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A Finite Element Computation of the Gravitational Radiation emitted by a Point-like object orbiting a Non-rotating Black Hole

Carlos F. Sopuerta, Pablo Laguna

TL;DR

This work introduces a time-domain finite-element method to compute gravitational radiation from a point-like object orbiting a Schwarzschild black hole, addressing a key challenge in extreme-mass-ratio binaries. By formulating the perturbation equations with CPM and ZM master functions and treating the point-particle source within the weak form, the authors achieve accurate energy, angular momentum fluxes, and waveforms while accommodating a moving, localized source via adaptive/moving mesh strategies. They validate the approach against established results across circular, elliptic, and parabolic orbits, achieving sub-percent level agreement in many cases and robust waveform production through effective high-frequency damping. The study demonstrates the viability of FEM for EMRB perturbations and highlights its potential for self-force calculations and extensions to more general spacetimes such as Kerr, leveraging adaptive meshing and gauge choices to manage complexity.

Abstract

The description of extreme-mass-ratio binary systems in the inspiral phase is a challenging problem in gravitational wave physics with significant relevance for the space interferometer LISA. The main difficulty lies in the evaluation of the effects of the small body's gravitational field on itself. To that end, an accurate computation of the perturbations produced by the small body with respect the background geometry of the large object, a massive black hole, is required. In this paper we present a new computational approach based on Finite Element Methods to solve the master equations describing perturbations of non-rotating black holes due to an orbiting point-like object. The numerical computations are carried out in the time domain by using evolution algorithms for wave-type equations. We show the accuracy of the method by comparing our calculations with previous results in the literature. Finally, we discuss the relevance of this method for achieving accurate descriptions of extreme-mass-ratio binaries.

A Finite Element Computation of the Gravitational Radiation emitted by a Point-like object orbiting a Non-rotating Black Hole

TL;DR

This work introduces a time-domain finite-element method to compute gravitational radiation from a point-like object orbiting a Schwarzschild black hole, addressing a key challenge in extreme-mass-ratio binaries. By formulating the perturbation equations with CPM and ZM master functions and treating the point-particle source within the weak form, the authors achieve accurate energy, angular momentum fluxes, and waveforms while accommodating a moving, localized source via adaptive/moving mesh strategies. They validate the approach against established results across circular, elliptic, and parabolic orbits, achieving sub-percent level agreement in many cases and robust waveform production through effective high-frequency damping. The study demonstrates the viability of FEM for EMRB perturbations and highlights its potential for self-force calculations and extensions to more general spacetimes such as Kerr, leveraging adaptive meshing and gauge choices to manage complexity.

Abstract

The description of extreme-mass-ratio binary systems in the inspiral phase is a challenging problem in gravitational wave physics with significant relevance for the space interferometer LISA. The main difficulty lies in the evaluation of the effects of the small body's gravitational field on itself. To that end, an accurate computation of the perturbations produced by the small body with respect the background geometry of the large object, a massive black hole, is required. In this paper we present a new computational approach based on Finite Element Methods to solve the master equations describing perturbations of non-rotating black holes due to an orbiting point-like object. The numerical computations are carried out in the time domain by using evolution algorithms for wave-type equations. We show the accuracy of the method by comparing our calculations with previous results in the literature. Finally, we discuss the relevance of this method for achieving accurate descriptions of extreme-mass-ratio binaries.

Paper Structure

This paper contains 14 sections, 95 equations, 7 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Summary of Perturbation Theory for non-rotating Black Holes
  3. The numerical framework
  4. The Mathematical Formulation of the Problem
  5. The Finite Element Discretization
  6. Domain Discretization and the FE functional spaces
  7. The weak form of the equations
  8. The FE discretization of the equation
  9. Evolution Algorithms
  10. Structure and Motion of the Mesh
  11. Results from the Simulations
  12. Conclusions and Discussion
  13. Motion of the Point-like Object
  14. Gauss-Legendre Quadratures

Figures (7)

  • Figure 1: One-dimensional Mesh.
  • Figure 2: Linear interpolation functions $M^{}_i(x)$ and $N^{}_i(x)\,.$
  • Figure 3: Nodal functions $n^{}_i(x)\,.$
  • Figure 4: Examples showing the structure of the Mesh for $(p^{}_\bullet,q^{}_\bullet) = (4,3)$: On the top we have the case of a Mesh where the particle is located at a node. On the bottom we have the case of a Mesh where the particle is always in the interior of an element.
  • Figure 5: Component $(\ell,m)=(2,2)$ of the waveform corresponding to circular orbits ($e=0$) with $p=7.9456\,$.
  • ...and 2 more figures