Three-dimensional BTZ Black Hole Research Articles

The effects of running gravitational coupling on three dimensional black holes

In the present work, we investigate the consequences of running gravitational coupling on the properties of the three-dimensional BTZ black hole. We take as starting point the functional form of gravitational coupling obtained in the context of asymptotic safe gravity theory. By using the standard scale setting relation where ksim xi /r^n, we compute the solution of the Einstein field equations. We get and analyze the horizon and the thermodynamic properties of this new class of black hole solutions. The impact of the scale-dependent parameter xi on the cosmological “constant” and metric functions are briefly discussed. We find that the null energy condition is also violated in this setup when scale-dependent gravity and Newton’s coupling (coming from the asymptotic safety scenario of gravity) are simultaneously taken into account.

Black hole and BTZ black string in the Einstein- SU(2) Skyrme model

We present novel analytic hairy black holes with a flat base manifold in the (3+1)-dimensional Einstein SU(2)-Skyrme system with negative cosmological constant. We also construct (3+1)-dimensional black strings in the Einstein $SU(2)$-non linear sigma model theory with negative cosmological constant. The geometry of these black strings is a three-dimensional charged BTZ black hole times a line, without any warp factor. The thermodynamics of these configurations (and its dependence on the discrete hairy parameter) is analyzed in details. A very rich phase diagram emerges.

Motion and collision of particles in a rotating linear dilaton black hole

We study the motion of particles in the background of a four-dimensional linear dilaton black hole. We solve analytically the equations of motion of the test particles and we describe their motion. We show that the dilaton black hole acts as a particle accelerator by analyzing the energy in the center of mass (CM) frame of two colliding particles in the vicinity of its horizon. In particular we find that there is a critical value of the angular momentum, which depends on the string coupling, and a particle with this critical angular momentum can reach the inner horizon with an arbitrarily high CM energy. This is known as the Ba\~nados, Silk and West (BSW) process. We also show that the motion and collisions of particles have a similar behavior to the three-dimensional BTZ black hole. In fact, the photons can plunge into the horizon or escape to infinity, and they can not be deflected, while for massive particles there are no confined orbits of first kind, like planetary or circular orbits.

Throat quantization of the Schwarzschild-Tangherlini(-AdS) black hole

By the throat quantization pioneered by Louko and Mäkelä, we derive the mass and area/entropy spectra for the Schwarzschild-Tangherlini-type asymptotically flat or AdS vacuum black hole in arbitrary dimensions. Using the WKB approximation for black holes with large mass, we show that area/entropy is equally spaced for asymptotically flat black holes, while mass is equally spaced for asymptotically AdS black holes. Exact spectra can be obtained for toroidal AdS black holes in arbitrary dimensions including the three-dimensional BTZ black hole.

Heat engine in the three-dimensional spacetime

We define a kind of heat engine via three-dimensional charged BTZ black holes. This case is quite subtle and needs to be more careful. The heat flow along the isochores does not equal to zero since the specific heat CV ≠ 0 and this point completely differs from the cases discussed before whose isochores and adiabats are identical. So one cannot simply apply the paradigm in the former literatures. However, if one introduces a new thermodynamic parameter associated with the renormalization length scale, the above problem can be solved. We obtain the analytical efficiency expression of the three-dimensional charged BTZ black hole heat engine for two different schemes. Moreover, we double check with the exact formula. Our result presents the first specific example for the sound correctness of the exact efficiency formula. We argue that the three-dimensional charged BTZ black hole can be viewed as a toy model for further investigation of holographic heat engine. Furthermore, we compare our result with that of the Carnot cycle and extend the former result to three-dimensional spacetime. In this sense, the result in this paper would be complementary to those obtained in four-dimensional spacetime or ever higher. Last but not the least, the heat engine efficiency discussed in this paper may serve as a criterion to discriminate the two thermodynamic approaches introduced in ref. [29] and our result seems to support the approach which introduces a new thermodynamic parameter R = r0.

Black hole remnant in asymptotic anti-de Sitter space

It is known that a solution of remnant were suggested for black hole ground state after surface gravity is corrected by loop quantum effect. On the other hand, a Schwarzschild black hole in asymptotic Anti-de Sitter space would tunnel into the thermal soliton solution known as the Hawking-Page phase transition. In this letter, we investigate the low temperature phase of three-dimensional BTZ black hole and four-dimensional AdS Schwarzschild black hole. We find that the thermal soliton is energetically favored than the remnant solution at low temperature in three dimensions, while Planck-size remnant is still possible in four dimensions. Though the BTZ remnant seems energetically disfavored, we argue that it is still possible to be found in the overcooled phase if strings were present and its implication is discussed.

Geometrical Method for Thermal Instability of Nonlinearly Charged BTZ Black Holes

We consider three-dimensional BTZ black holes with three models of nonlinear electrodynamics as source. Calculating heat capacity, we study the stability and phase transitions of these black holes. We show that Maxwell, logarithmic, and exponential theories yield only type one phase transition which is related to the root(s) of heat capacity, whereas, for correction form of nonlinear electrodynamics, heat capacity contains two roots and one divergence point. Next, we use geometrical approach for studying classical thermodynamical behavior of the system. We show that Weinhold and Ruppeiner metrics fail to provide fruitful results and the consequences of the Quevedo approach are not completely matched to the heat capacity results. Then, we employ a new metric for solving this problem. We show that this approach is successful and all divergencies of its Ricci scalar and phase transition points coincide. We also show that there is no phase transition for uncharged BTZ black holes.

Throat quantization of the Schwarzschild–Tangherlini(-AdS) black hole

Adopting the throat quantization pioneered by Louko and Mäkelä, we derive the mass and area spectra for the Schwarzschild–Tangherlini black hole and its anti-de Sitter (AdS) generalization in arbitrary dimensions. We find that the system can be quantized exactly in three special cases: the three-dimensional BTZ black hole, toroidal black holes in any dimension, and five-dimensional Schwarzshild–Tangherlini(-AdS) black holes. For the remaining cases the spectra are obtained for large mass using the WKB approximation. For asymptotically flat black holes, the area/entropy has an equally spaced spectrum, as expected from previous work. In the asymptotically AdS case on the other hand, it is the mass spectrum that is equally spaced. Our exact results for the BTZ black hole mass with Dirichlet boundary conditions are consistent with the spectra of the corresponding operators in the dual CFT.

On the many saddle points description of quantum black holes

Considering two dimensional gravity coupled to a CFT, we show that a semiclassical black hole can be described in terms of two Liouville theories matched at the horizon. The black hole exterior corresponds to a space-like while the interior to a time-like Liouville theory. This matching automatically implies that a semiclassical black hole has an infinite entropy. The path integral description of the time-like Liouville theory (the Black Hole interior) is studied and it is found that the correlation functions of the coupled CFT-gravity system are dominated by two (complex) saddle points, even in the semiclassical limit. We argue that this system can be interpreted as two interacting Bose–Einstein condensates constructed out of two degenerate quantum states. In AdS/CFT context, the same system is mapped into two interacting strings intersecting inside a three-dimensional BTZ black hole.

ON THE TIME TIMES TEMPERATURE BOUND

Recently Hod proposed a lower bound on the relaxation time of a perturbed thermodynamic system. For gravitational systems this bound transforms into a condition on the fundamental quasinormal frequency. We test the bound in some space–times whose quasinormal frequencies are calculated exactly, as the three-dimensional BTZ black hole, the D-dimensional de Sitter space–time, and the D-dimensional Nariai space–time. We find that for some of these space–times their fundamental quasinormal frequencies do not satisfy the bound proposed by Hod.

Generalized uncertainty principle and entropy of three-dimensional BTZ black hole

Applying the new equation of state density motivated by the generalized uncertainty relation in the quantum gravity field，we investigate the black hole entropy on the background of three-dimensionnal BTZ. When the λ with the Planck scale，and relative to the dimensions of spacetime introduced in generalized uncertainty relation has a fixed value，we obtain the Bekenstein-Hawking entropy of BTZ black hole and the correction term. Because we have used the new equation of state density，in our results the divergence term appearing in the brick-wall model is removed，at the same time with holding the small mass approximation. Thus the correction value of the Bekenstein-Hawking entropy of the black hole is derived from the quantum statistical view. It depends the understanding of the correction value of the black hole entropy.

Canonical entropy of three-dimensional BTZ black hole

Abstract Recently, Hawking radiation of the black hole has been studied using the tunnel effect method. It is found that the radiation spectrum of the black hole is not a strictly pure thermal spectrum. How does the departure from pure thermal spectrum affect the entropy? This is a very interesting problem. In this Letter, we calculate the partition function by energy spectrum obtained from tunnel effect. Using the partition function, we compute the black hole entropy and derive the expression of the black hole entropy after considering the radiation. And we derive the entropy of charged black hole. In our calculation, we consider not only the correction to the black hole entropy due to fluctuation of energy but also the effect of the change of the black hole charges on entropy. There is no other hypothesis. Our result is more reasonable. According to the fact that the black hole entropy is not divergent, we obtain the lower limit of Banados–Teitelboim–Zanelli black hole energy. That is, the least energy of Banados–Teitelboim–Zanelli black hole, which satisfies the stationary condition in thermodynamics.

Two-dimensional quantum black holes, branes in Banados-Teitelboim-Zanelli spacetime, and holography

We solve semiclassical Einstein equations in two dimensions with a massive source and we find a static, thermodynamically stable, quantum black hole solution in the Hartle-Hawking vacuum state. We then study the black hole geometry generated by a boundary mass sitting on a non-zero tension 1-brane embedded in a three-dimensional BTZ black hole. We show that the two geometries coincide and we extract, using holographic relations, information about the CFT living on the 1-brane. Finally, we show that the quantum black hole has the same temperature of the bulk BTZ, as expected from the holographic principle.

On the thermal description of the BTZ black holes

We investigate the limitations on the thermal description of three-dimensional BTZ black holes. We derive on physical grounds three basic mass scales that are relevant to characterize these limitations. The Planck mass in 2+1 dimensions indicates the limits where the black hole can emit Hawking's radiation. We show that the back reaction is meaningless for spinless BTZ black hole. For stationary BTZ black holes the nearly extreme case is analyzed showing that may occur a breakdown of its description as a thermal object.

Entropy of three-dimensional BTZ black holes

The entropies of scalar field and neutrino field are calculated in the background of three-dimensional BTZ black hole. Considering statistical physics, we propose not to consider the superraradiant modes for bosons (Fermion fields do not display supperradiance). In fact, the nonsuperradiant modes do contribute exactly the area entropy for both bosons and fermions. The result shows that the neutrino field entropy is 3/2 times the scalar one.

Logarithmic corrections to three-dimensional black holes and de Sitter spaces

We calculate logarithmic corrections to the Bekenstein–Hawking entropy for three-dimensional BTZ black hole with J=0 and Kerr–de Sitter (KdS) space with J=0 including the Schwarzschild–de Sitter (SdS) solution due to thermal fluctuations. It is found that there is no distinction between the event horizon of the BTZ black hole and the cosmological horizon of KdS space. We obtain the same correction to the Cardy formula for BTZ, KdS, and SdS cases. We discuss AdS/CFT and dS/ECFT correspondences in connection with logarithmic corrections.

A Zeta Function for the BTZ Black Hole

For the three-dimensional BTZ black hole we consider a Selberg type zeta function. We indicate how special values of its logarithm correspond to certain thermodynamic quantities associated with the black hole.

CHOPTUIK SCALING AND QUASINORMAL MODES IN THE ADS/CFT CORRESPONDENCE

The exact quasinormal frequencies for scalar perturbations of the three-dimensional BTZ black hole were presented1. The timescale for the decay of the scalar perturbation is given by the imaginary part of the quasinormal frequencies. Via the AdS/CFT correspondence, one then obtains a prediction of the timescale for return to equilibrium of the dual conformal field theory. By studying a two-particle collision process, an exact analytic expression for the Choptuik scaling parameter of the BTZ black hole was obtained2. A possible connection between the Choptuik scaling parameter and the imaginary part of the quasinormal frequencies was explored1.

EXACT RESULTS FOR THE BTZ BLACK HOLE

In this review, we summarize exact results for the three-dimensional BTZ black hole. We use rigorous mathematical results to clarify the general structure and properties of this black hole spacetime and its microscopic description. In particular, we study the formation of the black hole by point particle collisions, leading to an exact analytic determination of the Choptuik scaling parameter. We also show that a "No Hair Theorem" follows immediately from a mathematical theorem of hyperbolic geometry, due to Sullivan. A microscopic understanding of the Bekenstein–Hawking entropy, and decay rate for massless scalars, is shown to follow from standard results of conformal field theory.

Choptuik scaling and quasinormal modes in the anti–de Sitter space/conformal-field theory correspondence

We establish an exact connection between the Choptuik scaling parameter for the three-dimensional BTZ black hole, and the imaginary part of the quasinormal frequencies for scalar perturbations. Via the AdS/CFT correspondence, this leads to an interpretation of Choptuik scaling in terms of the timescale for return to equilibrium of the dual conformal field theory.