Self-force via $m$-mode regularization and 2+1D evolution: III. Gravitational field on Schwarzschild spacetime

TL;DR

This paper develops and demonstrates a 2+1D time-domain method for calculating the gravitational self-force in Schwarzschild spacetime, involving mode decomposition, puncture regularization, and addressing gauge instabilities.

Contribution

It introduces a practical implementation of a mode-sum regularization method for gravitational self-force calculations in Schwarzschild spacetime using 2+1D evolution.

Findings

Successfully evolved $m=0,1$ modes in 2+1D with gauge instability analysis.

Identified and proposed solutions for gauge instabilities in nonradiative modes.

Established a foundation for extending the method to Kerr spacetime in future work.

Abstract

This is the third in a series of papers aimed at developing a practical time-domain method for self-force calculations in Kerr spacetime. The key elements of the method are (i) removal of a singular part of the perturbation field with a suitable analytic "puncture", (ii) decomposition of the perturbation equations in azimuthal ($m$-)modes, taking advantage of the axial symmetry of the Kerr background, (iii) numerical evolution of the individual $m$-modes in 2+1-dimensions with a finite difference scheme, and (iv) reconstruction of the local self-force from the mode sum. Here we report a first implementation of the method to compute the gravitational self-force. We work in the Lorenz gauge, solving directly for the metric perturbation in 2+1-dimensions. The modes $m=0,1$ contain nonradiative pieces, whose time-domain evolution is hampered by certain gauge instabilities. We study this problem in detail and propose ways around it. In the current work we use the Schwarzschild geometry as a platform for development; in a forthcoming paper---the fourth in the series---we apply our method to the gravitational self-force in Kerr geometry.