Conservative Evolution of Black Hole Perturbations with Time-Symmetric Numerical Methods

Abstract

The scheduled launch of the LISA Mission in the next decade has called attention to the gravitational self-force problem. Despite an extensive body of theoretical work, long-time numerical computations of gravitational waves from extreme-massratio-inspirals remain challenging. This work proposes a class of numerical evolution schemes suitable to this problem based on Hermite integration. Their most important feature is time-reversal symmetry and unconditional stability, which enables these methods to preserve symplectic structure, energy, momentum and other Noether charges over long time periods. We apply Noether's theorem to the master fields of black hole perturbation theory on a hyperboloidal slice of Schwarzschild spacetime to show that there exist constants of evolution that numerical simulations must preserve. We demonstrate that time-symmetric integration schemes based on a 2-point Taylor expansion (such as Hermite integration) numerically conserve these quantities, unlike schemes based on a 1-point Taylor expansion (such as Runge-Kutta). This makes timesymmetric schemes ideal for long-time EMRI simulations.

Hyperboloidal approach for static spherically symmetric spacetimes: a didactical introduction and applications in black-hole physics

“…The application of the framework in black-hole perturbation theory has reached a mature stage, and it has established itself as an essential method in the study of wave propagation on a fixed background. Initially, the works focused on the development of numerical codes for time evolutions, benchmarked by the study of the late time decay of several fields propagating in black-hole spacetimes [31][32][33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. In this context, the hyperboloidal foliations also offer the correct tool for rigourous mathematical statements about the perturbations' late time decay and black-hole dynamical stability [49][50][51][52][53].…”

Worldtube puncture scheme for first- and second-order self-force calculations in the Fourier domain

“…Most of this progress in waveform modeling has been driven by calculations in the Fourier domain [2-7, 9, 10]. While there has been continued progress in time-domain calculations [11][12][13][14], and while it is possible to construct practical surrogate models [15] from a bank of time-domain waveforms, most development has been on Fourier methods that leverage the disparate time scales in small-mass-ratio binaries: the fast orbital time scale ∼ M and the slow time scale ∼ M/ε over which the system evolves. This separation of scales allows one to divide waveform generation into two steps: an expensive offline step in which one solves Fourier-domain field equa-tions on a grid of slowly evolving parameter values (e.g., eccentricity, semi-latus rectum, the mass and spin of the primary black hole, etc.…”