Cosmological Solutions, a New Wick-Rotation, and the First Law of Thermodynamics

Abstract

We present a modified implementation of the Euclidean action formalism suitable for studying the thermodynamics of a class of cosmological solutions containing Killing horizons. To obtain a real metric of definite signature, we perform a `triple Wick-rotation' by analytically continuing all space-like directions. The resulting Euclidean geometry is used to calculate the Euclidean on-shell action, which defines a thermodynamic potential. We show that for the vacuum de Sitter solution, planar solutions of Einstein-Maxwell theory and a previously found class of cosmological solutions of N = 2 supergravity, this thermodynamic potential can be used to define an internal energy which obeys the first law of thermodynamics. Our approach is complementary to, but consistent with the isolated horizon formalism. For planar Einstein-Maxwell solutions, we find dual solutions in Einstein-anti-Maxwell theory where the sign of the Maxwell term is reversed. These solutions are planar black holes, rather than cosmological solutions, but give rise, upon a standard Wick-rotation to the same Euclidean action and thermodynamic relations.

arXiv comment80 pages including 27 pages of appendices, 10 figures. Revised version: extended discussion of related results, several references added. Accepted for publication in JHEP

References in corpus (21)

• Black Hole Entropy is Noether Charge
• Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy
• Hawking Radiation as Tunneling
• Quasilocal Energy and Conserved Charges Derived from the Gravitational Action
• Lectures on holographic methods for condensed matter physics
• Isolated Horizons: Hamiltonian Evolution and the First Law
• Duality between Electric and Magnetic Black Holes
• Duality and the Signature of Space-Time
• Les Houches Lectures on De Sitter Space
• The BPS limit of rotating AdS black hole thermodynamics
• Holographic dual to charged SYK from 3D Gravity and Chern-Simons
• The Apparent Universe
• On Thermodynamics of AdS Black Holes with Scalar Hair
• Geometry of Killing spinors in neutral signature
• Special Geometry and Space-time Signature
• Phantom Metrics With Killing Spinors
• Black hole thermodynamics from a variational principle: Asymptotically conical backgrounds
• Five-dimensional vector multiplets in arbitrary signature
• Phantom Space-times in Fake Supergravity
• Non-extremal black hole solutions from the c-map
• Dynamics of the four kinds of Trapping Horizons and Existence of Hawking Radiation