Abstract
Motivated by the novel asymptotically global AdS$_4$ solutions with deforming horizon in [JHEP {\bf 1802}, 060 (2018)], we analyze the boundary metric with odd multipolar differential rotation and numerically construct a family of deforming solutions with tripolar differential rotation boundary, including two classes of solutions: solitons and black holes. We find that the maximal values of the rotation parameter $\varepsilon$, below which the stable large black hole solutions could exist, are not a constant for $T> T_{schw}=\sqrt{3}/2\pi\simeq0.2757$. When temperature is much higher than $ T_{schw}$, even though the norm of Killing vector $\partial_{t}$ keeps timelike for some regions of $\varepsilon<2$, solitons and black holes with tripolar differential rotation could be unstable and develop hair due to superradiance. As the temperature $T$ drops toward $T_{schw}$, we find that though there exists the spacelike Killing vector $\partial_{t}$ for some regions of $\varepsilon>2$, solitons and black holes still exist and do not develop hair due to superradiance. Moreover, for $T\leqslant T_{schw}$, the curves of entropy firstly combine into one curve and then separate into two curves again, in the case of each curve there are two solutions at a fixed value of $\varepsilon$. In addition, we study the deformations of horizon for black holes by using an isometric embedding in the hyperbolic three-dimensional space. Furthermore, we also study the quasinormal modes of the solitons and black holes, which have analogous behaviours to that of dipolar rotation and quadrupolar rotation.
Highlights
The uniqueness theorem of black holes [1,2,3,4] in classical general relativity has shown that the asymptotically flat black hole solutions with zero angular momentum in four dimensions are named as Schwarzschild black holes, which have a spherical event horizon
It is well known that in fourdimensional anti–de Sitter (AdS) spacetime, there exist some solutions with noncompact planar, hyperbolic horizons and compact horizons of arbitrary genus g
Because the asymptotically AdS black hole has a conformal boundary at infinity, we could deform the
Summary
The uniqueness theorem of black holes [1,2,3,4] in classical general relativity has shown that the asymptotically flat black hole solutions with zero angular momentum in four dimensions are named as Schwarzschild black holes, which have a spherical event horizon. In [32] authors try to numerically solve the Einstein equations and give a family of deforming solitons and black holes with even multipolar differential rotation boundary, which have the antisymmetric rotation profile with respect to reflections on the equatorial plane and the total angular momentum of black hole is 0. We attempt to numerically solve the Einstein equation and give a family of deforming solitons and black holes with odd multipolar differential rotation boundary, which has the symmetric rotation profile with respect to reflections on the equatorial plane. When temperature is much higher than Tschw 1⁄4 3=2π ≃ 0.2757, for which TSchw is the minimal temperature of the AdS4-Schwarzschild black hole, the norm of Killing vector ∂t keeps timelike for some regions of ε < 2; solitons and black holes could be unstable and develop hair due to super-radiance. The conclusions and discussions are given in the last section
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