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Starting from the Hawking--Page solutions, we consider the correspondingLorentzian cone metrics. These represent cone interior scale-invariant vacuumsolutions, defined in the chronological past of the scaling origin. We extendthe Lorentzian Hawking--Page solutions to the cone exterior region in the classof $(4+1)$-dimensional scale-invariant vacuum solutions with an $SO(3)\timesU(1)$ isometry, using the Kaluza--Klein reduction and the methods ofChristodoulou. We prove that each Lorentzian Hawking--Page solution hasextensions with a null curvature singularity, extensions with a spacelikecurvature singularity, and extensions with a null Cauchy horizon of Taub--NUTtype. These are all the possible extensions within our symmetry class. Theextensions to spacetimes with a null curvature singularity can be used toconstruct $(4+1)$-dimensional asymptotically flat vacuum spacetimes withlocally naked singularities, where the null curvature singularity is notpreceded by trapped surfaces. We prove the instability of such locally nakedsingularities using the blue-shift effect of Christodoulou.

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