Vacuum-dual static perfect fluid obeying p=−(n−3)ρ/(n+1) in n(⩾4) dimensions

Abstract

Abstract We obtain the general n ( ⩾ 4 ) -dimensional static solution with an ( n − 2 ) -dimensional Einstein base manifold for a perfect fluid obeying a linear equation of state p = − ( n − 3 ) ρ / ( n + 1 ) . It is a generalization of Semiz’s four-dimensional general solution with spherical symmetry and consists of two different classes. Through the Buchdahl transformation, the class-I and class-II solutions are dual to the topological Schwarzschild–Tangherlini-(A)dS solution and one of the Λ-vacuum direct-product solutions, respectively. While the metric of the spherically symmetric class-I solution is C ∞ at the Killing horizon for n = 4 and 5, it is C 1 for n ⩾ 6 and then the Killing horizon turns to be a parallelly propagated curvature singularity. For n = 4 and 5, the spherically symmetric class-I solution can be attached to the Schwarzschild–Tangherlini vacuum black hole with the same value of the mass parameter at the Killing horizon in a regular manner, namely without a lightlike massive thin-shell. This construction allows new configurations of an asymptotically (locally) flat black hole to emerge. If a static perfect fluid hovers outside a vacuum black hole, its energy density is negative. In contrast, if the dynamical region inside the event horizon of a vacuum black hole is replaced by the class-I solution, the corresponding matter field is an anisotropic fluid and may satisfy the null and strong energy conditions. While the latter configuration always involves a spacelike singularity inside the horizon for n = 4, it becomes a non-singular black hole of the big-bounce type for n = 5 if the ADM mass is larger than a critical value.