A Framework for Lorentz-Dirac Dynamics and Post-Newtonian Interaction of Radiating Point Charges

TL;DR

The paper addresses classical radiation reaction for radiating point charges by blending the covariant Lorentz-Dirac/Lorentz-Dirac formulation with a post-Newtonian perspective. It develops an electromagnetic analogue of PN/EOB methods by pairing a conservative Darwin Hamiltonian (1PN) with a leading 1.5PN dipole radiation-reaction term obtained via reduction of order, and validates energy balance in single-particle tests. The authors then implement this framework in an N-body setting and demonstrate dissipative binary dynamics—orbital decay, circularization, and monotonic Hamiltonian decrease consistent with dipole Larmor losses. Overall, the framework provides a tractable laboratory for studying dissipative, potentially chaotic electromagnetic dynamics with clear connections to gravitational radiation-reaction theory.

Abstract

We examine classical radiation reaction by combining the covariant Lorentz--Dirac formulation, its Landau--Lifshitz (LL) order reduction, and a post-Newtonian (PN) Hamiltonian treatment of interacting and radiating charges. After reviewing the LL reduction and its removal of runaway and preacceleration behavior, we verify energy balance in several relativistic single-particle scenarios by demonstrating agreement between the LL Larmor power and the loss of mechanical energy. We then construct an N-body framework based on the conservative Darwin Hamiltonian supplemented with the leading 1.5PN radiation--reaction term. Numerical simulations of charge-neutral binary systems of both symmetric and asymmetric mass configurations show orbital decay, circularization, and monotonic Hamiltonian decrease consistent with dipole radiative losses. The resulting framework provides a simple analogue of gravitational PN radiation reaction and a tractable system for studying dissipative and potentially chaotic electromagnetic dynamics.

Figures (6)

• Figure 1: Relativistic trajectory of a charged particle in a uniform magnetic field $\mathbf{B}=B\,\hat{\mathbf{z}}$, computed using the order–reduced Lorentz–Dirac equations. (a) Three–dimensional trajectory with initial (green) position. (b–e) Radius $r(\tau)$ measured from the relativistic guiding center and the spatial components $x(\tau)$, $y(\tau)$, and $z(\tau)$ at select time intervals. The motion shows damped cyclotron oscillations in the $x$–$y$ plane, while $z(\tau)$ stays constant for the chosen initial data. Parameters described in the main text.
• Figure 2: More examples: Landau--Lifshitz trajectories for three representative external-field configurations. Each row shows (left to right) the full 3D worldline projection followed by the coordinate evolution $x(\tau)$, $y(\tau)$, and $z(\tau)$.
• Figure 3: Two oppositely charged blobs, each containing ten particles, evolved with the conservative Darwin Hamiltonian at 1PN order. Panel (a) shows the three--dimensional trajectories (blue: positive charges; black: negative charges), and panel (b) the corresponding evolution of the Darwin Hamiltonian $H(\tau)$.
• Figure 4: Charge neutral binary of same mass starting from diametrically opposite locations with equal and opposite momenta, with radiation reaction from our trucated post Newtonian approach: (a) Full 3D trajectories; (b) parametric $x$–$y$ plane; (c–e) components $x(\tau)$, $y(\tau)$, $z(\tau)$; (f) Hamiltonian $H(\tau)$, decreasing due to radiation. The initial particle positions are shown in green, and the final positions at $\tau=\tau_f$ are shown in red.
• Figure 5: Charge neutral binary of extreme mass ratio, one heavy and one light, with radiation reaction from our trucated post Newtonian approach: (a) Full 3D trajectories; (b) parametric $x$–$y$ plane; (c–e) components $x(\tau)$, $y(\tau)$, $z(\tau)$; (f) Hamiltonian $H(\tau)$, decreasing due to radiation. The initial particle positions are shown in green, and the final positions at $\tau=\tau_f$ are shown in red.