Relativistic Mechanics and a Special Role for the Coulomb Potential

TL;DR

This paper demonstrates that extending nonrelativistic potentials to relativistic systems requires velocity- and acceleration-dependent forces, highlighting the unique role of the Coulomb potential in Lorentz invariance and classical electrodynamics.

Contribution

It shows that only the Coulomb potential allows a Lorentz-invariant extension of nonrelativistic mechanics at order v^2/c^2, emphasizing its special role.

Findings

Lorentz invariance requires Coulomb potential for Lorentz-invariant momentum

Velocity-dependent and acceleration-dependent forces are necessary for relativistic extension

Coulomb potential leads to the Darwin Lagrangian and classical electrodynamics

Abstract

It is shown that a nonrelativistic mechanical system involving a general nonrelativistic potential V(|r1-r2|) between point particles at positions r1 and r2 can be extended to a Lagrangian system which is invariant under Lorentz transformation through order v^2/c^2. However, this invariance requires the introduction of velocity-dependent and acceleration-dependent forces between particles. The textbook treatments of "relativistic mechanics" can be misleading; the discussions usually deal with only one particle experiencing prescribed forces and so make no mention of these additional velocity- and acceleration-dependent forces. A simple example for a situation analogous to a parallel-plate capacitor is analyzed for all the conservation laws of Galilean invariance or Lorentz invariance. For this system, Galilean invariance requires that the mechanical momentum is given by pmech=mv but places no restriction on the position-dependent potential function. On the other hand, Lorentz invariance requires that the mechanical momentum is given by pmech=mv/(1-v^2/c^2)^1/2, and in addition requires that the potential function is exactly the Coulomb potential V(|r1-r2|)=k/|r1-r2|. It is also noted that the transmission of the interparticle-force signal at the speed of light again suggests a special role for the Coulomb potential. A nonrelativistic particle system interacting through the Coulomb potential becomes the Darwin Lagrangian when extended to a system relativistic through order v^2/c^2, and then allows extension to classical electrodynamics as a fully Lorentz-invariant theory of interacting particles.