Hyperboloidal Slices and Artificial Cosmology for Numerical Relativity

Abstract

This preliminary report proposes integrating the Maxwell equations in Minkowski spacetime using coordinates where the spacelike surfaces are hyperboloids asymptotic to null cones at spatial infinity. The space coordinates are chosen so that Scri+ occurs at a finite coordinate and a smooth extension beyond Scri+ is obtained. The question addressed is whether a Cauchy evolution numerical integration program can be easily modified to compute this evolution. In the spirit of the von Neumann and Richtmyer artificial viscosity which thickens a shock by many orders of magnitude to facilitate numerical simulation, I propose artificial cosmology to thicken null infinity Scri+ to approximate it by a de Sitter cosmological horizon where, in conformally compactified presentation, it provides a shell of purely outgoing null cones where asymptotic waves can be read off as data on a spacelike pure outflow outer boundary. This should be simpler than finding Scri+ as an isolated null boundary or imposing outgoing wave conditions at a timelike boundary at finite radius.It is a pleasure to dedicate this work to my friend and colleague Stanley Deser as we celebrate his ancient works and his continuing contributions to a broad variety of problems in physics.

Binary Black Hole Coalescence

“…A couple of alternative methods of compactification include conformal compactification [140,203,204], and using asymptotically hyperboloidal or null slices [205,206].…”

Excising das All: Evolving Maxwell waves beyond scri

“…For use in later modifications we include a function W r in the definition of the metric functions in (4). For now we choose W 0 and then the following equations give the analytically continued and conformally regulated de Sitter metric described above:…”

“…Were such a slicing to be combined with a straightforward Cauchy evolution, the Courant time step condition would defeat the advantages of approximating retarded time, as the nearly infinite coordinate speed of light toward infinity would require infinitesimal time steps (since the Courant condition generally requires that t= x < 1=c for numerical stability). Thus hyperboloidal slicings effectively require a conformal compactification which makes the coordinate speed of light finite again [4].…”

“…We develop and implement the proposals made earlier in Refs. [4,5] and suggested above to obtain waveforms promptly at infinity by using hyperboloidal time slices in a Cauchy evolution, to avoid infinitesimal time steps through conformal compactification, to convert outer boundary conditions to excision boundary conditions by using an artificial cosmological constant, and to defend the physical domain against computational boundary errors by tilting the light cones further outward in the region beyond the outer horizon. The aim of this paper is to explore the application of this set of boundary treatment tools in a low cost application in order to see whether there are unexpected impediments to its application, and to get guidance for the best approaches for its application to larger problems.…”

Exact boundary conditions in numerical relativity using multiple grids: scalar field tests

“…To improve the quality of the outgoing radiation one should aim to keep the coordinate distance that the waves have to travel to a minimum, without making their speed go to zero. This can be achieved, for example, by employing asymptotically null slices [6][7][8][9][10], or by supplementing the Cauchy evolution with a characteristic code [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26], which extends all the way to infinity. See figures 1-3.…”