The Lazarus project: A pragmatic approach to binary black hole evolutions

TL;DR

The Lazarus Project presents a pragmatic hybrid framework for binary black hole mergers by stitching full 3D numerical relativity in the nonlinear intermediate phase to Kerr perturbation (close-limit) evolution for the late-time regime. It introduces invariant-based criteria and carefully constructed Kerr backgrounds to determine when perturbation theory is valid, and provides strategies to compute accurate Cauchy data for Teukolsky evolution via a consistent tetrad and coordinate matching. The approach is validated by applying it to a rotating Kerr hole, demonstrating minimal spurious radiation and convergence, thereby enabling complete waveform predictions through the post-orbital dynamics. This method offers a practical bridge between fully nonlinear simulations and perturbative analyses, enabling more timely and astrophysically relevant gravitational-wave predictions. The work also outlines pathways to incorporate improved full evolutions and to extend the framework to broader astrophysical scenarios.

Abstract

We present a detailed description of techniques developed to combine 3D numerical simulations and, subsequently, a single black hole close-limit approximation. This method has made it possible to compute the first complete waveforms covering the post-orbital dynamics of a binary black hole system with the numerical simulation covering the essential non-linear interaction before the close limit becomes applicable for the late time dynamics. To determine when close-limit perturbation theory is applicable we apply a combination of invariant a priori estimates and a posteriori consistency checks of the robustness of our results against exchange of linear and non-linear treatments near the interface. Once the numerically modeled binary system reaches a regime that can be treated as perturbations of the Kerr spacetime, we must approximately relate the numerical coordinates to the perturbative background coordinates. We also perform a rotation of a numerically defined tetrad to asymptotically reproduce the tetrad required in the perturbative treatment. We can then produce numerical Cauchy data for the close-limit evolution in the form of the Weyl scalar $ψ_4$ and its time derivative $\partial_tψ_4$ with both objects being first order coordinate and tetrad invariant. The Teukolsky equation in Boyer-Lindquist coordinates is adopted to further continue the evolution. To illustrate the application of these techniques we evolve a single Kerr hole and compute the spurious radiation as a measure of the error of the whole procedure. We also briefly discuss the extension of the project to make use of improved full numerical evolutions and outline the approach to a full understanding of astrophysical black hole binary systems which we can now pursue.

Figures (8)

• Figure 1: The eclectic approach: We represent the three phases of the binary black hole evolution and the corresponding techniques adapted to each phase. The full numerical (FN) evolution is located to cover the truly nonlinear dynamical interaction. The domain of perturbative evolution (CL) follows the FN domain allowing indefinite evolution. Waveforms are extracted at the dotted world line depicted on the right. Though such observers are located in the CL part of the spacetime they will experience all radiation arriving from the strong field dynamical FN region. In the far limit regime we envision to use the post-Newtonian (PN) approximation
• Figure 2: The benefit of our fish-eye coordinates compared against the typical isotropic coordinates. The $\cal{S}$ invariant, plotted here, gives an indication of the radiation moving out from an initial ISCO system after $10M$ of numerical evolution. In the strong field region up to $z=6$ the two coordinate systems are very similar. Moving outside that region though the Fish-eye coordinate cover a significantly larger region of the physical spacetime with fewer grid-points. The extra grid-points in isotropic coordinates are wasted by over-resolving the outer part of the radiation. In Fish-eye coordinates the wave is resolved more evenly.
• Figure 3: The 'speciality' invariant for binary black holes evolving from the 'ISCO' showing damped oscillations around unity, its Kerr value. The location of the horizon in these coordinates is roughly $2.5$. Its behavior at larger radius suggests radiation is beginning to leave the system.
• Figure 4: Energy radiated from two black holes from ISCO configuration for different transition times showing a plateau when reaching the 'linear' regime.
• Figure 5: Detail of the progressive waveform locking process for black holes at the location of the ISCO.