We analyze the gravitational perturbations induced by particles falling into a three dimensional, asymptotically AdS black hole geometry. More specifically, we solve the linearized perturbation equations obtained from the geodesic motion of a ring-like distribution of test particles in the BTZ background. This setup ensures that the U(1) symmetry of the background is preserved. The non-asymptotic flatness of the background raises difficulties in attributing the significance of energy and angular momentum to the conserved quantities of the test particles. This issue is well known but, to the best of our knowledge, has never been addressed in the literature. We confirm that the naive expressions for energy and angular momentum are the correct definitions. Finally, we put an asymptotically AdS version of the weak cosmic censorship to a test: by attempting to overspin the BTZ black hole with test particles it is found that the black hole cannot be spun-up past its extremal limit.
Recently it has been shown that quantum fields can be regular on the inner, Cauchy horizon of a rotating BTZ black hole, which appears to indicate a failure of strong cosmic censorship. We argue that, instead, what these results imply is that the inner horizon remains non-singular when leading-order backreaction of the quantum fields is computed, but, after next-order backreaction is accounted for, it will become singular. Then, strong cosmic censorship will be enforced in the BTZ black hole. We support our claims using a four-dimensional holographic dual of the system, which connects the inst…
We study the phenomena of mass inflation using the Ori model for a rotating BTZ black hole that is sufficiently far from extremality, and show that the right Cauchy horizon ($\mathcal{CH}^+_R$) of the BTZ black hole becomes singular. Motivated by the recent analysis of Dias, Reall, Santos [J. High Energy Phys. 12 (2019) 097], we choose the retrograde quasinormal modes to govern the decay of perturbations exterior to the black hole. The resulting model captures the violation of the strong cosmic censorship conjecture near extremality. On the other hand, far from extremality, the $\mathcal{CH}^+…
The strong cosmic censorship conjecture, whose validation asserts the deterministic nature of general relativity, has been studied for charged BTZ black holes in three dimensional general relativity, as well as for Nth order pure Lovelock gravity in d=2N+1 spacetime dimensions. Through both analytical and numerical routes, we have computed the ratio of the imaginary part of the quasi-normal mode frequencies with the surface gravity at the Cauchy horizon. The lowest of which corresponds to the key parameter associated with violation of strong cosmic censorship conjecture. Our results demonstrat…
As a charged spinning fermion drops into a charged rotating BTZ black hole, we investigate the laws of thermodynamics and weak cosmic censorship conjecture with and without pressure respectively. For the case without pressure, the first law, second law, as well as the weak cosmic censorship are found to be valid. While for the case with pressure, though the first law is still valid, the second law and the weak cosmic censorship conjecture are found to be violable, depending on the charge, angular momentum, AdS radius, and their variations. In addition, in both cases, the configurations of the …
We investigate the stability of the inner horizon of a rotating BTZ black hole. We show that linear perturbations arising from smooth initial data are arbitrarily differentiable at the inner horizon if the black hole is sufficiently close to extremality. This is demonstrated for scalar fields, for massive Chern-Simons fields, for Proca fields, and for massive spin-2 fields. Thus the strong cosmic censorship conjecture is violated by a near-extremal BTZ black hole in a large class of theories. However, we show that a weaker "rough" version of the conjecture is respected. We calculate the renorm…
As charged fermions drop into the BTZ black holes, we check the laws of thermodynamics and weak cosmic censorship conjecture in the normal phase space and extended phase space, where the cosmological parameter and renormalization length are regarded as extensive quantities. In the normal phase space, the first and second law of thermodynamics as well as the weak cosmic censorship are valid. While in the extended phase space, though the first law and weakcosmic censorship conjecture are still valid, the second law is dependent on the variation of the renormalization energy $dK$. In addition, in…
The weak cosmic censorship conjecture in the near-extremal BTZ black hole has been tested by the test particles and fields. It was claimed that this black hole could be overspun. In this paper, we review the thermodynamics and weak cosmic censorship conjecture in BTZ black holes by the scattering of the scalar field. The first law of thermodynamics in the non-extremal BTZ black hole is recovered. For the extremal and near-extremal black holes, due to the divergence of the variation of the entropy, we test the weak cosmic censorship conjecture by evaluating the minimum values of the function $f…
It is well known that ($2+1$) dimensional charged BTZ (Banados, Teitelboim, Zanelli) black hole can be overcharged by a charged scalar field and a charged particle in contrast to their analogues in ($3+1$) and higher dimensions. In this regard, it may play an important role to understand more deeply the properties of the ($2+1$)-dimensional charged black hole in the Einstein-Gauss-Bonnet (EGB) gravity. In this paper, we test the validity of the weak cosmic censorship conjecture for the ($2+1$)-dimensional charged black hole in novel EGB theory derived recently by Henniger et. al (2021). We sho…
Investigating the quasinormal modes of a massive scalar field on the 3-dimensional black string (3dBS), we study the strong cosmic censorship (SCC) conjecture for the 3dBS in the T-dual relationship with the 3-dimensional rotating anti-de-Sitter (BTZ) black hole. It is shown that even though geometries of the two spacetimes are quite different, such as asymptotically AdS for the BTZ black hole and asymptotically flat for the 3dBS, the BTZ black hole and the 3dBS share similar properties for the SCC. Concretely speaking, the SCC conjecture can be violated even for asymptotically flat spacetime,…
An exotic Banados-Teitelboim-Zanelli (BTZ) black hole has an angular momentum larger than its mass in three dimension (3D), which suggests the possibility that cosmic censorship could be violated if angular momentum is extracted by the Penrose process. In this paper, we propose a mass formula for the exotic BTZ black hole and show no violation of weak cosmic censorship in the gedanken process above by understanding properly its mass formula. Unlike the other black holes, the total energy of the exotic BTZ black hole is represented by the angular momentum instead of the mass, which supports a b…
We reexamine our proposed counterexample (gr-qc/0307102) to cosmic censorship in anti de Sitter (AdS) space, and find a gap in the construction. We mention some possible ways to close the gap, but at present the question of whether cosmic censorship is violated in AdS remains open.
In three dimensions, a phase transition occurs between the non-rotating BTZ black hole and the massless BTZ black hole. Further, introducing the mass of a conical singularity, we show that a transition between the non-rotating BTZ black hole and thermal AdS space is also possible.
We consider time-dependent solutions of the Einstein-Maxwell equations using anti-de Sitter (AdS) boundary conditions, and provide the first counterexample to the weak cosmic censorship conjecture in four spacetime dimensions. Our counterexample is entirely formulated in the Poincaré patch of AdS. We claim that our results have important consequences for quantum gravity, most notably to the weak gravity conjecture.
We review the relation between AdS spacetime in 1+2 dimensions and the BTZ black hole. Later we show that a ground state in AdS spacetime becomes a thermal state in the BTZ black hole. We show that this is true in the bulk and in the boundary of AdS spacetime. The existence of this thermal state is tantamount to say that the Unruh effect exists in AdS spacetime and becomes the Hawking effect for an eternal BTZ black hole. In order to make this we use the correspondence introduced in Algebraic Holography between algebras of quasi-local observables associated to wedges and doble cones regions in…
We show that cosmic censorship takes an exceptionally complex and interesting form in the case of five-dimensional AdS-Kerr black holes, due to the unusually distant relation obtaining in that case between the black hole parameters and the physical mass and angular momentum. One finds that, in this case, censorship is less restrictive than one might hope: it apparently allows some rather bizarre behaviour, and in particular does not prohibit arbitrarily large angular momenta. We find however that most of the unwelcome geometries permitted by censorship can be eliminated by requiring stability …
We study the problem of construction of global isometric embedding for spherically symmetric black holes with negative cosmological constant in various dimensions. Firstly, we show that there is no such embedding for 4D RN-AdS black hole in 6D flat ambient space, completing the classification which we started earlier. Then we construct an explicit embedding of non-spinning BTZ black hole in 6D flat ambient space. Using this embedding as an anzats, we then construct a global explicit embedding of (d)-dimensional Schwarzschild-AdS black hole in a flat (d+3)-dimensional ambient space.
We investigate the laws of thermodynamics and weak cosmic censorship conjecture in the normal and extended phase space of a Kerr-AdS black hole by analyzing the energy-momentum relation of the absorbed fermion dropping into the horizon. In the normal phase space, the first law, second law as well as the weak cosmic censorship conjecture are found to be valid in all the initial states of the black hole. However, in the extended phase space, although the first law and weak cosmic censorship conjecture are still valid, the second law becomes more subtle. We find that the validity or violation of …
For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated to be equal to the Bekenstein-Hawking entropy. In this paper, the statistical entropy of the BTZ black hole in the higher curvature gravity is calculated and shown to be equal to the one derived by using the Noether charge method. This suggests that the equivalence of the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the b…
The first law of black hole thermodynamics in the extended phase space is prevailing recently. However, the second law as well as the weak cosmic censorship conjecture has not been investigated extensively. In this paper, we investigate the laws of thermodynamics and the weak cosmic censorship conjecture of an AdS black hole with a global monopole in the extended phase space under a charged particle absorption. It is shown that the first law of thermodynamics is valid, while the second law is violated for the extremal and near-extremal black holes. Moreover, we find the weak cosmic censorship …
We show that a noncommutative massless scalar probe can dress a naked singularity in $AdS_3$ spacetime, consistent with the weak cosmic censorship. The dressing occurs at high energies, which is typical at the Planck scale. Using a noncommutative duality, we show that the dressed singularity has the geometry of a rotating BTZ black hole which satisfies all the laws of black hole thermodynamics. We calculate the entropy and the quasi-normal modes of the dressed singularity and show that the corresponding spacetime can be quantum mechanically complete. The noncommutative duality also gives rise …
We find an exact nonstatic charged BTZ-like solutions, in ($N$+1)-dimensional Einstein gravity in the presence of negative cosmological constant and a nonlinear Maxwell field defined by a power $s$ of the Maxwell invariant, which describes the gravitational collapse of charged null fluid in an anti-de Sitter background. Considering the situation that a charged null fluid injects into the initially an anti-de Sitter spacetime, we show that a black hole form rather than a naked singularity, irrespective of spacetime dimensions, from gravitational collapse in accordance with cosmic censorship con…
Considering the cosmological constant as the pressure, we mainly study the laws of thermodynamics and weak cosmic censorship conjecture in the Reissner-Nordström-AdS black hole surrounded by quintessence dark energy under charged particle absorption. The first law of thermodynamics is found to be valid as a particle is absorbed by the black hole. The second law however is found to be violated for the extremal and near-extremal black holes since the entropy of these black hole decrease. Moreover, we find that the extremal black hole do not change it configuration in the extended phase space, im…
The greybody factors in BTZ black holes are evaluated from 2D CFT in the spirit of AdS$_3$/CFT correspondence. The initial state of black holes in the usual calculation of greybody factors by effective CFT is described as Poincaré vacuum state in 2D CFT. The normalization factor which cannot be fixed in the effective CFT without appealing to string theory is shown to be determined by the normalized bulk-to-boundary Green function. The relation among the greybody factors in different dimensional black holes is exhibited. Two kinds of $(h,{\bar h})=(1,1)$ operators which couple with the boundary…
Numerical evidence suggests that the superradiant instability of Kerr-AdS black holes and "black resonators" evolves to smaller and smaller scales towards a "limiting" black resonator. Using $AdS_4$ supergravity, we argue that this putative endpoint to the superradiant instability does not exist.
With the Hamilton-Jacobi equation, we obtain the energy-momentum relation of a charged particle as it is absorbed by the Gauss-Bonnet AdS black hole. On the basis of the energy-momentum relation at the event horizon, we investigate the first law, second law, and weak cosmic censorship conjecture in both the normal phase space and extended phase space. Our results show that the first law, second law as well as the weak cosmic censorship conjecture are valid in the normal phase space for all the initial states are black holes. However, in the extended phase space, the second law is violated for …
The strong cosmic censorship conjecture states (SCCC) that one cannot extend spacetime beyond the Cauchy horizon with a square-integrable connection. This conjecture was postulated to save the deterministic nature of the most successful theory of gravitation, general relativity. In order to explore the validation/violation of the SCCC for the charged anti-de Sitter black hole spacetime, we compute the ratio of the imaginary part of the quasinormal mode frequencies and the surface gravity at the Cauchy horizon both analytically and numerically. The lowest value of which defines the key paramete…
We consider vacuum solutions of four dimensional general relativity with $Λ< 0$. We numerically construct stationary solutions that asymptotically approach a boundary metric with differential rotation. Smooth solutions only exist up to a critical rotation. We thus argue that increasing the differential rotation by a finite amount will cause the curvature to grow without bound. This holds for both zero and nonzero temperature, and both compact and noncompact boundaries. However, the boundary metric always develops an ergoregion before reaching the critical rotation, which probably means that th…
In this paper, we study the $AdS/BCFT$ construction in $AdS_3$ and the BTZ black hole spacetime. We find a new solution to the equation for the End of the World (EoW) brane. The induced metric on this solution is that of $dS_2$ with an isometry group identical to that of $AdS_2$ and the symmetry group of the corresponding BCFT. It also leads to a corresponding EoW brane in the non-rotating BTZ background which is consistent with the periodic identifications of the non-rotating BTZ black hole. The new solution generalizes easily to higher dimensions.
We examine the achronal averaged null energy condition~(ANEC) for a class of conformal field theories~(CFT) at strong coupling in curved spacetime. By applying the AdS/CFT duality, we find holographic models which violate the achronal ANEC for $3+1$ and $4+1$-dimensional boundary theories. In our model, the bulk spacetime is an asymptotically AdS vacuum bubble solution with neither causality violation nor singularities. The conformal boundary of our bubble solution is asymptotically flat and is causally proper in the sense that a "fastest null geodesics" connecting any two points on the bounda…
Recently Hertog, Horowitz, and Maeda (HHM) (hep-th/0310054) have proposed that cosmic censorship can be violated in the AdS/CFT context. They argue that for certain initial data there is insufficient energy available to make a black hole whose horizon is big enough to cloak the singularity that forms. We have investigated this proposal in the models HHM discuss and have thus far been unable to find initial data that provably satisfy this criterion, despite our development of an improved lower bound on the size of the singular region. This is consistent with recent numerical results (hep-th/040…
We show that the 3D charged Banados-Teitelboim-Zanelli (BTZ) black hole solution interpolates between two different 2D AdS spacetimes: a near-extremal, near-horizon AdS_2 geometry with constant dilaton and U(1) field and an asymptotic AdS_2 geometry with a linear dilaton. Thus, the charged BTZ black hole can be considered as interpolating between the two different formulations proposed until now for AdS_2 quantum gravity. In both cases the theory is the chiral half of a 2D CFT and describes, respectively, Brown-Hennaux-like boundary deformations and near-horizon excitations. The central charge…
We investigate thermodynamics of the BTZ black hole in new massive gravity explicitly. For $m^2\ell^2>1/2$ with $m^2$ the mass parameter of fourth-order terms and $\ell^2$ AdS$_3$ curvature radius, the Hawking-Page phase transition occurs between the BTZ black hole and AdS (thermal) soliton. For $m^2\ell^2<1/2$, however, this transition unlikely occurs but a phase transition between the BTZ black hole and the massless BTZ black hole is possible to occur. We may call the latter as the inverse Hawking-Page phase transition and this transition is favored in the new massive gravity.
By dropping particles into black hole, we have employed the recently new assumption [1] that the change of the black hole mass(enthalpy) should be the same amount as the energy of an infalling particle($ω= dM$), to carefully test the laws of thermodynamics and the weak cosmic censorship conjecture of a conformal anomaly corrected AdS black hole in different phase spaces. Using the energy-momentum relation, the result shows that the first law, second law and weak cosmic censorship conjecture of black hole are all valid in the normal phase space, no violations occur. In the extended phase space,…
By applying Newman's method, the AdS_3 rotating black hole solution is "derived" from the nonrotating black hole solution of Banados, Teitelboim and Zanelli (BTZ). The rotating BTZ solution derived in this fashion is given in "Boyer-Lindquist-type" coordinates whereas the form of the solution originally given by BTZ is given in a kind of an "unfamiliar" coordinates which are related to each other by a transformation of time coordinate alone. The relative physical meaning between these two coordinates is carefully studied by evaluating angular momentum per unit mass, angular velocity, surface g…
We study the Lamb shift of a two-level atom arising from its coupling to the conformal massless scalar field, which satisfies the Dirichlet boundary conditions, in the Hartle-Hawking vacuum in the BTZ spacetime, and find that the Lamb shift in the BTZ spacetime is structurally similar to that of a uniformly accelerated atom near a perfectly reflecting boundary in (2+1)-dimensional flat spacetime. Our results show that the Lamb shift is suppressed in the BTZ spacetime as compared to that in the flat spacetime as long as the transition wavelength of the atom is much larger than $AdS$ radius of t…
We consider gedanken experiments to destroy an extremal or near-extremal BTZ black hole by throwing matter into the horizon. These black holes are vacuum solutions to (2+1)-dimensional gravity theories, and are asymptotically $\mathrm{AdS}_3$. Provided the null energy condition for the falling matter, we prove the following---(i) in a Mielke-Baekler model without ghost fields, when torsion is present, an extremal BTZ black hole can be overspun and becomes a naked conical singularity; (ii) in 3-dimensional Einstein gravity and chiral gravity, which both live in torsionless limits of Mielke-Baek…
Based on the new version of the gedanken experiment proposed by Sorce and Wald, we investigate the weak cosmic censorship conjecture (WCCC) for a Reissner-Nordström-Anti-de Sitter (RN-AdS) black hole under the perturbation of extra matter fields. Firstly, we propose that the cosmological constant can be effectively derived from the matter fields and its value varies with the matter fields perturbing the black hole. Meanwhile, we assume that the perturbation satisfies the stability condition. This condition means that after a long time of the perturbation, the black hole solution also belongs t…
Present-day temperature $T_0$ of cosmic microwave background has been precisely measured by the FIRAS experiment. We identify that the early dark energy (EDE) (non-negligible around matter-radiation equality) scenario can remain compatible with the FIRAS result, while lifting the Hubble constant $H_0$. We perform Monte Carlo Markov chain analysis to confirm our observations. We also present an $α$-attractor Anti-de Sitter (AdS) model of EDE, in which the AdS depth is consistently varied in the Monte Carlo Markov chain analysis. We found that our datasets weakly hinted the existence of an AdS p…
In [1] the horizon fluffs proposal is put forward to identify microstates of generic non-extremal three-dimensional Banados-Teitelboim-Zanelli (BTZ) black holes. The proposal is that black hole microstates, which were dubbed as horizon fluffs, are states labelled by the conserved charges associated with non-trivial diffeomorphisms on the near horizon geometry and these states are not in the coadjoint orbits of the asymptotic Virasoro algebra at the Brown-Henneaux central charge, associated with the BTZ black holes. It is also known that AdS3 Einstein gravity has more general black hole solutio…
We propose a 1+1 dimensional CFT dual structure for quantum gravity and matter on the extended 2+1 dimensional BTZ black hole, realized as a quotient of the Poincare patch of AdS$_3$. The quotient spacetime includes regions beyond the singularity, "whiskers", containing timelike and lightlike closed curves, which at first sight seem unphysical. The spacetime includes the usual AdS-asymptotic boundaries outside the horizons as well as boundary components inside the whiskers. We show that local boundary correlators with some endpoints in the whisker regions: (i) are a protected class of amplitud…
We study linear scalar perturbations of black holes in two-dimensional (2D) gravity models with a particular emphasis on Jackiw-Teitelboim (JT) gravity. We obtain an exact expression of the quasinormal mode frequencies for single horizon black holes in JT gravity and then verify it numerically using the Horowitz-Hubeny method. For a 2D Reissner-Nordström like solution, we find that the massless scalar wave equation reduces to the confluent Heun equation using which we calculate the Hawking spectra. Finally, we consider the dimensionally reduced Bañados-Teitelboim-Zanelli (BTZ) black hole and o…
In this work, we study gravity duals of conformal field theories with boundaries known as AdS/BCFT correspondence, put forward by Takayanagi [1], within a scalar-tensor theory, as proposed by Horndeski in 4D [2]. We consider the case of 3D gravity dual to 2D BCFT, take a Gibbons-Hawking surface term modified by Horndeski's theory and find the corresponding 3D (BTZ) black hole solutions. We analyze the effects of Horndeski's gravity on the profile of the extra boundary for the black hole by using an approximate analytic solution. Performing a holographic renormalization, we have calculated the …
For the BTZ black hole in the Einstein gravity, a statistical entropy has been calculated. It is equal to the Bekenstein-Hawking entropy. In this article, its statistical entropy in the higer curvature gravity is calculated and shown to be equal to one derived by using the Noether charge method. This suggests that the equivalence between the geometrical and statistical entropies of the black hole is retained in the general diffeomorphism invariant theories of gravity. A relation between the cosmic censorship conjecture and the unitarity of the conformal field theory on the boundary of AdS_3 is…
According to the horizon fluff proposal microstates of a generic black hole belong to a certain subset of near horizon soft hairs that cannot be extended beyond the near horizon region. In [1,2] it was shown how the horizon fluff proposal works for AdS3 black holes. In this work we clarify further this picture by showing that BTZ black hole microstates are in general among the coherent states in the Hilbert space associated with conic spaces or their Virasoro descendants, provided we impose a (Bohr-type) quantization condition on the angular deficit. Thus BTZ black holes may be viewed as conde…
Recent work has argued that the strong cosmic censorship (SCC) conjecture is violated by near-extremal Reissner-Nordström de Sitter (RNdS) black holes but respected by Kerr-de Sitter black holes. It has also been shown that the conjecture is violated by near-extremal BTZ black holes. The latter result relies on a coincidence between "exterior" and "interior" quasinormal frequencies. If this coincidence were to occur also for RNdS or Kerr-dS then it would significantly modify the conclusions of earlier work. In this paper, it is demonstrated that this coincidence does not occur for RNdS or Kerr…
In this paper, we study the thermodynamics and the weak cosmic censorship conjecture of the RN-AdS black hole surrounded by the quintessence under the scattering of a charged complex scalar field. With scalar field scattering, the variation of the black hole is calculated in the extended and normal phase spaces. In the extended phase space, the cosmological constant and the normalization parameter are considered as thermodynamic variables, and the first law of thermodynamics is valid, but the second law of thermodynamics is not valid. In the normal phase space, the cosmological constant and th…
Employing the operator method, we obtain log-square quasinormal modes and frequencies of a graviton around the BTZ black hole at the tricritical point of the generalized massive gravity. The log-square quasinormal frequencies are also obtained by considering a finite temperature conformal field theory. This shows the AdS/LCFT correspondence at the tricritical point approximately. We discuss a truncation process to the unitary theory on the BTZ black hole background.
In gr-qc/9902061 it was shown that (n+1)-dimensional asymptotically anti-de-Sitter spacetimes obeying natural causality conditions exhibit topological censorship. We use this fact in this paper to derive in arbitrary dimension relations between the topology of the timelike boundary-at-infinity, $\scri$, and that of the spacetime interior to this boundary. We prove as a simple corollary of topological censorship that any asymptotically anti-de Sitter spacetime with a disconnected boundary-at-infinity necessarily contains black hole horizons which screen the boundary components from each other. …
We investigate the connection between the BTZ black holes and the near-horizon geometry of higher-dimensional black holes. Under mild conditions, we show that (i) if a black hole has a global structure of the type of the non-extremal Reissner-Nordstrom black holes, its near-horizon geometry is $AdS_2$ times a sphere, and further (ii) if such a black hole is obtained from a boosted black string by dimensional reduction, the near-horizon geometry of the latter contains a BTZ black hole. Because of these facts, the calculation of the Bekenstein-Hawking entropy and the absorption cross-sections of…
In this paper, we investigate the thermodynamics and weak cosmic censorship conjecture in a RN-AdS black hole with the cloud of strings and quintessence by the scattering of a scalar field. The variations of the thermodynamic variables are calculated in the normal and extended phase spaces. In the normal phase space, where the cosmological constant is considered as a constant, the first and second laws of thermodynamics are satisfied. In the extended phase space, where the cosmological constant and the parameters related to the cloud of strings and quintessence are treated as variables, the fi…
The greybody factors for spin 1/2 particles in the BTZ black holes are discussed from 2D CFT in bulk-boundary correspondence. It is found that the initial state of spin 1/2 particle in the BTZ black holes can be described by the Poincaré vacuum state in boundary 2D CFT, and the nonlinear coordinate transformation causes the thermalization of the Poincaré vacuum state. For special case, our results for the greybody factors agree with the semiclassical calculation.
This paper is the continuation of [8]. We essentially prove that the familly of strongly causal spacetimes defined in [8] associated to generic achronal subsets in Ein contains all the examples of BTZ multi black-holes. It provides new elements for the global description of these multi black-holes. We also prove that any strongly causal spacetime locally modelled on the anti-de Sitter space admits a well-defined maximal strongly causal conformal boundary locally modelled on Ein.
We show that the nonlinear $σ-$model in an asymptotically $AdS_3$ space-time admits a novel local symmetry. The field action is assumed to be quartic in the nonlinear $σ-$model fields and minimally coupled to gravity. The local symmetry transformation simultaneously twists the nonlinear $σ-$model fields and changes the space-time metric, and it can be used to map an extremal $BTZ$ black hole to infinitely many hairy black hole solutions.
