New black holes of vacuum Einstein equations with hyperscaling violation and Nil geometry horizons

In this paper, we present a new solution of the vacuum Einstein equations in five dimensions which is a static black hole with hyperscaling violation and with a three-dimensional horizon modeled by one of the eight Thurston geometries, namely the Nil geometry. This homogeneous geometry is nontrivial in the sense that it is neither of constant curvature nor a product of constant curvature manifolds. Using the Hamiltonian formalism, we identify the mass and entropy of the black hole solution. Curiously enough, in spite of the fact that the entropy turns out to be negative, the mass is positive and the first law of thermodynamics holds. We also discuss the extension in higher dimensions.