Covariant Wick rotation: action, entropy, and holonomies

Given an arbitrary Lorentzian metric gab and a nowhere vanishing, timelike vector field ua, one can construct a class of metrics g^ ab which have Euclidean signature in a specific domain, with a transition to Lorentzian regime occurring on some hypersurface Σ orthogonal to ua. Geometry associated with g^ 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 g^ 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.