Eur. Phys. J. C (2023) 83: 194
https://doi.org/10.1140/epjc/s10052-023-11340-1

Regular Article - Theoretical Physics

Covariant Wick rotation: action, entropy, and holonomies

¹ Institute of Mathematical Sciences, Homi Bhabha National Institute (HBNI), IV Cross Road, C. I. T. Campus, Taramani, 600 113, Chennai, India
² Centre for Strings, Gravitation and Cosmology, Department of Physics, Indian Institute of Technology Madras, 600 036, Chennai, India

^a raghvendra@imsc.res.in

Received: 3 December 2022
Accepted: 19 February 2023
Published online: 6 March 2023

Abstract

Given an arbitrary Lorentzian metric $g_{\mathit{ab}}$ and a nowhere vanishing, timelike vector field $u^a$, one can construct a class of metrics ${\widehat{g}}_{\mathit{ab}}$ which have Euclidean signature in a specific domain, with a transition to Lorentzian regime occurring on some hypersurface $\mathrm{\Sigma }$ orthogonal to $u^a$. Geometry associated with ${\widehat{g}}_{\mathit{ab}}$ has been shown to yield some remarkable insights for classical and quantum gravity. In this work, we focus on studying the implications of this geometry for thermal effects in curved spacetimes and compare and contrast the results with those obtained through conventional Euclidean methods. We show that the expression for entropy computed using ${\widehat{g}}_{\mathit{ab}}$ for simple field theories and Lanczos–Lovelock actions differ from Wald entropy by additional terms depending on extrinsic curvature. We also compute the holonomy associated with loops lying partially or wholly in the Euclidean regime in terms of extrinsic curvature and acceleration and compare it with the well-known expression for temperature.