A Note on the Instability of Lorentzian Taub-NUT-Space

I show that there are no SU(2)-invariant (time-dependent) tensorial perturbations of Lorentzian Taub-NUT space. It follows that the spacetime is unstable at the linear level against generic perturbations. I speculate that this fact is responsible for so far unsuccessful attempts to define a sensible thermodynamics for NUT-charged spacetimes.Comment: 13 pages, no figure

The Einstein-Klein-Gordon-AdS system for general boundary conditions

We construct unique local solutions for the spherically-symmetric Einstein–Klein–Gordon–anti-de Sitter (AdS) system subject to a large class of initial and boundary conditions including some considered in the context of the AdS-CFT correspondence. The proof relies on estimates developed for the linear wave equation by the second author and involves a careful renormalization of the dynamical variables, including a renormalization of the well-known Hawking mass. For some of the boundary conditions considered this system is expected to exhibit rich global dynamics, including the existence of hairy black holes. This paper furnishes a starting point for such global investigations. </jats:p

A scattering theory construction of dynamical vacuum black holes

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full “functional” degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift

Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes

We initiate the study of the spherically symmetric Einstein-Klein-Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H^2-type norm measuring the size of the Klein-Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r-weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system.Comment: 50 pages, v2: minor changes, to appear in Annales Henri Poincar\'

Semi-classical stability of AdS NUT instantons

The semi-classical stability of several AdS NUT instantons is studied. Throughout, the notion of stability is that of stability at the one-loop level of Euclidean Quantum Gravity. Instabilities manifest themselves as negative eigenmodes of a modified Lichnerowicz Laplacian acting on the transverse traceless perturbations. An instability is found for one branch of the AdS-Taub-Bolt family of metrics and it is argued that the other branch is stable. It is also argued that the AdS-Taub-NUT family of metrics are stable. A component of the continuous spectrum of the modified Lichnerowicz operator on all three families of metrics is found.Comment: 18 pages, 3 figures; references adde

The Simplicial Ricci Tensor

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman's proof of the Poincare conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area -- an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension.Comment: 19 pages, 2 figure