Conflict Between Classical Mechanics and Electromagnetism: The Harmonic Oscillator in Equilibrium with a Bath

TL;DR

The paper discusses the fundamental differences between classical mechanics and electromagnetism in the context of harmonic oscillators, emphasizing the role of Lorentz-invariant zero-point radiation as the true classical equilibrium spectrum.

Contribution

It highlights that classical electromagnetic equilibrium involves Lorentz-invariant zero-point radiation, contrasting with the nonrelativistic Rayleigh-Jeans spectrum often assumed in physics texts.

Findings

Harmonic oscillators in electromagnetic baths are in equilibrium with Lorentz-invariant radiation.

Classical zero-point radiation is the true equilibrium spectrum, not Rayleigh-Jeans.

Modern physics texts incorrectly assume Rayleigh-Jeans spectrum as equilibrium in classical radiation.

Abstract

It is pointed out that an electric charge oscillating in a one-dimensional purely-harmonic potential is in detailed balance at its harmonics with a radiation bath whose energy $U_{rad}$ per normal mode is linear in frequency $\omega$, $U_{rad}=const\times\omega,$ and hence is Lorentz invariant, as seems appropriate for relativistic electromagnetism. The oscillating charge is NOT in equilibrium with the Rayleigh-Jeans spectrum which arises from energy-sharing equipartition ideas which are valid only in nonrelativistic mechanics. Here we explore the contrasting behavior of harmonic oscillators connected to baths in classical mechanics and electromagnetism. It is emphasized that modern physics text are in error in suggesting that the Rayleigh-Jeans spectrum corresponds to the equilibrium spectrum of random classical radiation, and in ignoring Lorentz-invariant classical zero-point radiation which is indeed a classical equilibrium spectrum.