Towards the classification of static vacuum spacetimes with negative cosmological constant

Abstract

We present a systematic study of static solutions of the vacuum Einstein equations with negative cosmological constant which asymptotically approach the generalized Kottler (``Schwarzschild--anti-de Sitter'') solution, within (mainly) a conformal framework. We show connectedness of conformal infinity for appropriately regular such space-times. We give an explicit expression for the Hamiltonian mass of the (not necessarily static) metrics within the class considered; in the static case we show that they have a finite and well defined Hawking mass. We prove inequalities relating the mass and the horizon area of the (static) metrics considered to those of appropriate reference generalized Kottler metrics. Those inequalities yield an inequality which is opposite to the conjectured generalized Penrose inequality. They can thus be used to prove a uniqueness theorem for the generalized Kottler black holes if the generalized Penrose inequality can be established.

the discussion of our results includes now some solutions of Horowitz and Myers; typos corrected here and there; a shortened version of this version will appear in Journal of Mathematical Physics

References in corpus (8)

• Asymptotically Anti-de Sitter Space-times: Conserved Quantities
• The AdS/CFT Correspondence Conjecture and Topological Censorship
• On the structure of solutions to the static vacuum Einstein equations
• On stationary vacuum solutions to the Einstein equations
• Boundary conditions at spatial infinity from a Hamiltonian point of view
• Proof of the Riemannian Penrose Conjecture Using the Positive Mass Theorem
• Connectedness Of The Boundary In The AdS/CFT Correspondence
• Bounded Area Theorems for Higher Genus Black Holes

Cited by in corpus (60)

• Entropy production in collisions of gravitational shock waves and of heavy ions
• Present status of the Penrose inequality
• Hawking radiation from AdS black holes
• The bending of light and the cosmological constant
• On Black Hole Scalar Hair in Asymptotically Anti de Sitter Spacetimes
• An Alexandrov-Fenchel-type inequality in hyperbolic space with an application to a Penrose inequality
• From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
• A uniqueness theorem for the adS soliton
• Is AdS stable?
• Critical Trapped Surfaces Formation in the Collision of Ultrarelativistic Charges in (A)dS
• Generalized inverse mean curvature flows in spacetime
• Non-trivial, static, geodesically complete, vacuum space-times with a negative cosmological constant
• On the Geometry and Mass of Static, Asymptotically AdS Spacetimes, and the Uniqueness of the AdS Soliton
• On the mass of static metrics with positive cosmological constant - I
• General Bounds on Holographic Complexity
• The cosmological constant and the energy of gravitational radiation
• On the uniqueness of the AdS space-time in higher dimensions
• On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications
• Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant
• Classification of Kerr-de Sitter-like spacetimes with conformally flat Scri
• The mass of asymptotically anti-de Sitter space-times
• On the mass of static metrics with positive cosmological constant -- II
• Quantum reverse hypercontractivity: its tensorization and application to strong converses
• The Trautman-Bondi mass of initial data sets
• The hyperbolic positive energy theorem
• Non-trivial, static, geodesically complete space-times with a negative cosmological constant II. $n\ge 5$
• On the mass aspect function and positive energy theorems for asymptotically hyperbolic manifolds
• The mass of asymptotically hyperbolic Riemannian manifolds
• Non-genericity of the Nariai solutions: I. Asymptotics and spatially homogeneous perturbations
• Hamiltonian dynamics in the space of asymptotically Kerr-de Sitter spacetimes
• On the canonical energy of weak gravitational fields with a cosmological constant $\Lambda \in \mathbb{R}$
• On static three-manifolds with positive scalar curvature
• The rigid Horowitz-Myers conjecture
• Energy in higher-dimensional spacetimes
• Quantum Backreaction on Chronology Horizons
• Complex scalar field in strictly stationary Einstein-Maxwell-axion-dilaton spacetime with negative cosmological constant
• Uniqueness of the extremal Schwarzschild de Sitter spacetime
• On the energy of the Horowitz-Myers metrics
• Generalizations and challenges for the spacetime block-diagonalization
• The mass of spacelike hypersurfaces in asymptotically anti-de Sitter space-times
• Brendle's inequality on static manifolds
• Uniqueness and energy bounds for static AdS metrics
• A Penrose inequality for asymptotically locally hyperbolic graphs
• Vacuum energy is non-positive for (2+1)-dimensional holographic CFTs
• The Penrose inequality for asymptotically locally hyperbolic spaces with nonpositive mass
• The Penrose Inequality
• Static Black Hole Uniqueness for nonpositive masses
• Positive mass theorems for asymptotically hyperbolic Riemannian manifolds with boundary
• Uniqueness of extremal charged black holes in de Sitter
• A new energy upper bound for AdS black holes inspired by free field theory
• Static black hole uniqueness and Penrose inequality
• On the topology of initial data sets with higher genus ends
• Symmetry results for Serrin-type problems in doubly connected domains
• The Positive Energy Theorem for Asymptotically Hyperboloidal Initial Data Sets With Toroidal Infinity and Related Rigidity Results
• Black Hole and Equipotential Photon Surface Uniqueness in 4-dimensional Asymptotically Flat Electrostatic Electro-Vacuum Spacetimes
• Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity
• Hyperbolic positive energy theorems
• Report on GRG18, Session A3, Mathematical Studies of the Field Equations
• Uniqueness of AdS spacetime in any dimensions
• Uniqueness of the AdS Spacetime among Static Vacua with prescribed Null infinity