Abstract
Using the inverse scattering method to solve the five-dimensional vacuum Einstein equations, we construct an asymptotically flat four-soliton solution as a stationary and bi-axisymmetric solution. We impose certain boundary conditions on this solution so that it includes a rotating black hole whose horizon-cross section is topologically a lens space of L(2,1). The solution has nine parameters but three only is physically independent due to the constraint equations. The remaining degrees of freedom correspond to the mass and two independent angular momenta of the black hole. We analyze a few simple cases in detail, in particular, the static case with two zero-angular momenta and the stationary case with a single non-zero angular momentum.
Highlights
The studies on higher-dimensional black hole solutions to Einstein’s equations have played roles in the microscopic derivation of Bekenstein-Hawking entropy [1], and the realistic black hole production at an accelerator in the scenario of large extra dimensions [2]
A regular vacuum black hole solution with the horizon of lens space topology has been difficult to find in spite of a few trials, since the resultant solutions always suffer from naked singularities
In order that the four-soliton solution describes a physically interesting solution, we need to impose suitable boundary conditions at infinity, on the horizon, and on a symmetry of axis: (i) The spacetime is asymptotically flat at infinity. (ii) The spacetime has a smooth horizon whose spatial topology is the lens space Lð2; 1Þ 1⁄4 S3=Z2. (iii) The spacetime has no curvature singularities, no conical singularities, no Dirac-Misner strings, and besides, no orbifold singularities at isolated points on the axis
Summary
The studies on higher-dimensional black hole solutions to Einstein’s equations have played roles in the microscopic derivation of Bekenstein-Hawking entropy [1], and the realistic black hole production at an accelerator in the scenario of large extra dimensions [2]. Based on the well-known framework of the construction for supersymmetric solutions in the bosonic sector of five-dimensional minimal supergravity developed by Gauntlett et al [24], Kundhuri and Lucietti [25] succeeded in the derivation of the first regular exact solution of an asymptotically flat black lens with the horizon topology of Lð2; 1Þ 1⁄4 S3=Z2. K is the integration constant which is determined from the requirements of the absence from conical singularities at infinity Note that this metric is exactly the same as that of the seed solution used for the derivation of the rotating black ring by the ISM. Ð7Þ where the matrix Γ0 1⁄4 ðΓ0klÞ is obtained by putting Ci 1⁄4 0 in Eq (6)
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