Approximate Analytical Solutions to the Initial Data Problem of Black Hole Binary Systems

TL;DR

This paper introduces approximate analytical initial data solutions for black hole binary systems, simplifying numerical simulations by reducing constraint violations below typical discretization errors, especially for head-on equal-mass black hole collisions.

Contribution

The work provides a new analytical method for generating initial data for black hole binaries that are easier to implement and more accurate than traditional numerical solutions.

Findings

Analytical solutions have constraint violations below discretization errors.

Superposition methods produce adequate initial data at moderate resolutions.

Attenuated superposition improves data quality for finer resolutions.

Abstract

We present approximate analytical solutions to the Hamiltonian and momentum constraint equations, corresponding to systems composed of two black holes with arbitrary linear and angular momentum. The analytical nature of these initial data solutions makes them easier to implement in numerical evolutions than the traditional numerical approach of solving the elliptic equations derived from the Einstein constraints. Although in general the problem of setting up initial conditions for black hole binary simulations is complicated by the presence of singularities, we show that the methods presented in this work provide initial data with $l_1$ and $l_\infty$ norms of violation of the constraint equations falling below those of the truncation error (residual error due to discretization) present in finite difference codes for the range of grid resolutions currently used. Thus, these data sets are suitable for use in evolution codes. Detailed results are presented for the case of a head-on collision of two equal-mass M black holes with specific angular momentum 0.5M at an initial separation of 10M. A straightforward superposition method yields data adequate for resolutions of $h=M/4$, and an "attenuated" superposition yields data usable to resolutions at least as fine as $h=M/8$. In addition, the attenuated approximate data may be more tractable in a full (computational) exact solution to the initial value problem.