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

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.

46 pages, 18 figures, 8 tables

References in corpus (19)

• Intermediate and Extreme Mass-Ratio Inspirals -- Astrophysics, Science Applications and Detection using LISA
• Two timescale analysis of extreme mass ratio inspirals in Kerr. I. Orbital Motion
• Gravitational self-force correction to the innermost stable circular orbit of a Schwarzschild black hole
• Hyperboloidal foliations and scri-fixing
• A Rigorous Derivation of Electromagnetic Self-force
• Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism
• Osculating orbits in Schwarzschild spacetime, with an application to extreme mass-ratio inspirals
• Intermediate-mass-ratio black hole binaries: intertwining numerical and perturbative techniques
• Scalar-field perturbations from a particle orbiting a black hole using numerical evolution in 2+1 dimensions
• Scalar self-force on eccentric geodesics in Schwarzschild spacetime: a time-domain computation
• Self-Force Calculations with Matched Expansions and Quasinormal Mode Sums
• m-Mode Regularization Scheme for the Self Force in Kerr Spacetime
• Gravitational perturbations and metric reconstruction: Method of extended homogeneous solutions applied to eccentric orbits on a Schwarzschild black hole
• Spacelike matching to null infinity
• Mode-sum regularization of the scalar self-force: Formulation in terms of a tetrad decomposition of the singular field
• Self-forces from generalized Killing fields
• Quasi-local contribution to the scalar self-force: Geodesic Motion
• Computational Efficiency of Frequency-- and Time--Domain Calculations of Extreme Mass--Ratio Binaries: Equatorial Orbits
• A new method to integrate (2+1)-wave equations with Dirac's delta functions as sources

Cited by in corpus (1)

• Forced motion near black holes