Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the tracefree matter case

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4-dimensional anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang-Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.Comment: 25 page

Zero rest-mass fields and the Newman-Penrose constants on flat space

Zero rest-mass fields of spin 1 (the electromagnetic field) and spin 2 propagating on flat space and their corresponding Newman–Penrose (NP) constants are studied near spatial infinity. The aim of this analysis is to clarify the correspondence between data for these fields on a spacelike hypersurface and the value of their corresponding NP constants at future and past null infinity. To do so, Friedrich’s framework of the cylinder at spatial infinity is employed to show that, expanding the initial data in terms spherical harmonics and powers of the geodesic spatial distance ρ to spatial infinity, the NP constants correspond to the data for the second highest possible spherical harmonic at a fixed order in ρ. In addition, it is shown that for generic initial data within the class considered in this article, there is no natural correspondence between the NP constants at future and past null infinity—for both the Maxwell and spin-2 field. However, if the initial data are time-symmetric, then the NP constants at future and past null infinity have the same information

New spinorial approach to mass inequalities for black holes in general relativity

A new spinorial strategy for the construction of geometric inequalities involving the Arnowitt-Deser-Misner mass of black hole systems in general relativity is presented. This approach is based on a second order elliptic equation (the approximate twistor equation) for a valence 1 Weyl spinor. This has the advantage over other spinorial approaches to the construction of geometric inequalities based on the Sen-Witten-Dirac equation that it allows us to specify boundary conditions for the two components of the spinor. This greater control on the boundary data has the potential of giving rise to new geometric inequalities involving the mass. In particular, it is shown that the mass is bounded from below by an integral functional over a marginally outer trapped surface (MOTS) which depends on a freely specifiable valence 1 spinor. From this main inequality, by choosing the free data in an appropriate way, one obtains a new nontrivial bounds of the mass in terms of the inner expansion of the MOTS. The analysis makes use of a new formalism for the 1 + 1 + 2 decomposition of spinorial equations

The conformal Einstein field equations and the local extension of future null infinity

We make use of an improved existence result for the characteristic initial value problem for the conformal Einstein equations to show that given initial data on two null hypersurfaces ⋆ and ′⋆ such that the conformal factor (but not its gradient) vanishes on a section of ⋆, one recovers a portion of null infinity. This result combined with the theory of the hyperboloidal initial value problem for the conformal Einstein field equations allows us to show the semi-global stability of the Minkowski spacetime from characteristic initial data

Killing spinor data on distorted black hole horizons and the uniqueness of stationary vacuum black holes

We make use of the black hole holograph construction of [I. R\'acz, Stationary black holes as holographs, Class. Quantum Grav. 31, 035006 (2014)] to analyse the existence of Killing spinors in the domain of dependence of the horizons of distorted black holes. In particular, we provide conditions on the bifurcation sphere ensuring the existence of a Killing spinor. These conditions can be understood as restrictions on the curvature of the bifurcation sphere and ensure the existence of an axial Killing vector on the 2-surface. We obtain the most general 2-dimensional metric on the bifurcation sphere for which these curvature conditions are satisfied. Remarkably, these conditions are found to be so restrictive that, in the considered particular case, the free data on the bifurcation surface (determining a distorted black hole spacetime) is completely determined by them. In addition, we formulate further conditions on the bifurcation sphere ensuring that the Killing vector associated to the Killing spinor is Hermitian. Once the existence of a Hermitian Killing vector is guaranteed, one can use a characterisation of the Kerr spacetime due to Mars to identify the particular subfamily of 2-metrics giving rise to a member of the Kerr family in the black hole holograph construction. Our analysis sheds light on the role of asymptotic flatness and curvature conditions on the bifurcation sphere in the context of the problem of uniqueness of stationary black holes. The Petrov type of the considered distorted black hole spacetimes is also determined.Comment: 39 pages, 1 figur