Self force via m-mode regularization and 2+1D evolution: Foundations and a scalar-field implementation on Schwarzschild

TL;DR

This paper develops and implements an $m$-mode regularization method for calculating the scalar-field self-force on Schwarzschild spacetime, crucial for modeling extreme mass-ratio inspirals relevant to gravitational wave detection.

Contribution

It introduces a practical $m$-mode regularization scheme with a 4th-order puncture and demonstrates its implementation for scalar fields on Schwarzschild, advancing computational techniques for self-force calculations.

Findings

Mode sum for dissipative SF converges exponentially.

Mode sum for conservative SF converges with a power law.

Large $m$ contributions decay as $m^{-n}$ or $m^{-n+1}$ depending on puncture order.

Abstract

To model the radiative evolution of extreme mass-ratio binary inspirals (a key target of the LISA mission), the community needs efficient methods for computation of the gravitational self-force (SF) on the Kerr spacetime. Here we further develop a practical `$m$-mode regularization' scheme for SF calculations, and give details of a first implementation. The key steps in the method are (i) removal of a singular part of the perturbation field with a suitable `puncture' to leave a sufficiently regular residual within a finite worldtube surrounding the particle's worldline, (ii) decomposition in azimuthal ($m$-)modes, (iii) numerical evolution of the $m$-modes in 2+1D with a finite difference scheme, and (iv) reconstruction of the SF from the mode sum. The method relies on a judicious choice of puncture, based on the Detweiler--Whiting decomposition. We give a working definition for the `order' of the puncture, and show how it determines the convergence rate of the $m$-mode sum. The dissipative piece of the SF displays an exponentially convergent mode sum, while the $m$-mode sum for the conservative piece converges with a power law. In the latter case the individual modal contributions fall off at large $m$ as $m^{-n}$ for even $n$ and as $m^{-n+1}$ for odd $n$, where $n$ is the puncture order. We describe an $m$-mode implementation with a 4th-order puncture to compute the scalar-field SF along circular geodesics on Schwarzschild. In a forthcoming companion paper we extend the calculation to the Kerr spacetime.