Axisymmetric stationary solutions with arbitrary multipole moments

In this paper, the problem of finding an axisymmetric stationary spacetime from a specified set of multipole moments, is studied. The condition on the multipole moments, for existence of a solution, is formulated as a convergence condition on a power series formed from the multipole moments. The methods in this paper can also be used to give approximate solutions to any order as well as estimates on each term of the resulting power series.Comment: 12 page

Calculation of, and bounds for, the multipole moments of stationary spacetimes

In this paper the multipole moments of stationary asymptotically flat spacetimes are considered. We show how the tensorial recursion of Geroch and Hansen can be replaced by a scalar recursion on R^2. We also give a bound on the multipole moments. This gives a proof of the "necessary part" of a long standing conjecture due to Geroch.Comment: 11 page

The "non-Kerrness" of domains of outer communication of black holes and exteriors of stars

In this article we construct a geometric invariant for initial data sets for the vacuum Einstein field equations $(\mathcal{S},h_{ab},K_{ab})$, such that $\mathcal{S}$ is a 3-dimensional manifold with an asymptotically Euclidean end and an inner boundary $\partial \mathcal{S}$ with the topology of the 2-sphere. The hypersurface $\mathcal{S}$ can be though of being in the domain of outer communication of a black hole or in the exterior of a star. The geometric invariant vanishes if and only if $(\mathcal{S},h_{ab},K_{ab})$ is an initial data set for the Kerr spacetime. The construction makes use of the notion of Killing spinors and of an expression for a \emph{Killing spinor candidate} which can be constructed out of concomitants of the Weyl tensor.Comment: 13 page

Static spacetimes with prescribed multipole moments; a proof of a conjecture by Geroch

In this paper we give sufficient conditions on a sequence of multipole moments for a static spacetime to exist with precisely these moments. The proof is constructive in the sense that a metric having prescribed multipole moments up to a given order can be calculated. Since these sufficient conditions agree with already known necessary conditions, this completes the proof of a long standing conjecture due to Geroch.Comment: 29 page

Report on workshop A1: Exact solutions and their interpretation

I report on the communications and posters presented on exact solutions and their interpretation at the GRG18 Conference, Sydney.Comment: 9 pages, no figures. Many typos corrected. Report submitted to the Proceedings of GR18. To appear in CQ

Constructing "non-Kerrness" on compact domains

Given a compact domain of a 3-dimensional hypersurface on a vacuum spacetime, a scalar (the "non-Kerrness") is constructed by solving a Dirichlet problem for a second order elliptic system. If such scalar vanishes, and a set of conditions are satisfied at a point, then the domain of dependence of the compact domain is isometric to a portion of a member of the Kerr family of solutions to the Einstein field equations. This construction is expected to be of relevance in the analysis of numerical simulations of black hole spacetimes

Stability for linearized gravity on the Kerr spacetime

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the Kerr black hole exterior. The estimates are valid for the full, subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky Master Equation holds. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors arXiv:1708.07385. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity. The proof relies on three key steps. First, there are energy decay estimates for the Teukolsky equation, proved by applying weighted multiplier estimates to a system of spin-weighted wave equations derived from the Teukolsky equation, and making use of the pigeonhole principle for the resulting hierarchy of weighted energy estimates. Second, working in the outgoing radiation gauge, the linearized Einstein equations are written as a system of transport equations, driven by one of the Teukolsky scalars. Third, expansions for the relevant curvature, connection, and metric components can be made near null infinity. An analysis of the dynamics on future null infinity, together with the Teukolsky Starobinsky Identity plays an important role in the argument

Decay of solutions to the Maxwell equation on the Schwarzschild background

A new Morawetz or integrated local energy decay estimate for Maxwell test fields on the exterior of a Schwarzschild black hole spacetime is proved. The proof makes use of a new superenergy tensor $H_{ab}$ defined in terms of the Maxwell field and its first derivatives. The superenergy tensor, although not conserved, yields a conserved higher order energy current $H_{ab} (\partial_t)^b$. The tensor $H_{ab}$ vanishes for the static Coulomb field, and the Morawetz estimate proved here therefore yields integrated decay for the Maxwell field to the Coulomb solution on the Schwarzschild exterior.Comment: 15 pages, updated reference

On the construction of a geometric invariant measuring the deviation from Kerr data

This article contains a detailed and rigorous proof of the construction of a geometric invariant for initial data sets for the Einstein vacuum field equations. This geometric invariant vanishes if and only if the initial data set corresponds to data for the Kerr spacetime, and thus, it characterises this type of data. The construction presented is valid for boosted and non-boosted initial data sets which are, in a sense, asymptotically Schwarzschildean. As a preliminary step to the construction of the geometric invariant, an analysis of a characterisation of the Kerr spacetime in terms of Killing spinors is carried out. A space spinor split of the (spacetime) Killing spinor equation is performed, to obtain a set of three conditions ensuring the existence of a Killing spinor of the development of the initial data set. In order to construct the geometric invariant, we introduce the notion of approximate Killing spinors. These spinors are symmetric valence 2 spinors intrinsic to the initial hypersurface and satisfy a certain second order elliptic equation ---the approximate Killing spinor equation. This equation arises as the Euler-Lagrange equation of a non-negative integral functional. This functional constitutes part of our geometric invariant ---however, the whole functional does not come from a variational principle. The asymptotic behaviour of solutions to the approximate Killing spinor equation is studied and an existence theorem is presented.Comment: 36 pages. Updated references. Technical details correcte