Hyperboloidal slices for the wave equation of Kerr–Schild metrics and numerical applications

103

Section: Decay Rates For S = 0 Perturbationssupporting

confidence: 89%

Numerical solution of the 2 + 1 Teukolsky equation on a hyperboloidal and horizon penetrating foliation of Kerr and application to late-time decays

Harms

Bernuzzi

Bruegmann

2013

103

“…This definition allows us to write the characteristic variables (36) in a natural way, avoiding complicated coupling between the transverse and longitudinal blocks of the principal symbol as would occur if we were to use a projection operator defined like (38) but with γ ik instead of (g −1 ) ik . Notice the drawback of this strategy is that the representation of the lightspeeds, β s ± α 1 + (v s ) 2 , is slightly more complicated than usual.…”

Section: Dual Foliation Shells Adapted Ghgmentioning

confidence: 99%

The evolution of hyperboloidal data with the dual foliation formalism: mathematical analysis and wave equation tests

Hilditch¹,

Harms²,

Bugner³

et al. 2018

A long-standing problem in numerical relativity is the satisfactory treatment of future null-infinity. We propose an approach for the evolution of hyperboloidal initial data in which the outer boundary of the computational domain is placed at infinity. The main idea is to apply the 'dual foliation' formalism in combination with hyperboloidal coordinates and the generalized harmonic gauge formulation. The strength of the present approach is that, following the ideas of Zenginoglu, a hyperboloidal layer can be naturally attached to a central region using standard coordinates of numerical relativity applications. Employing a generalization of the standard hyperboloidal slices, developed by Calabrese et. al., we find that all formally singular terms take a trivial limit as we head to null-infinity. A byproduct is a numerical approach for hyperboloidal evolution of nonlinear wave equations violating the null-condition. The height-function method, used often for fixed background spacetimes, is generalized in such a way that the slices can be dynamically 'waggled' to maintain the desired outgoing coordinate lightspeed precisely. This is achieved by dynamically solving the eikonal equation. As a first numerical test of the new approach we solve the 3D flat space scalar wave equation. The simulations, performed with the pseudospectral bamps code, show that outgoing waves are cleanly absorbed at null-infinity and that errors converge away rapidly as resolution is increased.PACS numbers: 04.25.D-, 95.30.Sf

“…leading via (27) with ω = σ to φ Kerr ≡ 2 √ M . Then all coefficients denoted in Fraktur letters as well as {H 1 , .…”

Section: Numerical Solution Of the Constraint Equationsmentioning

confidence: 99%

“…Note that for the hyperboloidal slices used in[26,27], no specific requirements in regard of the asymptotics of the mean extrinsic curvature K were specified.…”

mentioning

confidence: 99%

Initial data for perturbed Kerr black holes on hyperboloidal slices

Schinkel¹,

Macedo²,

Ansorg³

2014

Abstract. We construct initial data corresponding to a single perturbed Kerr black hole in vacuum. These data are defined on specific hyperboloidal ("ACMC-") slices on which the mean extrinsic curvature K asymptotically approaches a constant at future null infinity I + . More precisely, we require that K obeys the Taylor expansionwhere K 0 is a constant and σ describes a compactified spatial coordinate such that I + is represented by σ = 0. We excise the singular interior of the black hole and assume a marginally outer trapped surface as inner boundary of the computational domain. The momentum and Hamiltonian constraints are solved by means of pseudo-spectral methods and we find exponential rates of convergence of our numerical solutions. Some physical properties of the initial data are studied with the calculation of the Bondi Mass, together with a multipole decomposition of the horizon. We probe the standard picture of gravitational collapse by assessing a family of Penrose-like inequalities and discuss in particular their rigidity aspects. Dynamical evolutions are planned in a future project.