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

TL;DR

This paper introduces a modified Wick-rotation method to study the thermodynamics of cosmological solutions with horizons, establishing a first law of thermodynamics for these solutions using Euclidean action formalism.

Contribution

It presents a new triple Wick-rotation technique and demonstrates its application to cosmological solutions, linking Euclidean action to thermodynamic properties and the first law.

Findings

Thermodynamic potential can define internal energy obeying the first law.

The approach is consistent with the isolated horizon formalism.

Dual solutions in Einstein-anti-Maxwell theory are identified.

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.