$\alpha-$surfaces and global coordinates in black hole spacetimes

TL;DR

The paper introduces a Lorentzian approach to analyze the periodicity of maximally extended black hole spacetimes, applicable to Kerr black holes without Euclidean methods, using complex coordinates and $ ext{alpha}$-surfaces.

Contribution

It develops a Lorentzian framework for studying black hole spacetime periodicity that avoids Euclidean manifolds and complexification of angular momentum, applicable to Kerr black holes.

Findings

Applicable to Kerr black holes without Euclidean methods

Uses complex coordinates associated with $ ext{alpha}$-surfaces

Provides a coordinate system covering the entire exterior geometry

Abstract

We present a discussion of the periodicity in imaginary time of maximally extended black hole spacetimes without reference to Euclidean manifolds. As motivation, we first demonstrate our approach for the Rindler geometry in flat space before then applying it for the case of Schwarzschild time in the maximally extended Kruskal geometry. One notable advantage of our approach is that it can be utilized in the Kerr case, again without reference to a Euclidean manifold, and without complexifying the angular momentum parameter, $a$. One unusual feature of our application in the Kerr case is that, even for the purely Lorentzian geometry, it is developed in terms of explicitly complex coordinates which are associated with distinct families of $\alpha-$surfaces. Moreover, coordinatization of these gives a single set of coordinates which cover the entire geometry outside the inner horizon.