Kerr-de Sitter Space Research Articles

Asymptotics for the wave equation on differential forms on Kerr–de Sitter space

We study asymptotics for solutions of Maxwell's equations, in fact of the Hodge-de Rham equation $(d+\delta)u=0$ without restriction on the form degree, on a geometric class of stationary spacetimes with a warped product type structure (without any symmetry assumptions), which in particular include Schwarzschild-de Sitter spaces of all spacetime dimensions $n\geq 4$. We prove that solutions decay exponentially to $0$ or to stationary states in every form degree, and give an interpretation of the stationary states in terms of cohomological information of the spacetime. We also study the wave equation on differential forms and in particular prove analogous results on Schwarzschild-de Sitter spacetimes. We demonstrate the stability of our analysis and deduce asymptotics and decay for solutions of Maxwell's equations, the Hodge-de Rham equation and the wave equation on differential forms on Kerr-de Sitter spacetimes with small angular momentum.

Resonance expansions for tensor-valued waves on asymptotically Kerr–de Sitter spaces

In recent joint work with András Vasy [25], we analyze the low energy behavior of differential form-valued waves on black hole spacetimes. In order to deduce symptotics and decay from this, one in addition needs high energy estimates for the wave operator acting on sections of the form bundle. The present paper provides these on perturbations of Schwarzschild–de Sitter spaces in all spacetime dimensions n ≥ 4 . In fact, we prove exponential decay, up to a finite-dimensional space of resonances, of waves valued in any finite rank subbundle of the tensor bundle, which in particular includes differential forms and symmetric tensors. As the main technical tool for working on vector bundles that do not have a natural positive definite inner product, we introduce pseudodifferential inner products, which are inner products depending on the position in phase space.

Semilinear wave equations on asymptotically de Sitter, Kerr–de Sitter and Minkowski spacetimes

In this paper we show the small data solvability of suitable semilinear wave and Klein-Gordon equations on geometric classes of spaces, which include so-called asymptotically de Sitter and Kerr-de Sitter spaces, as well as asymptotically Minkowski spaces. These spaces allow general infinities, called conformal infinity in the asymptotically de Sitter setting; the Minkowski type setting is that of non-trapping Lorentzian scattering metrics introduced by Baskin, Vasy and Wunsch. Our results are obtained by showing the global Fredholm property, and indeed invertibility, of the underlying linear operator on suitable L^2-based function spaces, which also possess appropriate algebra or more complicated multiplicative properties. The linear framework is based on the b-analysis, in the sense of Melrose, introduced in this context by Vasy to describe the asymptotic behavior of solutions of linear equations. An interesting feature of the analysis is that resonances, namely poles of the inverse of the Mellin transformed b-normal operator, which are `quantum' (not purely symbolic) objects, play an important role.

Global Analysis of Quasilinear Wave Equations on Asymptotically Kerr-de Sitter Spaces

We extend the semilinear framework developed by the two authors and the non-trapping quasilinear theory developed by the first author to solve quasilinear wave equations with normally hyperbolic trapping. The most well-known example that fits into our general framework is wave-type equations on Kerr-de Sitter space. The key advance is an adaptation of the Nash-Moser iteration to our framework.

Quasilinear waves and trapping: Kerr-de Sitter space

In these notes, we will describe recent work on globally solving quasilinear wave equations in the presence of trapped rays, on Kerr-de Sitter space, and obtaining the asymptotic behavior of solutions. For the associated linear problem without trapping, one would consider a global, non-elliptic, Fredholm framework; in the presence of trapping the same framework is available for spaces of growing functions only. In order to solve the quasilinear problem we thus combine these frameworks with the normally hyperbolic trapping results of Dyatlov and a Nash-Moser iteration scheme.

Microlocal analysis of asymptotically hyperbolic and Kerr-de Sitter spaces (with an appendix by Semyon Dyatlov)

In this paper we develop a general, systematic, microlocal framework for the Fredholm analysis of non-elliptic problems, including high energy (or semiclassical) estimates, which is stable under perturbations. This framework, described in Sect. 2, resides on a compact manifold without boundary, hence in the standard setting of microlocal analysis. Many natural applications arise in the setting of non-Riemannian b-metrics in the context of Melrose’s b-structures. These include asymptotically de Sitter-type metrics on a blow-up of the natural compactification, Kerr-de Sitter-type metrics, as well as asymptotically Minkowski metrics. The simplest application is a new approach to analysis on Riemannian or Lorentzian (or indeed, possibly of other signature) conformally compact spaces (such as asymptotically hyperbolic or de Sitter spaces), including a new construction of the meromorphic extension of the resolvent of the Laplacian in the Riemannian case, as well as high energy estimates for the spectral parameter in strips of the complex plane. These results are also available in a follow-up paper which is more expository in nature (Vasy in Uhlmann, G. (ed.) Inverse Problems and Applications. Inside Out II, 2012). The appendix written by Dyatlov relates his analysis of resonances on exact Kerr-de Sitter space (which then was used to analyze the wave equation in that setting) to the more general method described here.

Statistical mechanics of three-dimensional Kerr-de Sitter space

The statistical entropy computation of the (2+1)-dimensional Kerr-de Sitter space in the context of the classical Virasoro algebra for an asymptotic isometry group has been a mystery since first, the degeneracy of the states has the right value only at the infinite boundary which is casually disconnected from our universe; second, the analyses were based on the unproven Cardy's formula for complex central charge and conformal weight. In this paper, I consider the entropy in Carlip's ‘would-be gauge’ degrees of freedom approach instead. I find that it agrees with the Bekenstein–Hawking entropy and the above problems do not occur. Implications to the dS/CFT are noted.

Holography in three-dimensional Kerr–de Sitter space with a gravitational Chern–Simons term

The holographic description of the three-dimensional Kerr–de Sitter space with a gravitational Chern–Simons term is studied, in the context of dS/CFT correspondence. The space has only one (cosmological) event horizon and its mass and angular momentum are identified from the holographic energy–momentum tensor at the asymptotic infinity. The thermodynamic entropy of the cosmological horizon is computed directly from the first law of thermodynamics, with the conventional Hawking temperature, and it is found that the usual Gibbons–Hawking entropy is modified. It is remarked that, due to the gravitational Chern–Simons term, (a) the results go beyond the analytic continuation from AdS, (b) the maximum-mass/N-bound conjecture may be violated and (c) the three-dimensional cosmology is chiral. A statistical mechanical computation of the entropy, from a Cardy-like formula for a dual CFT at the asymptotic boundary, is discussed. Some remarks on the technical differences in the Chern–Simons energy–momentum tensor, from the literature, are also made.

Statistical entropy of three-dimensional q-deformed Kerr–de Sitter space

Abstract A quantum deformation of three-dimensional de Sitter space was proposed in hep-th/0407188 . We use this to calculate the entropy of Kerr–de Sitter space, using a canonical ensemble, and find agreement with the semiclassical result.

CFT description of three-dimensional Kerr–de Sitter spacetime

We describe three-dimensional Kerr–de Sitter space using similar methods as recently applied to the BTZ black hole. A rigorous form of the classical connection between gravity in three dimensions and two-dimensional conformal field theory is employed, where the fundamental degrees of freedom are described in terms of two dependent SL(2, C) currents. In contrast to the BTZ case, however, quantization does not give the Bekenstein–Hawking entropy connected to the cosmological horizon of Kerr–de Sitter space.

Statistical entropy of three-dimensional Kerr–de Sitter space

I consider the (2+1)-dimensional Kerr–de Sitter space and its statistical entropy computation. It is shown that this space has only one (cosmological) event horizon, and there is a phase transition between the stable horizon and the evaporating horizon at a point M 2= 1 3 J 2/l 2 together with a lower bound of the horizon temperature. Then I compute the statistical entropy of the space by using a recently developed formulation of the Chern–Simons theory with boundaries, and extended Cardy's formula. This is in agreement with the thermodynamics formula.

Further investigations of the Kerr-de Sitter space.

Some recursion relations pertaining to the radial functions of Dirac, electromagnetic, and gravitational fields in the Kerr--de Sitter universe are proved. The behavior of the tortoise coordinate is investigated and it is shown that super radiance can also occur in this space.