Numerical space-times near space-like and null infinity. The spin-2 system on Minkowski space

Abstract

In this paper we demonstrate for the first time that it is possible to solve numerically the Cauchy problem for the linearisation of the general conformal field equations near spacelike infinity, which is only well-defined in Friedrich's cylinder picture. We have restricted ourselves here to the "core" of the equations -the spin-2 system -propagating on Minkowski space. We compute the numerical solutions for various classes of initial data, do convergence tests and also compare to exact solutions. We also choose initial data which intentionally violate the smoothness conditions and then check the analytical predictions about singularities. This paper is the first step in a long-term investigation of the use of conformal methods in numerical relativity. arXiv:1207.5854v2 [gr-qc]

Peeling or not peeling—is that the question?

“…To keep their numerical grids finite, most relativists who develop numerical 3+1 codes for the standard Cauchy problem to calculate binary black hole merger (say) wave forms use cut-off procedures and essentially ignore space-like and null infinity, thus also the finer details of the asymptotic behaviour there. The radiation field, originally defined at null infinity, is calculated only approximately at a finite, somewhat arbitrary location (see, however, the work by F. Beyer et al [7] and J. Frauendiener and J. Hennig [36] which takes first steps towards calculating entire solutions determined by asymptotically flat Cauchy data). This cut-off deletes a neighbourhood of J ± ∪ i 0 which is of infinite extent as measured in terms of affine parameters on the outgoing null geodesics.…”

Numerical initial boundary value problem for the generalized conformal field equations

“…This still remains as just a conjecture and hence it would be intriguing to probe this question numerically by evolving sets of initial data that do and do not satisfy the necessary conditions of the conjecture. There have already been numerical studies of linearly perturbed space-times which incorporate space-like infinity, see for example [34,37,38], while in [35,36] we have studied simpler systems which show similar behaviours near space-like infinity.…”

“…The finite IBVP at space-like infinity is based on asymptotically Euclidean hyper-surfaces which extend out to spacelike infinity represented as a cylinder (see [34][35][36][37][38] for some studies of this approach in very simplified situations). Initial data are obtained by solving the conformal constraint equations and the solution is global covering the entire past and future of the space-time evolving from the initial data, see Fig.…”

Spectral methods for the spin-2 equation near the cylinder at spatial infinity

“…(iii) For p ≥ 2, = p and all admissible m, the coefficients a n,p;p,m (τ ) have logarithmic singularities at τ = ±1. More precisely, the coefficients split into a part with polynomial (and thus smooth) dependence of τ and a singular part of the form Finally, we notice the following result providing the link between the formal expansions of Ansatz (9) and actual solutions to the spin-2 equations -see [25] for a proof: Proposition 2. In a neighbourhood of I (including I ± ) the solutions to equations (4a)-(4e) and (8a)-(8c) are of the form…”