We studied the eigenvalue problem of scalar fields in the (2+1)-dimensional BTZ black hole spacetime. The Dirichlet boundary condition at infinity and the Dirichlet or the Neumann boundary condition at the horizon are imposed. Eigenvalues for normal modes are characterized by the principal quantum number $(0<n)$ and the azimuthal quantum number $(-\infty<m< \infty)$. Effects to eigenvalues of the black hole rotation and of the scalar field mass are studied explicitly. Relation of the black hole rotation to the super-radiant instability is discussed.
Eigenfunctions for normal modes of scalar fields in BTZ black hole spacetime are studied. Orthonormal relations among them are derived. Quantization for scalar fields is done and particle number, energy and angular momentum are expressed by the creation and annihilation operators. Allowed physical normal mode region is studied on the basis of the no zero mode theorem. Its implication to the statistical mechanics is also studied.
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 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…
The thermodynamic equilibrium states of a static thin ring shell in a (2+1)-dimensional spacetime with a negative cosmological constant are analyzed. Inside the ring, the spacetime is pure anti-de Sitter (AdS), whereas outside it is a Bañados-Teitelbom-Zanell$ (BTZ) spacetime and thus asymptotically AdS. The first law of thermodynamics applied to the thin shell, plus one equation of state for the shell's pressure and another for its temperature, leads to a shell's entropy, which is a function of its gravitational radius alone. A simple example for this gravitational entropy, namely, a power la…
Various observations from cosmic microwave background radiation (CMBR), type Ia supernova and baryon acoustic oscillations (BAO) are strongly suggestive of an accelerated expansion of the universe which can be explained by the presence of mysterious energy known as dark energy. The quintessential matter coupled with gravity minimally is considered one of the possible candidates to represents the presence of such dark energy in our universe. In view of this scenario, we study the geodesic of massless particles as well as massive particles around a (2+1) dimensional BTZ black hole (BH) spacetime…
We investigate the Teichmüller parameters for a Euclidean multiple BTZ black hole spacetime. To induce a complex structure in the asymptotic boundary of such a spacetime, we consider the limit in which two black holes are at a large distance from each other. In this limit, we can approximately determine the period matrix (i.e., the Teichmüller parameters) for the spacetime boundary by using a pinching parameter. The Teichmüller parameters are essential in describing the partition function for the boundary conformal field theory (CFT). We provide an interpretation of the partition function for …
An exact solution of $(2+1)$-dimensional Einstein gravity with cosmological constant is studied. The corresponding spacetime is interpreted as an accelerating BTZ spacetime. The proper acceleration, horizon structure, temperature and entropy are presented in detail. The metric being studied is very similar to the one studied by Astorino in arXiv:1101.2616, but the range of parameters is different which results in significant changes in the causal structures.
An algebraic global isometric embedding of the nonrotating BTZ black hole is presented. The ambient spacetime is $\mathbb{M}^{2,3}$, the 3+2 dimensional flat spacetime. We also present the analogous embedding for the Euclidean BTZ spacetime and by performing a kind of double analytic continuation construct a 1-parameter family of embeddings of cosmological AdS spacetime into $\mathbb{M}^{2,3}$ which coincide asymptotically with the embedded BTZ manifold of the appropriate mass. Finally we note that the family of embeddings of cosmological AdS$_{n}$ into $\mathbb{M}^{2,n}$ generalises to higher…
We consider the duality of the quasilocal black hole thermodynamics, explicitly the quasilocal black hole thermodynamic first law, in BTZ black hole solution as a special one of the three-dimensional low energy effective string theory.
Minkowski space is the local model of 3 dimensionnal flat spacetimes. Recent progress in the description of globally hyperbolic flat spacetimes showed strong link between Lorentzian geometry and Teichm{ü}ller space. We notice that Lorentzian generalisations of conical singularities are useful for the endeavours of descripting flat spacetimes, creating stronger links with hyperbolic geometry and compactifying spacetimes. In particular massive particles and extreme BTZ singular lines arise naturally. This paper is three-fold. First, prove background local properties which will be useful for futu…
We compute the transition rate of an Unruh-DeWitt detector coupled both to a ground state and to a KMS state of a massless, conformally coupled scalar field on a static BTZ black hole with Robin boundary conditions. We observe that, although the anti-Hawking effect is manifest for the ground state, this is not the case for the KMS state. In addition, we show that our analysis applies with minor modifications also to the anti-Unruh effect on Rindler-AdS$_3$ spacetime.
We calculated the Dirac Particles' Hawking radiation from the outer horizon of BTZ black hole via tunneling formalism. Applying WKB approximation to the Dirac equation in (2+1) dimensional BTZ spacetime background, we obtain the radiation spectrum for fermions and Hawking temperature of BTZ black hole. The results obtained by taking the fermion tunneling into account is consistent with the previous literatures.
We investigate the existence and stability of both the timelike and null circular orbits for a (2+1) dimensional charged BTZ black hole in Einstein-nonlinear Maxwell gravity with a negative cosmological constant. The stability analysis of orbits are performed to study the possibility of chaos in geodesic motion for a special case of black hole so-called conformally invariant Maxwell spacetime. The computations of both proper time Lyapunov exponent ($λ_{p}$) and coordinate time Lyapunov exponent ($λ_{c}$) are useful to determine the stability of these circular orbits. We observe the behavior of…
We analyse vacuum decay in rotating BTZ black hole spacetimes with the thin wall approximation. Possible parameter regions for the vacuum decay are clarified. We find that the nucleation rate is dominated by the bounce solution with the static shell configuration. The nucleation rate of the static shell decreases with the mass of the initial black hole. For a larger/smaller value of the initial black hole, the nucleation rate can be smaller/larger than that of the Coleman De Luccia vacuum decay in the pure AdS spacetime. Through the vacuum decay, the black hole gains its mass and loses the hor…
A simple algebraic global isometric embedding is presented for the nonrotating BTZ black hole and its counterpart of Euclidean signature. The image of the embedding, in Minkowski space of two extra dimensions, is the interection of two quadric hypersurfaces. Furthermore an embedding into $AdS_4$ or $H_4$ is also obtained, showing that the spacetime is of embedding class one with respect to maximally symmetric space of negative curvature. The rotating solution of Euclidean signature is also shown to admit a quadratic algebraic embedding, but seemingly requires more than two extra dimensions.
The spinless Bañados-Teiltelboim-Zanelli (BTZ) spacetime is considered in the quantum theory context. Specially, we study the case of negative mass parameter using quantum test particles obeying the Klein-Gordon and Dirac equations. We study if this classical singular spacetime, with a naked singularity at the origin, remains singular when tested with quantum particles. The need of additional information near the origin is confirmed for massive scalar particles and all the possible boundary conditions necessary to turn the spatial portion of the wave operator self-adjoint are found. When teste…
The effects of spacetime quantization on black hole and big bang/big crunch singularities can be studied using new tools from (2+1)-dimensional quantum gravity. I investigate effects of spacetime quantization on singularities of the (2+1)-dimensional BTZ black hole and the (2+1)-dimensional torus universe. Hosoya has considered the BTZ black hole, and using a ``quantum generalized affine parameter'' (QGAP), has shown that, for some specific paths, quantum effects ``smear'' the singularity. Using generic gaussian wave functions, I show that both BTZ black hole and the torus universe contain fam…
We investigate the first law of thermodynamics in the case of the (2+1)-dimensional BTZ black holes and Kerr-de Sitter spacetimes, in particular, we focus on the integral mass formulas. It is found that by assuming the cosmological constant as a variable state parameter, both the differential and integral mass formulas of the first law of black hole thermodynamics in the asymptotic flat spacetimes can be directly extended to those of rotating black holes in anti-de Sitter and de Sitter backgrounds. It should be pointed that these formulas come into existence in any dimensions also.
We study properties of resonant solutions to the scalar wave equation in several axially symmetric spacetimes. We prove that non-axial resonant modes do not exist neither in the Lanczos dust cylinder, the $(2+1)$ extreme BTZ spacetime nor in a class of simple rotating wormhole solutions. Moreover, we find unstable solutions to the wave equation in the Lanczos dust cylinder and in the $r^2 <0$ region of the extreme $(2+1)$ BTZ spacetime, two solutions that possess closed timelike curves. Similarities with previous results obtained for the Kerr spacetime are explored.
In this paper we will show that using the cosmological constant as a new thermodynamical state variable, the differential and integral mass formulas of the first law of thermodynamics for asymptotic flat spacetimes can be extended to (2+1) dimensional charged BTZ black holes and charged de Sitter space.
Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. To incorporate DSR into general relativity, one could use "Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. For a black hole, there are two natural orthonormal frames, the stationary one hovering above it and freely falling one along geodesics. Since the rainbow metric is the metric that the radiated particles "see", a more natural orthonormal frame is the one anchored to the particles. And the cases with the stationary orthonormal frame ha…
Recently it has been shown that a 2+1 dimensional black hole can be created by a collapse of two colliding massless particles in otherwise empty anti-de Sitter space. Here we generalize this construction to the case of a non-zero impact parameter. The resulting spacetime, which may be regarded as a Gott universe in anti-de Sitter background, contains closed timelike curves. By treating these as singular we are able to interpret our solution as a rotating black hole, hence providing a link between the Gott universe and the BTZ black hole. When analyzing the spacetime we see how the full causal …
It is shown that 2+1 dimensional anti-de Sitter spacetimes are Lorentz-flat. This means, in particular, that any simply-connected patch of the BTZ black hole solution can be endowed with a Lorentz connection that is locally pure gauge. The result can be naturally extended to a wider class of black hole geometries and point particles in three-dimensional spacetime.
We evaluate the energy distribution associated with the (2+1)-dimensional rotating BTZ black hole. The energy-momentum complexes of Landau-Lifshitz and Weinberg are employed for this computation. Both prescriptions give exactly the same form of energy distribution. Therefore, these results provide evidence in support of the claim that, for a given gravitational background, different energy-momentum complexes can give identical results in three dimensions, as it is the case in four dimensions.
Quantum singularities considered in the 3D BTZ spacetime by Pitelli and Letelier (Phys. Rev. D77: 124030, 2008) is extended to charged BTZ and 3D Einstein-Maxwell-dilaton gravity spacetimes. The occurence of naked singularities in the Einstein-Maxwell extension of the BTZ spacetime both in linear and non-linear electrodynamics as well as in the Einstein-Maxwell-dilaton gravity spacetimes are analysed with the quantum test fields obeying the Klein-Gordon and Dirac equations. We show that with the inclusion of the matter fields; the conical geometry near r=0 is removed and restricted classes of …
As is already known, a spacetime horizon acts like a boundary of a thermal system an we can associate with it notions as temperature and entropy. Following the work of M. Akbar, in this paper we will show how it is possible to interpret the field equation of a charged BTZ black hole near horizon as a thermodynamic identity dE=TdS+PdA+PhidQ, where Phi is the electric potential and Q is the electric charge of BTZ black hole. These results indicate that the field equations for the charged BTZ black hole possess intrinsic thermodynamic properties near horizon.
We investigate the correlation harvesting protocol for mutual information between two Unruh-DeWitt detectors in a static BTZ black hole spacetime. Here, the effects coming from communication and change in proper separation of the detectors are set to be negligible so that only a black hole affects the extracted mutual information. We find that, unlike the entanglement harvesting scenario, harvested mutual information is zero only when a detector reaches an event horizon, and that although the Hawking effect and gravitational redshift both affect the extraction of mutual information, it is extr…
In three dimensional spacetime with negative cosmology constant, the general relativity can be written as two copies of SO$(2,1)$ Chern-Simons theory. On a manifold with boundary the Chern-Simons theory induces a conformal field theory--WZW theory on the boundary. In this paper, it is show that with suitable boundary condition for BTZ black hole, the WZW theory can reduce to a massless scalar field on the horizon.
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.
In this paper we will show that using the cosmological constant as a new thermodynamical state variable, the differential and integral mass formulas of the first law of thermodynamics for asymptotic flat spacetimes can be extended to be used at the two horizons of the (2+1) dimensional BTZ black hole. We also extend this equations to the stringy description of the BTZ black hole, in which two new systems that resemble the right and left modes of effective string theory, are defined in terms of the inner and outer horizons.
The entanglement entropy approach to study the dependence of entropy upon the location of degrees of freedom (dof) (near/far) from the horizon is discussed in this article. We try to understand the physical deviation of the area law for the excited states by incorporating the logarithmic and power law corrections. We show that the dof near the horizon gives contribution to the total entropy of the system in the ground state and away from the event horizon gives contribution to the excited state.
This study is purposed to derive the equation of motion for geodesics in vicinity of spacetime of a (2 + 1)-dimensional charged BTZ black hole. In this paper, we solve geodesics for both massive and massless particles in terms of Weierstrass elliptic and Kleinian sigma hyper-elliptic functions. Then we determine different trajectories of motion for particles in terms of conserved energy and angular momentum and also using effective potential.
In this paper a Banados, Teitelboim and Zanelli (BTZ) black hole is constructed from an exact solution of the Einstein field equations in a (2+1)-dimensional anti-de Sitter spacetime in the context of noncommutative geometry. The BTZ black hole turns out to have two horizons, no horizon or a single horizon corresponding to a minimal mass. Certain thermodynamical properties are investigated, including Hawking temperature, entropy and heat capacity. Also discussed is the geodesic structure of BTZ black holes for both massless and massive particles. In particular, it is shown that bound orbits fo…
We propose an approach for constructing spatial slices of (3+1) spacetimes with cosmological constant but without a matter content, which yields (2+1) vacuum with $Λ$ solutions. The reduction mechanism from (3+1) to (2+1) gravity is supported on a criterion in which the Weyl tensor components are required to vanish together with a dimensional reduction via an appropriate foliation. By using an adequate reduction mechanism from the Plebański-Carter[A] solution in (3+1) gravity, the (2+1) BTZ solution can be obtained.
In this paper, we study static vacuum solutions of quantum gravity at a fixed Lifshitz point in (2+1) dimensions, and present all the diagonal solutions in closed forms in the infrared limit. The exact solutions represent spacetimes with very rich structures: they can represent generalized BTZ black holes, Lifshitz space-times or Lifshitz solitons, in which the spacetimes are free of any kind of space-time singularities, depending on the choices of the free parameters of the solutions. We also find several classes of exact static non-diagonal solutions, which represent similar space-time struc…
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 be…
In this paper, we study the phase transition and the entropy spectrum of BTZ black hole obtained in a model of three-dimensional gravity with torsion. By calculating the heat capacity we find that the BTZ black hole we considered will experience phase transition at some critical point. This indicates that the critical behaviors of black holes do not only depend on the spacetime metric, but have to do with the theory of gravity under consideration. In addition we derived the entropy spectrum of the BTZ black hole according to the quasinormal modes(QNMs) and the adiabatic invariance. It shows th…
Motivated by recent developments of BTZ black holes and interesting results of massive gravity, we investigate massive BTZ black holes in the presence of Maxwell and Born-Infeld (BI) electrodynamics. We study geometrical properties such as type of singularity and asymptotical behavior as well as thermodynamic structure of the solutions through canonical ensemble. We show that despite the existence of massive term, obtained solutions are asymptotically (a)dS and have a curvature singularity at the origin. Then, we regard varying cosmological constant and examine the Van der Waals like behavior …
In order to classify and understand the spacetime structure, investigation of the geodesic motion of massive and massless particles is a key tool. So the geodesic equation is a central equation of gravitating systems and the subject of geodesics in the black hole dictionary attracted much attention. In this paper, we give a full description of geodesic motions in three-dimensional spacetime. We investigate the geodesics near charged BTZ black holes and then generalize our prescriptions to the case of massive gravity. We show that electric charge is a critical parameter for categorizing the geo…
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 present a detailed study of gravitational lensing around a rotating BTZ black hole in (2+1) dimensional gravity. The study of orbits for massless test particle around this BH spacetime is performed to describe the nature of cosmological constant in lower dimensions. We study the effect of cosmological constant on the photon orbit in view of other critical parameters. The bending angle of light is studied in view of different values of cosmological constant for direct and retrograde motion of test particles. It is being observed that the bending angle slightly decreases as the value of cosmo…
We consider a real, massive scalar field in BTZ spacetime, a 2+1-dimensional black hole solution of the Einstein's field equations with a negative cosmological constant. First, we analyze the space of classical solutions in a mode decomposition and we characterize the collection of all admissible boundary conditions of Robin type which can be imposed at infinity. Secondly, we investigate whether, for a given boundary condition, there exists a ground state by constructing explicitly its two-point function. We demonstrate that for a subclass of the boundary conditions it is possible to construct…
The observation that the 2+1 dimensional BTZ black hole can be obtained as a quotient space of anti-de Sitter space leads one to ask what causal behaviour other such quotient spaces can display. In this paper we answer this question in 2+1 and 3+1 dimensions when the identification group has one generator. Among other things we find that there does not exist any 3+1 generalization of the rotating BTZ hole. However, the non-rotating generalization exists and exhibits some unexpected properties. For example, it turns out to be non-static and to possess a non-trivial apparent horizon.
We investigate the thermodynamic equilibrium states of a rotating thin shell, i.e., a ring, in a (2+1)-dimensional spacetime with a negative cosmological constant. The inner and outer regions with respect to the shell are given by the vacuum anti-de Sitter and the rotating Bañados-Teitelbom-Zanelli spacetimes, respectively. The first law of thermodynamics on the thin shell, together with three equations of state for the pressure, the local inverse temperature and the thermodynamic angular velocity of the shell, yields the entropy of the shell, which is shown to depend only on its gravitational…
The (single) black hole solutions of Bañados, Teitelboim and Zanelli (BTZ) in 2+1 dimensional anti-de Sitter space are generalized to an arbitrary number $n$ of such black holes. The resulting multi-black-hole (MBH) spacetime is locally isometric to anti-de Sitter space, and globally it is obtained from the latter as a quotient space by means of suitable identifications. The MBH spacetime has $n$ asymptotically anti-de Sitter exterior regions, each of which has the geometry of a single BTZ black hole. These exterior regions are separated by $n$ horizons from a common interior region. This inte…
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…
The regular black hole solution arising as a spherically symmetric vacuum solution of Born-Infeld gravity possesses an asymptotic interior structure which is very well described by a four dimensional generalization of the non-rotating BTZ metric. According to this picture no singularity exists, and instead, infalling observers experience a constant curvature manifold as they travel towards future null infinity. This is characterized by the BTZ event horizon. The exterior structure of the black hole is also studied, and it is shown that it corresponds to the Schwarzschild solution provided the …
Lower dimensional gravity has the potential of providing non-trivial and valuable insight into some of the conceptual issues arising in four dimensional relativistic gravitational analysis. The asymptotically anti-de Sitter (2+1) dimensional spacetime described by Ba$\tilde{n}$ados, Teitelboim and Zanelli (BTZ) which admits a black hole solution, has become a source of fascination for theoretical physicists in recent years. By suitably choosing the form of the mass function $m(r)$, we derive a new class of solutions for the interior of an isotropic star corresponding to the exterior (2+1) asym…
We investigate the Noether symmetries of the Lagrangian for the stationary rotating BTZ-type three-dimensional spacetimes in $f(R)$ theory of gravity. A detailed analysis of Noether symmetries of (2+1)-dimensional rotating BTZ-type black hole spacetime model is presented. Applying the Noether symmetry approach, the first integrals (constants of motion) for each of Noether symmetries are obtained to look for the exact solutions. After solving the first integral equations depending on the form of the function $f(R)$, we derived some new (2+1)-dimensional rotating BTZ-type black hole solutions. W…
It is well-known that unlike its chargeless version the charged Banados-Teitelboim-Zanelli (BTZ) black hole solution in 2+1- dimensional spacetime is singular. We construct a charged, regular extension of the BTZ black hole solution by employing nonlinear Born-Infeld electrodynamics, supplemented with the Hoffmann term and gluing different spacetimes. The role of the latter term is to divide spacetime in a natural way into two regions by a circle and eliminate the inner singularity. Thermodynamics of such a black hole is investigated by Kaluza-Klein reduction to the 1+1-dimensional dilaton gra…
