What is the Shape of a Black Hole?

Abstract

A brief, and certainly not exhaustive, survey is provided of some recent results and conjectures in four and higher spacetime dimensions, such as the Hoop Conjecture, relating the geometry of event horizons to dynamical quantities such as the total energy or mass of the spacetime containng the black hole. As a measure of the size of a hoop one may take Birkhoff's invariant based on sweepouts by circles or higher dimensional analogues or one may take the length of the shortest non-trivial closed geodesic. Also discussed are whether one can hear the shape of a black hole and to what extent one may associate with it a volume.

11 pages, no figures Written version of an invited lecture talk given at the the International Fall Workshop on Geometry and Physics held In ICMAT, MADRID (Spain), from August 31 to September 3 2011 and to appear in the Proceedings

References in corpus (3)

• Black Hole Enthalpy and an Entropy Inequality for the Thermodynamic Volume
• Global embedding of the Kerr black hole event horizon into hyperbolic 3-space
• Birkhoff's invariant and Thorne's Hoop Conjecture

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• Lower-Dimensional Black Hole Chemistry
• Thermodynamics of Kerr-AdS black holes in the restricted phase space
• Reverse Hawking-Page Phase Transition in de Sitter Black Holes
• On the volume inside old black holes
• Mass and Thermodynamic Volume in Lifshitz Spacetimes
• The Vector Volume and Black Holes
• Geometric model of black hole quantum $N$-portrait, extradimensions and thermodynamics
• Thermodynamics of rotating black holes and black rings: phase transitions and thermodynamic volume
• Isolated black holes without ${\mathbb{Z}}_2$ isometry
• Geometric inequalities for black holes
• Entropy in the interior of a Kerr black hole
• Inequality between size and angular momentum for bodies
• Formation of dynamically transversely trapping surfaces and the stretched hoop conjecture
• On the shape of rotating black-holes
• Thermodynamic Volume of Cosmological Solitons
• Volume dependent extension of Kerr-Newman black hole thermodynamics
• The CR Volume for Black Holes and the Corresponding Entropy Variation: A Review
• Volumetric Maxima to be Attained by a Nonstatic Black Hole
• Quantum Shape Effects