Binary Black Hole Mergers in 3d Numerical Relativity

TL;DR

This work tackles 3D numerical relativity for binary black hole inspiral and merger by combining the 3+1 ADM formulation with maximal slicing and vanishing shift, implemented on $\mathbb{R}^3$ using a time-independent conformal rescaling based on puncture data. The authors construct initial data with two black holes carrying linear momentum and spin via Bowen–York extrinsic curvature on conformally flat slices and evolve them with $g_{ab}=\psi^4 \delta_{ab}$, $K_{ab}=\psi^{-2}\bar{K}_{ab}$, applying $\psi = u+\sum_i m_i/(2r_i)$ and $\phi=1+\sum_i m_i/(2r_i)$ to manage the inner singularities. Implemented in the BAM code with fixed mesh refinement, the method achieves evolution up to $t\approx 22\,M$ and demonstrates the apparent horizon transitioning from two disjoint surfaces to a single surface, marking the first 3D evolution through a brief merger phase. The study provides convergence evidence (roughly order $1.5$ away from punctures) and shows the viability of puncture-based 3D evolutions while highlighting grid-stretching and boundary-condition challenges that motivate shifting to nonzero shift and apparent horizon boundary conditions, as well as porting to the Cactus framework for longer runs and potential gravitational wave extraction.

Abstract

The standard approach to the numerical evolution of black hole data using the ADM formulation with maximal slicing and vanishing shift is extended to non-symmetric black hole data containing black holes with linear momentum and spin by using a time-independent conformal rescaling based on the puncture representation of the black holes. We give an example for a concrete three dimensional numerical implementation. The main result of the simulations is that this approach allows for the first time to evolve through a brief period of the merger phase of the black hole inspiral.

Figures (9)

• Figure 1: Convergence results for geodesically sliced Schwarzschild at $t = 2$. The discretized Hamiltonian $H(h)$ is shown for both numerical and analytical data.
• Figure 2: Basic features of the evolution with maximal slicing and vanishing shift for a general data set: explosion of the rescaled metric and collapse of the lapse for $t = 0, 2, \ldots, 20$.
• Figure 3: The rescaled metric and the Hamiltonian constraint at $t = 0$ (general data).
• Figure 4: The rescaled metric and the Hamiltonian constraint at $t = 10$ (general data).
• Figure 5: The rescaled metric and the Hamiltonian constraint at $t = 20$ (general data).