Extensions of Lorentzian Hawking--Page Solutions with Null Singularities, Spacelike Singularities, and Cauchy horizons of Taub--NUT type

TL;DR

This paper extends Lorentzian Hawking--Page solutions in 4+1 dimensions to include null and spacelike singularities and Cauchy horizons, revealing their structure and instability, with implications for naked singularities.

Contribution

It constructs and classifies all possible extensions of Lorentzian Hawking--Page solutions with various singularities and horizons within a symmetry class, using Christodoulou's methods.

Findings

Existence of extensions with null curvature singularities

Existence of extensions with spacelike curvature singularities

Existence of extensions with Taub--NUT type Cauchy horizons

Abstract

Starting from the Hawking--Page solutions, we consider the corresponding Lorentzian cone metrics. These represent cone interior scale-invariant vacuum solutions, defined in the chronological past of the scaling origin. We extend the Lorentzian Hawking--Page solutions to the cone exterior region in the class of $(4+1)$-dimensional scale-invariant vacuum solutions with an $SO(3)\times U(1)$ isometry, using the Kaluza--Klein reduction and the methods of Christodoulou. We prove that each Lorentzian Hawking--Page solution has extensions with a null curvature singularity, extensions with a spacelike curvature singularity, and extensions with a null Cauchy horizon of Taub--NUT type. These are all the possible extensions within our symmetry class. The extensions to spacetimes with a null curvature singularity can be used to construct $(4+1)$-dimensional asymptotically flat vacuum spacetimes with locally naked singularities, where the null curvature singularity is not preceded by trapped surfaces. We prove the instability of such locally naked singularities using the blue-shift effect of Christodoulou.