Numerical investigations of the asymptotics of solutions to the evolutionary form of the constraints

Abstract

A systematic investigation of the asymptotic behavior of near Schwarzschild vacuum initial data sets is carried out by making use of numerical solutions to the evolutionary form of the constraints. The decay rate of the monopole part of the trace of the tensorial projection of the extrinsic curvature is found to be of critical importance in controlling asymptotic flatness of initial data configurations.

Asymptotically flat vacuum initial data sets from a modified parabolic-hyperbolic formulation of the Einstein vacuum constraint equations

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In this paper we continue earlier investigations [10,12,19] of evolutionary formulations of the Einstein vacuum constraint equations originally introduced by Rácz. Motivated by the strong evidence from these works that the resulting vacuum initial data sets are generically not asymptotically flat we analyse the asymptotics of the solutions of a modified formulation by a combination of analytical and numerical techniques.

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“…With no control over the asymptotics it is therefore possible that the method generates initial data sets that lack a physical interpretation. Exactly this issue has been explored recently in [10,12,19]. It was confirmed that generic solutions of these equations are not asymptotically flat (this notion is defined in Section 4 below).…”

“…The system Eqs. (3.1)-(3.3) has been used in several works among which are [12,19,22,28,35]. The particular choice of how to split the fields into free data and unknowns is however not the only possibility.…”

“…In this paper here we provide strong evidence (by a combination of analytic and numerical techniques, see below) that a small change of how the free data for Rácz's parabolic-hyperbolic formulation are specified is sufficient to resolve this issue and to generate vacuum initial data sets which are generically asymptotically flat. In this same spirit, a completely different modification was suggested in [19]. As in [10,12], we restrict most of our attention to Σ being the exterior region of an isolated gravitational source and we mostly assume that Σ is foliated by 2-spheres.…”

On the construction of Riemannian three-spaces with smooth inverse mean curvature foliation

“…Restricting now attention to asymptotically flat configurations recall first that a threemetric h ij is asymptotically flat if in the asymptotic region it approaches the Euclidean metric not slower than ρ −1 . This condition, in virtue of the results in subsection 2.2.1 of [7], can also be rephrased in terms of the variables N , N A , γ AB by requiring them to fall off as…”