# Scalar self-force on eccentric geodesics in Schwarzschild spacetime: a time-domain computation

## TL;DR

This paper presents a numerical time-domain method to compute the scalar self-force on particles in eccentric orbits around Schwarzschild black holes, using mode-sum regularization and high-order finite differences.

## Contribution

It introduces a fourth-order convergent finite-difference scheme combined with mode-sum regularization for accurate self-force calculations on eccentric geodesics.

## Key findings

- Validated the numerical method with various tests.
- Provided results for mildly eccentric and zoom-whirl orbits.
- Demonstrated the method's accuracy and applicability.

## Abstract

We calculate the self-force acting on a particle with scalar charge moving on a generic geodesic around a Schwarzschild black hole. This calculation requires an accurate computation of the retarded scalar field produced by the moving charge; this is done numerically with the help of a fourth-order convergent finite-difference scheme formulated in the time domain. The calculation also requires a regularization procedure, because the retarded field is singular on the particle's world line; this is handled mode-by-mode via the mode-sum regularization scheme first introduced by Barack and Ori. This paper presents the numerical method, various numerical tests, and a sample of results for mildly eccentric orbits as well as ``zoom-whirl'' orbits.