ωF^* has a similar relation to the force acting on a moving magnetic pole.
Text from Project Gutenberg, public domain in the United States.