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

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.

6 pages, 5 postscript figures. Minor changes and some points clarified. Accepted for publication in Phys. Rev. D

Cited by in corpus (25)

• Numerical Relativity and Compact Binaries
• Numerical Relativity: A review
• Binary black-hole evolutions of excision and puncture data
• Binary black holes in circular orbits. I. A global spacetime approach
• Grazing Collisions of Black Holes via the Excision of Singularities
• Conformally curved binary black hole initial data including tidal deformations and outgoing radiation
• NR/HEP: roadmap for the future
• Conformal Thin-Sandwich Solver for Generic Initial Data
• Status of black-hole-binary simulations for gravitational-wave detection
• Black hole head-on collisions and gravitational waves with fixed mesh-refinement and dynamic singularity excision
• Reducing spurious gravitational radiation in binary-black-hole simulations by using conformally curved initial data
• Solving the Initial Value Problem of two Black Holes
• Improved initial data for black hole binaries by asymptotic matching of post-Newtonian and perturbed black hole solutions
• Beyond the Bowen-York extrinsic curvature for spinning black holes
• Physics and Initial Data for Multiple Black Hole Spacetimes
• Circumbinary Disk Accretion into Spinning Black Hole Binaries
• Puncture Initial Data for Black-Hole Binaries with High Spins and High Boosts
• Initial data for neutron star binaries with arbitrary spins
• Chaos in Pseudo-Newtonian Black Holes with Halos
• Generic Tracking of Multiple Apparent Horizons with Level Flow
• Generic initial data for binary boson stars
• Including realistic tidal deformations in binary black-hole initial data
• Hydro-without-Hydro Framework for Simulations of Black Hole-Neutron Star Binaries
• Binary black hole evolutions of approximate puncture initial data
• Momentum constraint relaxation