Wick Rotation in the Tangent Space

TL;DR

This paper introduces a novel method for Wick rotation in curved spacetime using tangent space, enabling real Euclidean metrics without static assumptions, and applies it to compute black hole temperatures.

Contribution

It proposes a tangent space Wick rotation approach that works in general curved spacetimes without requiring static or Killing vector fields.

Findings

Allows Wick rotation in non-static spacetimes

Enables computation of Hawking temperature in Kerr black holes

Ensures absence of conical singularities in Euclidean space

Abstract

Wick rotation is usually performed by rotating the time coordinate to imaginary values. In a general curved spacetime, the notion of a time coordinate is ambiguous. We note here, that within the tetrad formalism of general relativity, it is possible to perform a Wick rotation directly in the tangent space using considerably less structure: a timelike, future pointing vector field, which need not be Killing or hypersurface orthogonal. This method has the advantage of yielding real Euclidean metrics, even in spacetimes which are not static. When applied to a black hole exterior, the null generators of the event horizon reduce to points in the Euclidean spacetime. Requiring that the Wick rotated holonomy of the null generators be trivial ensures the absence of a `conical singularity' in the Euclidean space. To illustrate the basic idea, we use the tangent space Wick rotation to compute the Hawking temperature by Euclidean methods in a few spacetimes including the Kerr black hole.