Addendum: Existence and absence of Killing horizons in static solutions with symmetries (2024 Class. Quantum Grav. 41 245013)

Abstract

In a recent paper (Maeda and Martínez 2024 Class. Quantum Grav.41 245013), we discussed regular extensions of static solutions, or more precisely solutions admitting a hypersurface-orthogonal Killing vector, with an -dimensional Einstein base manifold beyond a non-degenerate Killing horizon in general relativity in dimensions. Under the assumption that components of the energy-momentum tensor for the matter field, which are interpreted in static regions as the energy density ρ, radial pressure , and tangential pressure p2, satisfy linear relations and near the horizon, we proved proposition 6 asserting that solutions for can be extended beyond the Killing horizon and the metric in the regular single-null coordinates is at least there. Although we implicitly assumed there that the values of and in the extended dynamical region remain unchanged, regular extensions are possible even with different values as long as holds there. We present a concrete example of this regular extension in the four-dimensional spherically symmetric case using the Semiz class-I and Whittaker perfect-fluid solutions.