Four-force
The Lorentz four-vector F^μ = dp^μ/dτ; the relativistic generalisation of Newton's force. For a particle of constant rest mass, F^μ = m·a^μ. The time component is the rate of energy transfer (power); the spatial components reduce to Newton's second law in the low-β limit.
Definition
Four-force is the proper-time derivative of four-momentum: F^μ = dp^μ/dτ, the relativistic generalisation of Newton's second law. For a particle of constant rest mass m, the four-momentum factorises as p^μ = m u^μ, and the four-force collapses to F^μ = m·a^μ — manifestly Lorentz-covariant. The time component F⁰ = (1/c)(dE/dτ) tracks the rate at which energy is transferred to the particle, the relativistic generalisation of mechanical power. The spatial components F^i = γ(dp^i/dt) reduce to the ordinary Newtonian force F = ma in the low-β limit and inherit Lorentz transformation properties under boosts.
Because four-force is the τ-derivative of four-momentum, and four-momentum has fixed Minkowski norm m²c² for constant rest mass, the four-force is orthogonal to the four-velocity: F^μ u_μ = 0. This orthogonality is the relativistic version of the work-energy theorem — the time component F⁰ is determined by the spatial components F⁰ = (γ/c) F·v, ensuring that no algebraic freedom violates the constant-rest-mass condition. The Lorentz force on a charged particle, F^μ = qF^μν u_ν, is the canonical example: contracting the antisymmetric electromagnetic field tensor with the four-velocity yields a four-force automatically orthogonal to u, automatically Lorentz-covariant, automatically reducing to qE + qv×B in the lab frame. Four-force is the dynamical bridge from spacetime geometry to forces with definite frame-mixing rules.