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A binary black hole metric approximation from inspiral to merger

Luciano Combi, Sean M. Ressler

TL;DR

This work presents a semi-analytic binary black hole metric that spans inspiral to merger by combining a boosted Kerr-Schild superposition with PN trajectories up to $4PN$ and a merger-interpolation to NR-informed remnant properties. The approach, validated against full NR evolutions in two GRMHD contexts (uniform gas and magnetized circumbinary disk), yields good agreement in horizon properties and fluid dynamics while offering substantial computational speed-ups. Key contributions include a general, open-source SKS implementation, open integration with GRMHD codes, and demonstrated applicability to regimes difficult for full NR simulations, such as large separations or small mass ratios. The metric enables fast, accurate GRMHD/ray-tracing studies of BBH spacetimes with flexible spin, eccentricity, and mass-ratio configurations, enhancing predictive capability for multimessenger astrophysics.

Abstract

We present a semi-analytic binary black hole (BBH) metric approximation that models the entire evolution of the system from inspiral to merger. The metric is constructed as a boosted Kerr-Schild superposition following post-Newtonian (PN) trajectories at the fourth PN order in the inspiral phase. During merger, we interpolate the binary metric in time to a single black hole remnant with properties obtained from numerical relativity {(NR)} fitting formulas. The new metric can model binary black holes with arbitrary spin direction, mass ratio, and eccentricity at any stage of their evolution in a fast and computationally efficient way. We analyze the properties of our new metric and compare it with a full numerical relativity evolution. We show that Hamiltonian constraints are well-behaved even at merger and that the mass and spin measured self-consistently on the black hole horizon deviate in average only $\sim 10-5 \%$ compared to the full numerical evolution. We perform General Relativistic magneto-hydrodynamical (GRMHD) simulations for two cases: merging black holes in a uniform gas, and inspiralling black holes accreting from a magnetized circumbinary disk. We demonstrate that, in both cases, the properties of the gas, such as the accretion rate, are remarkably similar between the two approaches, with small average differences. We show that the approximate metric has several computational advantages over numerical relativity evolution. The numerical implementation of the metric is now open-source and optimized for numerical work.

A binary black hole metric approximation from inspiral to merger

TL;DR

This work presents a semi-analytic binary black hole metric that spans inspiral to merger by combining a boosted Kerr-Schild superposition with PN trajectories up to and a merger-interpolation to NR-informed remnant properties. The approach, validated against full NR evolutions in two GRMHD contexts (uniform gas and magnetized circumbinary disk), yields good agreement in horizon properties and fluid dynamics while offering substantial computational speed-ups. Key contributions include a general, open-source SKS implementation, open integration with GRMHD codes, and demonstrated applicability to regimes difficult for full NR simulations, such as large separations or small mass ratios. The metric enables fast, accurate GRMHD/ray-tracing studies of BBH spacetimes with flexible spin, eccentricity, and mass-ratio configurations, enhancing predictive capability for multimessenger astrophysics.

Abstract

We present a semi-analytic binary black hole (BBH) metric approximation that models the entire evolution of the system from inspiral to merger. The metric is constructed as a boosted Kerr-Schild superposition following post-Newtonian (PN) trajectories at the fourth PN order in the inspiral phase. During merger, we interpolate the binary metric in time to a single black hole remnant with properties obtained from numerical relativity {(NR)} fitting formulas. The new metric can model binary black holes with arbitrary spin direction, mass ratio, and eccentricity at any stage of their evolution in a fast and computationally efficient way. We analyze the properties of our new metric and compare it with a full numerical relativity evolution. We show that Hamiltonian constraints are well-behaved even at merger and that the mass and spin measured self-consistently on the black hole horizon deviate in average only compared to the full numerical evolution. We perform General Relativistic magneto-hydrodynamical (GRMHD) simulations for two cases: merging black holes in a uniform gas, and inspiralling black holes accreting from a magnetized circumbinary disk. We demonstrate that, in both cases, the properties of the gas, such as the accretion rate, are remarkably similar between the two approaches, with small average differences. We show that the approximate metric has several computational advantages over numerical relativity evolution. The numerical implementation of the metric is now open-source and optimized for numerical work.
Paper Structure (27 sections, 25 equations, 11 figures, 1 table)

This paper contains 27 sections, 25 equations, 11 figures, 1 table.

Table of Contents

  1. Introduction
  2. Building a binary black hole metric approximation
  3. Kerr-Schild metric with arbitrary spin
  4. Boosted superposition of black holes
  5. Black hole trajectories
  6. Transition to merger
  7. Comparing numerical relativity and the SKS approximation
  8. Numerical implementation of the SKS metric
  9. Numerical implementation of Einstein's equations
  10. Merging black holes on a uniform unmagnetized gas
  11. Initial data
  12. Simulation set-up
  13. Spacetime properties
  14. Hamiltonian constraints and metric
  15. Apparent horizon formation and quasi-local properties
  16. ...and 12 more sections

Figures (11)

  • Figure 1: Lapse function in the equatorial plane for Run-SKS (left) and Run-BSSN (right). The SKS metric shows the lapse deformed by the boost while the puncture gauge maintains it symmetry.
  • Figure 2: Absolute value of the constraint violations $\mathcal{H}$ in the equatorial plane for Run-SKS (left) and Run-BSSN (right)
  • Figure 3: Evolution of Hamiltonian constraint for the superposed metric (SKS) and the numerically evolved metric (BSSN).
  • Figure 4: Evolution of the $x$ component of a BH trajectory for Run-SKS and Run-BSSN. The SKS run starts dephasing on the last orbits as expected, given that the BH trajectories are obtained from a PN evolution.
  • Figure 5: Apparent horizons (black surfaces) and their trajectory at the beginning of the simulation and at merger, where a final common horizon is formed (grey surface) for Run-SKS (left) and Run-BSSN (right).
  • ...and 6 more figures