Abstract
We study static black hole solutions in Einstein and Einstein–Gauss–Bonnet gravity with the topology of the product of two spheres, mathbf{S^{n} times S^{n}}, in higher dimensions. There is an unusual new feature of the Gauss–Bonnet black hole: the avoidance of a non-central naked singularity prescribes a mass range for the black hole in terms of Lambda >0. For an Einstein–Gauss–Bonnet black hole a limited window of negative values for Lambda is also permitted. This topology encompasses black strings, branes, and generalized Nariai metrics. We also give new solutions with the product of two spheres of constant curvature.
Highlights
The first question that arises is: what equation should describe gravitational dynamics in higher dimensions? Should it be the Einstein equation or should it be its natural generalization, the Lovelock equation? The Einstein equation is linear in Riemann, while the Lovelock equation concerns a homogeneous polynomial – yet it has the remarkable property that the resulting equation still remains second order quasilinear
The question is: should the equation be Einstein–Lovelock or pure Lovelock? It has been shown that a pure Lovelock equation has the unique distinguishing property that vacuum for static spacetime in all odd D = 2N + 1 dimensions being
For N = 1 Einstein gravity, it is kinematic in D = 3, and it becomes dynamic in the even dimension D = 4
Summary
The first question that arises is: what equation should describe gravitational dynamics in higher dimensions? Should it be the Einstein equation or should it be its natural generalization, the Lovelock equation? The Einstein equation is linear in Riemann, while the Lovelock equation concerns a homogeneous polynomial – yet it has the remarkable property that the resulting equation still remains second order quasilinear. The first interesting solution with this generalization was obtained for an E–GB black hole by Dotti and Gleiser [14] with the horizon space being a Weyl constant Einstein space, as realized by the product of two spheres. This is the case we will concern ourselves with in this paper. A (d = d1 + d2 + 2)-dimensional spacetime harbors a static black hole with topology of two spheres Sd1 × Sd2 for Einstein and Sd0 × Sd0 with d1 = d2 = d0 for GB and E–GB gravity In the latter case we show that the horizon space Sd0 × Sd0 has constant Weyl curvature.
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