Geometric aspects of covariant Wick rotation

TL;DR

This paper explores the geometric properties of metrics derived from Lorentzian spacetimes undergoing covariant Wick rotation, revealing new insights into classical and quantum gravity, especially regarding holonomy and entropy in curved spacetimes with horizons.

Contribution

It provides general results on the geometry of Wick-rotated metrics with non-zero acceleration, including holonomy and entropy calculations, applicable to physically relevant spacetimes.

Findings

Holonomy in Euclidean regime depends on extrinsic curvature and acceleration.

Entropy calculations show foliation-dependent corrections in gravity theories.

Results apply to spherically symmetric and maximally symmetric spacetimes.

Abstract

We discuss the generic geometric properties of metrics $\widehat {g}_{ab}$ constructed from Lorentzian metric $g_{ab}$ and a nowhere vanishing, hypersurface orthogonal, timelike vector field $u^a$. The metric ${\widehat g}_{ab}$ has Euclidean signature in a certain domain, with the transition to Lorentzian signature occurring at some hypersurface $\Sigma$ orthogonal to $u^a$. Geometry associated with ${\widehat g}_{ab}$ has recently been shown to yield remarkable new insights for classical and quantum gravity. In this work, we prove several general results applicable in physically relevant spacetimes for congruences $u^i$ with non-zero acceleration $a^i$. We present as examples the cases of dynamical spherically symmetric spacetimes and spacetimes with maximal symmetry. We also investigate this formalism within the context of thermal effects in curved spacetimes with horizons. Specifically, we discuss: (i) the Holonomy of loops lying partially or wholly in the Euclidean regime. We show that the contribution of the Euclidean domain to holonomy is completely determined by extrinsic curvature $K_{ab}$ of $\Sigma$ and acceleration $a^i$. (ii) We also compute entropy using this formalism for simple field theories and obtain foliation dependent corrections for the Lanczos-Lovelock gravity, Bekenstein-Hawking entropy relation in four spacetime dimensions.