We study the spectrum of the worldsheet theory of the bosonic closed string in the massless and extremal rotating BTZ black holes. We use a hyperbolic Wakimoto representation of the SL(2,R) currents to construct vertex operators for the string modes on these backgrounds. We argue that there are tachyons in the twisted sector, but these are not localised near the horizon. We study the relation to the null orbifold in the limit of vanishing cosmological constant. We also discuss the problem of extending this analysis to the supersymmetric case.
In this note we compute mutual information between two intervals in CFTs dual to extremal BTZ (UV CFT) and near horizon limit of extremal BTZ (IR CFT) using the replica technique in some limiting regimes, which can be compared with holographic description.
We study the near horizon 2d gravity theory which captures the near extremal thermodynamics of BTZ with excess mass $δM$ and excess angular momentum $δJ =\mathcal{L}\, δM$ over extremality. We find that the Jackiw-Teitelboim theory is able to capture such departures from extremality with an additional parameter $\mathcal{L}$ relating it to the near extremal BTZ configuration. We are able to show this by obtaining new simultaneous near horizon near extremal limits of the BTZ geometry parametrized by $\mathcal{L}$. The resulting Jackiw-Teitelboim theory captures the near extremal thermodynamics …
In this note, we use entanglement entropy as a tool to explore the universal properties of CFTs dual to extremal BTZ black holes. We demonstrate that the entanglement entropies computed in the CFTs at the boundary of the extremal BTZ and the boundary of the near-horizon limit of the extremal BTZ are in perfect agreement and have the form appropriate to a two-dimensional CFT with only the chiral part excited and the anti-chiral in the ground state. Furthermore, we analyze the universal limits of the entanglement entropy and recover the correct value of the thermal entropy for large entangling i…
We study the spectrum of tensor perturbations on extremal BTZ black holes in topologically massive gravity for arbitrary values of the coefficient of the Chern-Simons term, $μ$. Imposing proper boundary conditions at the boundary of the space and at the horizon, we find that the spectrum contains quasi-normal modes.
We consider the one-loop corrected geometry and thermodynamics of a rotating BTZ black hole by way of a dimensionally reduced dilaton model. The analysis begins with a comprehensive study of the non-extremal solution after which two different methods are invoked to study the extremal case. The first approach considers the extremal limit of the non-extremal calculations, whereas the second treatment is based on the following conjecture: extremal and non-extremal black holes ae qualitatively distinct entities. We show that only the latter method yields regularity and consistency at the one-loop …
We show that quasinormal modes cannot exist in the extremal BTZ black hole. For this purpose, we consider propagations of a minimally coupled scalar and a single massive graviton obtained from the cosmological topologically massive gravity on the extremal BTZ black hole. The would-be quasinormal modes for a scalar and graviton could not exist because it is impossible to make an ingoing flux into the extremal (degenerate) horizon. This is consistent with the argument that there is no propagating dynamics in the self-dual orbifold of AdS(3) which is just the near-horizon limit of the extremal BT…
Any unitary compact two-dimensional CFT with $c>1$ and no symmetries beyond Virasoro has a parametrically large density of primary states at large spin for $\bar{h}>\bar{h}_\text{extr}\sim \frac{c-1}{24}$, of a universal form determined by modular invariance. By including the contribution of light primary operators and multi-twist composites constructed from them in the modular bootstrap, we find that $\bar{h}_\text{extr}$ receives corrections in a large spin expansion, which we compute at finite $c$. The analysis uses a formulation of the modular S-transform as a Fourier transform acting on t…
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.
Recently, the AdS/CFT conjecture of Maldacena has been investigated in Lorentzian signature by Balasubramanian et. al. We extend this investigation to Lorentzian BTZ black hole spacetimes, and study the bulk and boundary behaviour of massive scalar fields both in the non-extremal and extremal case. Using the bulk-boundary correspondence, we also evaluate the two-point correlator of operators coupling to the scalar field at the boundary of the spacetime, and find that it satisfies thermal periodic boundary conditions relevant to the Hawking temperature of the BTZ black hole.
We discuss the microscopic states of the extremal BTZ black holes. Degeneracy of the primary states corresponding to the extremal BTZ black holes in the boundary N=(4,4) SCFT is obtained by utilizing the elliptic genus and the unitary representation theory of N=4 SCA. The degeneracy is consistent with the Bekenstein-Hawking entropy.
Out-of-time-order correlators (OTOCs) that capture maximally chaotic properties of a black hole are determined by scattering processes near the horizon. This prompts the question to what extent OTOCs display chaotic behaviour in horizonless microstate geometries. This question is complicated by the fact that Lyapunov growth of OTOCs requires nonzero temperature, whereas constructions of microstate geometries have been mostly restricted to extremal black holes. In this paper, we compute OTOCs for a class of extremal black holes, namely maximally rotating BTZ black holes, and show that on averag…
Motivated by several pieces of evidence, in order to show that extreme black holes cannot be obtained as limits of non-extremal black holes, in this article we calculate explicitly quasinormal modes for Bañados, Teitelboim and Zanelli (BTZ) extremal black hole and we showed that the imaginary part of the frequency is zero. We obtain exact result for the scalar an fermionic perturbations. We also showed that the frequency is bounded from below for the existence of the normal modes (non-dissipative modes).
In this paper we study the classical dynamics of long strings in AdS3, generalizing our previous study, arXiv:1410.3424 [hep-th], to rotating strings and BTZ black holes. As in the non-rotating case, BTZ black holes are generated in the large tension limit, in which string back-reaction must be taken into account. When back-reaction is properly accounted for, collapsing heavy, physical, rotating strings do not generate naked singularities but only BTZ black holes, including extremal ones. The rotating string must contain world-sheet excitations in order to have consistent equations of motion; …
We find solution to the metric function f(r)=0 of charged BTZ black hole making use of the Lambert function. The condition of extremal charged BTZ black hole is determined by a non-linear relation of M_e(Q)=Q^2(1-\ln Q^2). Then, we study the entropy of extremal charged BTZ black hole using the entropy function approach. It is shown that this formalism works with a proper normalization of charge Q for charged BTZ black hole because AdS_2 x S^1 represents near-horizon geometry of the extremal charged BTZ black hole. Finally, we introduce the Wald's Noether formalism to reproduce the entropy of t…
Integrability of classical strings in the BTZ black hole enables the construction and study of classical string propagation in this background. We first apply the dressing method to obtain classical string solutions in the BTZ black hole. We dress time like geodesics in the BTZ black hole and obtain open string solutions which are pinned on the boundary at a single point and whose end points move on time like geodesics. These strings upon regularising their charge and spins have a dispersion relation similar to that of giant magnons. We then dress space like geodesics which start and end on th…
We study the spectrum of bosonic string theory on rotating BTZ black holes, using a SL(2,R) WZW model. Previously, Natsuume and Satoh have analyzed strings on BTZ black holes using orbifold techniques. We show how an appropriate spectral flow in the WZW model can be used to generate the twisted sectors, emphasizing how the spectral flow works in the hyperbolic basis natural for the BTZ black hole. We discuss the projection condition which leads to the quantization condition for the allowed quantum numbers for the string excitations, and its connection to the anomaly in the corresponding conser…
We obtain the BTZ black hole (AdS$_3 \times$S$^3$) as a non-dilatonic solution from type 0B string theory. Analyzing the perturbation around this black hole background, we show that the tachyon is not a propagating mode.
In a (2+1)-dimensional spacetime with a negative cosmological constant, the thermodynamics and the entropy of an extremal rotating thin shell, i.e., an extremal rotating ring, are investigated. The outer and inner regions are taken to be the Bañados-Teitelbom-Zanelli (BTZ) spacetime and the vacuum ground state anti-de Sitter (AdS) spacetime, respectively. By applying the first law of thermodynamics to the extremal shell one shows that its entropy is an arbitrary function of the gravitational area $A_+$ alone, $S=S(A_+)$. When the shell approaches its own gravitational radius $r_+$ and turns in…
We study real-time propagation of a massive scalar field on the extremal BTZ black hole spacetime, focusing on the Aretakis instability of the event horizon. We obtain a simple time-domain expression for the $\textrm{AdS}_3$ retarded Green function with Dirichlet boundary conditions and construct the corresponding time-domain BTZ retarded Green function using the method of images. The field decays at different rates on and off the horizon, indicating that transverse derivatives grow with time on the horizon (Aretakis instability). We solve the null geodesic equation in full generality and show…
We study the wave equation for a massive scalar in three-dimensional AdS-black hole spacetimes to understand the unitarity issues in a semiclassical way. Here we introduce four interesting spacetimes: the non-rotating BTZ black hole (NBTZ), pure AdS spacetime (PADS), massless BTZ black hole (MBTZ), and extremal BTZ black hole (EBTZ). Our method is based on the potential analysis and solving the wave equation to find the condition for the frequency $ω$ exactly. In the NBTZ case, one finds the quasinormal (complex and discrete) modes which signals for a non-unitary evolution. Real and discrete m…
Strings near BTZ black holes: A Carrollian Chronicle
Birla Institute of Technology and Science, Pilani Campus（比拉理工学院皮拉尼校区）；Indian Institute of Technology Kanpur（坎普尔印度理工学院）；Indian Institute of Science Education and Research, Mohali（莫哈利印度科学教育与研究学院）；Tata Institute of Fundamental Research（塔塔基础研究所）
We study the boundary CFT under supertranslation in terms of the AdS/CFT correspondence. In particular, we probe the soft hair BTZ background by several open string configurations as follows. The Ryu-Takayanaki formula of holographic entanglement remains intact under supertranslation. The U-shape string probe indicates that the flat part of meson potential is altered by the soft charge. The trailing string solution shows that the hair parameter plays the role of external field in the drag force. The study of hanging string in the Brownian motion suggests the transport variables, such as fricti…
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…
We study string theory on the extended spacetime of the BTZ black hole, as described by an orbifold of the SL(2,R) WZW model. The full spacetime has an infinite number of disconnected boundary components, each corresponding to a dual CFT. We discuss the computation of bulk and boundary correlation functions for operators inserted on different components. String theory correlation functions are obtained by analytic continuation from an orbifold of the SL(2,C)/SU(2) coset model. This yields two-point functions for general operators, including those describing strings that wind around the horizon…
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}^+…
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 …
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…
We discuss realizations of the SL(2,R) current algebra in the hyperbolic basis using free scalar fields. It has been previously shown by Satoh how such a realization can be used to describe the principal continuous representations of SL(2,R). We extend this work by introducing another realization that corresponds to the principal discrete representations of SL(2,R). We show that in these realizations spectral flow can be interpreted as twisting of a free scalar field. Finally, we discuss how these realizations can be obtained from the BTZ Lagrangian.
Analysing closed string tachyon condensation will improve our understanding of spacetime in string theory. We study the string spectrum on a BTZ black hole spacetime supported by NS-NS flux, which provides a calculable example where we would expect to find a quasi-localised tachyon. We find that there is a winding tachyon when the black hole horizon is smaller than the string scale, as expected. However, due to effects of the NS-NS B field, this tachyon is not localised in the region where the spatial circle is string scale. We also discuss the relation to the Milne orbifold in the limit near …
Using a thin shell, the first law of thermodynamics, and a unified approach, we study the thermodymanics and find the entropy of a (2+1)-dimensional extremal rotating Bañados-Teitelbom-Zanelli (BTZ) black hole. The shell in (2+1) dimensions, i.e., a ring, is taken to be circularly symmetric and rotating, with the inner region being a ground state of the anti-de Sitter (AdS) spacetime and the outer region being the rotating BTZ spacetime. The extremal BTZ rotating black hole can be obtained in three different ways depending on the way the shell approaches its own gravitational or horizon radius…
Some thermodynamic properties of the Bañados-Teitelboim-Zanelli (BTZ) black hole are studied to get the effective dimension of its corresponding statistical model. For this purpose, we make use of the geometrical approach to the thermodynamics: Considering the black hole as a thermodynamic system with two thermodynamic variables (the mass $M$ and the angular momemtum $J$), we obtain two-dimensional Riemannian thermodynamic geometry described by positive definite Ruppeiner metric. From the thermodynamic curvature we find that the extremal limit is the critical point. The effective spatial dimen…
We compute holographically the energy loss of a moving quark in various states of the D1-D5 CFT. In the dual bulk geometries, the quark is the end of a trailing string, and the profile of this string determines the drag force exerted by the medium on the quark. We find no drag force when the CFT state has no momentum, and a nontrivial force for any value of the velocity (even at rest) when the string extends in the supersymmetric D1-D5-P black-hole geometry, or a horizonless microstate geometry thereof. As the length of the throat of the microstate geometry increases, the drag force approaches…
$AdS_3$ space-time admits a foliation by two-dimensional twisted conjugacy classes, stable under the identification subgroup yielding the non-rotating massive BTZ black hole. Each leaf constitutes a classical solution of the space-time Dirac-Born-Infeld action, describing an open D-string in $AdS_3$ or a D-string winding around the black hole. We first describe two nonequivalent maximal extensions of the non-rotating massive BTZ space-time and observe that in one of them, each D-string worldsheet admits an action of a two-parameter subgroup ($\ca \cn$) of $\SL$. We then construct non-formal, $…
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…
We analyze near-Hagedorn thermodynamics of strings in the WZW $AdS_3$ model. We compute the thermal spectrum of all primaries and find the thermal scalar explicitly in the string spectrum using CFT twist techniques. Then we use the link to the Euclidean WZW BTZ black hole and write down the Euclidean BTZ spectrum. We give a Hamiltonian interpretation of the thermal partition function of angular orbifolds where we find a reappearance of discrete states that dominate the partition function. Using these results, we discuss the nature of the thermal scalar in the WZW BTZ model. As a slight general…
In this paper we compute the decay rates of massless scalar waves excited by a star circularly orbiting around the non-extremal (general) and extremal BTZ black holes. These decay rates are compared with the corresponding quantities computed in the corresponding dual conformal field theories respectively. We find that matches are achieved in both cases.
This companion paper to arXiv:2109.00065 explores the interpretation of $AdS_3/CFT_2$ duality for $k=(R_{AdS}/\lstr)^2<1$ in terms of "non-critical little string theory". We review the underlying fivebrane structure of 1/2-BPS backgrounds for string theory in both the critical dimension and for non-critical models. D-branes bound to the fivebranes in these backgrounds are associated to nonabelian little strings, and flesh out a picture of the correspondence transition at $k=1$ wherein the nonabelian little string excitations lose their Hagedorn entropy, allowing the Hagedorn entropy of fundame…
We study the dynamical stability of the BTZ black string against fermonic and gravitational perturbations. The BTZ black string is not always stable against these perturbations. There exist threshold values for $m^2$ related to the compactification of the extra dimension for fermonic perturbation, scalar part of the gravitational perturbation and the tensor perturbation, respectively. Above the threshold values, perturbations are stable; while below these thresholds, perturbations can be unstable. We find that this non-trivial stability behavior qualitatively agrees with that predicted by a th…
In this paper, we study the equation of circular loops with time-dependant tension in the BTZ black hole background. We obtain various cases where cosmic string loops finally collapse to form black holes. Also, we study effect of the BTZ black hole mass and angular momentum on the evolution of cosmic string loops. We find the critical values of initial radii as a limit for the cosmic string loops collapsing to form black holes.
This is a set of introductory lecture notes on black holes in string theory. After reviewing some aspects of string theory such as dualities, brane solutions, supersymmetric and non-extremal intersection rules, we analyze in detail extremal and non-extremal 5d black holes. We first present the D-brane counting for extremal black holes. Then we show that 4d and 5d non-extremal black holes can be mapped to the BTZ black hole (times a compact manifold) by means of dualities. The validity of these dualities is analyzed in detail. We present an analysis of the same system in the spirit of the adS/C…
In this paper we study the first law of thermodynamics for the (2+1) dimensional charged BTZ black hole considering a pair of thermodinamical systems constructed with the two horizons of this solution. We show that these two systems are similar to the right and left movers of string theory and that the temperature associated with the black hole is the harmonic mean of the temperatures associated with these two systems.
We provide a realization of the AdS$_2$/CFT$_1$ correspondence in terms of asymptotic symmetries of the AdS$_2\times$S$^1$ and AdS$_2\times$S$^2$ geometries arising in near-extremal BTZ and Reissner-Nordström black holes. Cardy's formula exactly accounts for the deviation of the Bekenstein-Hawking entropy from extremality. We also argue that this result can be extended to more general black holes near extremality.
We extend our recent discussion of four-dimensional black holes bound to a two-brane to include a negative cosmological constant on the brane. We find that for large masses, the solutions are precisely BTZ black holes on the brane, and BTZ `black strings' in the bulk. For smaller masses, there are localized black holes which look like BTZ with corrections that fall off exponentially. We compute when the maximum entropy configuration changes from the black string to the black hole. We also present exact solutions describing rotating black holes on two-branes which are either asymptotically flat…
