Evolution of scalar fields surrounding black holes on compactified constant mean curvature hypersurfaces

Abstract

Motivated by the goal for high accuracy modeling of gravitational radiation emitted by isolated systems, recently, there has been renewed interest in the numerical solution of the hyperboloidal initial value problem for Einstein's field equations in which the outer boundary of the numerical grid is placed at null infinity. In this article, we numerically implement the tetrad-based approach presented in [J.M. Bardeen, O. Sarbach, and L.T. Buchman, Phys. Rev. D 83, 104045 (2011)] for a spherically symmetric, minimally coupled, self-gravitating scalar field. When this field is massless, the evolution system reduces to a regular, first-order symmetric hyperbolic system of equations for the conformally rescaled scalar field which is coupled to a set of singular elliptic constraints for the metric coefficients. We show how to solve this system based on a numerical finite-difference approximation, obtaining stable numerical evolutions for initial black hole configurations which are surrounded by a spherical shell of scalar field, part of which disperses to infinity and part of which is accreted by the black hole. As a non-trivial test, we study the tail decay of the scalar field along different curves, including one along the marginally trapped tube, one describing the world line of a timelike observer at a finite radius outside the horizon, and one corresponding to a generator of null infinity. Our results are in perfect agreement with the usual power-law decay discussed in previous work. This article also contains a detailed analysis for the asymptotic behavior and regularity of the lapse, conformal factor, extrinsic curvature and the Misner-Sharp mass function along constant mean curvature slices.

Updated version which includes a discussion of the Newman-Penrose scalar and which has some typos corrected. 33 pages, 19 figures, to appear in Physical Review D

References in corpus (9)

• LIGO: The Laser Interferometer Gravitational-Wave Observatory
• Hyperboloidal foliations and scri-fixing
• Extraction of Gravitational Waves in Numerical Relativity
• Hyperboloidal evolution with the Einstein equations
• Regularity of the Einstein Equations at Future Null Infinity
• Improved outer boundary conditions for Einstein's field equations
• Towards absorbing outer boundaries in General Relativity
• Tetrad formalism for numerical relativity on conformally compactified constant mean curvature hypersurfaces
• Black hole initial data on hyperboloidal slices