# Curing the Self-Force Runaway Problem in Finite-Difference Integration

## TL;DR

This paper introduces a finite-difference numerical method to effectively eliminate the runaway problem in electromagnetic self-force equations, enabling accurate simulations of charged particle motion under external forces.

## Contribution

A novel finite-difference approach is developed to suppress runaway solutions in self-force equations, improving numerical stability and physical accuracy.

## Key findings

- Complete suppression of runaway modes demonstrated
- Accurate modeling of radiation-reaction effects achieved
- Method validated with Gaussian and Sin^4 external forces

## Abstract

The electromagnetic self-force equation of motion is known to be afflicted by the so-called runaway problem. A similar problem arises in the semiclassical Einstein's field equation and plagues the self-consistent semiclassical evolution of spacetime. Motivated to overcome the latter challenge, we first address the former (which is conceptually simpler), and present a pragmatic finite-difference method designed to numerically integrate the self-force equation of motion while curing the runaway problem. We restrict our attention here to a charged point-like mass in a one-dimensional motion, under a prescribed time-dependent external force $F_{ext}(t)$. We demonstrate the implementation of our method using two different examples of external force: a Gaussian and a Sin^4 function. In each of these examples we compare our numerical results with those obtained by two other methods (a Dirac-type solution and a reduction-of-order solution). Both external-force examples demonstrate a complete suppression of the undesired runaway mode, along with an accurate account of the radiation-reaction effect at the physically relevant time scale, thereby illustrating the effectiveness of our method in curing the self-force runaway problem.