Starting from the Hawking–Page solutions of [14], 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 in [5]. 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 in [6].
