Non-extremal Black Holes Research Articles

Thermodynamics of extremal rotating thin shells in an extremal BTZ spacetime and the extremal black hole entropy

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 Ba\~{n}ados-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 into an extremal rotating BTZ black hole, it is found that the entropy of the spacetime remains such a function of $A_+$. It is thus vindicated, that extremal black holes, here extremal BTZ black holes, have different properties from the corresponding nonextremal black holes, which have the Bekenstein-Hawking entropy $S(A_+)= \frac{A_+}{4G}$, where $G$ is the gravitational constant. It is argued that for the extremal case $0\leq S(A_+)\leq \frac{A_+}{4G}$. Thus, rather than having just two entropies for extremal black holes, as previous results debated, 0 and $\frac{A_+}{4G}$, it is shown that extremal black holes may have a continuous range of entropies, limited by precisely those two entropies. Surely, the entropy that a particular extremal black hole picks must depend on past processes, notably on how it was formed. It is also found a remarkable relation between the third law of thermodynamics and the impossibility for a massive body to reach the velocity of light. In the procedure, it becomes clear that there are two distinct angular velocities for the shell, the mechanical and thermodynamic angular velocities. In passing, we clarify, for a static spacetime with a thermal shell, the meaning of the Tolman temperature formula at a generic radius and at the shell. (Abridged version).

Area Products for H± in AdS Space

We derive the thermodynamic products particularly area (or entropy) products of ${\cal H}^{\pm}$ for certain class of black holes in AdS space. We show by explicit and exact calculations that more complicated function of event horizon area and Cauchy horizon area is indeed mass-independent. This mass-independent quantities indicate that they \emph{could turn out to be an "universal" quantity} provided that they depends only on the quantized angular momentum, quantized charges, and cosmological constant respectively. Furthermore, these area (or entropy) product relations for several class of black holes in AdS space gives us strong indication to understanding the nature of non-extremal black hole entropy (both inner and outer) at the microscopic level. Moreover, we compute the Penrose's famous \emph{Cosmic Censorship Inequality} (which requires Cosmic-Censorship hypothesis) for these class of black holes in AdS space . Local thermodynamic stability has been discussed for these black holes and under certain condition certain black holes displayed second order phase transition.

Hawking Radiation-Quasinormal Modes Correspondence for Large AdS Black Holes

It is well-known that the nonstrictly thermal character of the Hawking radiation spectrum generates a natural correspondence between Hawking radiation and black hole quasinormal modes. This main issue has been analyzed in the framework of Schwarzschild black holes, Kerr black holes, and nonextremal Reissner-Nordstrom black holes. In this paper, by introducing the effective temperature, we reanalyze the nonstrictly thermal character of large AdS black holes. The results show that the effective mass corresponding to the effective temperature is approximatively the average one in any dimension. And the other effective quantities can also be obtained. Based on the known forms of frequency in quasinormal modes, we reanalyze the asymptotic frequencies of the large AdS black hole in three and five dimensions. Then we get the formulas of the Bekenstein-Hawking entropy and the horizon’s area quantization with functions of the quantum “overtone” number n.

Linear stability of the non-extreme Kerr black hole

It is proven that for smooth initial data with compact support outside the event horizon, the solution of every azimuthal mode of the Teukolsky equation for general spin decays pointwise in time.

BMS type symmetries at null-infinity and near horizon of non-extremal black holes

In this paper we consider a generally covariant theory of gravity, and extend the generalized off-shell ADT current such that it becomes conserved for field dependent (asymptotically) Killing vector field. Then we define the extended off-shell ADT current and the extended off-shell ADT charge. Consequently, we define the conserved charge perturbation by integrating from the extended off-shell ADT charge over a spacelike codimension two surface. Eventually, we use the presented formalism to find the conserved charge perturbation of an asymptotically flat spacetime. The conserved charge perturbation we obtain is exactly matched with the result of Ref. (Barnich and Troessaert, 12:105 2011). These charges are as representations of the $$\hbox {BMS}_4$$ symmetry algebra. Also, we find that the near horizon conserved charges of a non-extremal black hole with extended symmetries are the Noether charges. For this case our result is also exactly matched with that of Ref. (Donnay et al., arXiv:1607.05703 [hep-th], 2016).

The Heisenberg algebra as near horizon symmetry of the black flower solutions of Chern–Simons-like theories of gravity

In this paper we study the near horizon symmetry algebra of the non-extremal black hole solutions of the Chern–Simons-like theories of gravity, which are stationary but are not necessarily spherically symmetric. We define the extended off-shell ADT current which is an extension of the generalized ADT current. We use the extended off-shell ADT current to define quasi-local conserved charges such that they are conserved for Killing vectors and asymptotically Killing vectors which depend on dynamical fields of the considered theory. We apply this formalism to the Generalized Minimal Massive Gravity (GMMG) and obtain conserved charges of a spacetime which describes near horizon geometry of non-extremal black holes. Eventually, we find the algebra of conserved charges in Fourier modes. It is interesting that, similar to the Einstein gravity in the presence of negative cosmological constant, for the GMMG model also we obtain the Heisenberg algebra as the near horizon symmetry algebra of the black flower solutions. Also the vacuum state and all descendants of the vacuum have the same energy. Thus these zero energy excitations on the horizon appear as soft hairs on the black hole.

Schwinger effect, Hawking radiation and Unruh effect

We revisit the Schwinger effect in de Sitter (ds), anti-de Sitter (Ads) spaces and charged black holes, and explore the interplay between quantum electrodynamics (QED) and the quantum gravity effect at one-loop level. We then advance a thermal interpretation of the Schwinger effect in curved spacetimes. Finally, we show that the Schwinger effect in a near-extremal black hole differs from Hawking radiation of charged particles in a nonextremal black hole and is factorized into those in an Ads space and a Rindler space with the surface gravity for acceleration.

Extended symmetries at the black hole horizon

We prove that non-extremal black holes in four-dimensional general relativity exhibit an infinite-dimensional symmetry in their near horizon region. By prescribing a physically sensible set of boundary conditions at the horizon, we derive the algebra of asymptotic Killing vectors, which is shown to be infinite-dimensional and includes, in particular, two sets of supertranslations and two mutually commuting copies of the Virasoro algebra. We define the surface charges associated to the asymptotic diffeomorphisms that preserve the boundary conditions and discuss the subtleties of this definition, such as the integrability conditions and the correct definition of the Dirac brackets. When evaluated on the stationary solutions, the only non-vanishing charges are the zero-modes. One of them reproduces the Bekenstein-Hawking entropy of Kerr black holes. We also study the extremal limit, recovering the NHEK geometry. In this singular case, where the algebra of charges and the integrability conditions get modified, we find that the computation of the zero-modes correctly reproduces the black hole entropy. Furthermore, we analyze the case of three spacetime dimensions, in which the integrability conditions notably simplify and the field equations can be solved analytically to produce a family of exact solutions that realize the boundary conditions explicitly. We examine other features, such as the form of the algebra in the extremal limit and the relation to other works in the literature.

Black holes with vector hair

In this paper, we consider Einstein gravity coupled to a vector field, either minimally or non-minimally, together with a vector potential of the type $V=2\Lambda_0+\ft 12 m^2 A^2+\gamma_4 A^4$. For a simpler non-minimally coupled theory with $\Lambda_0=m=\gamma_4=0$, we obtain both extremal and non-extremal black hole solutions that are asymptotic to Minkowski space-times. We study the global properties of the solutions and derive the first law of thermodynamics using Wald formalism. We find that the thermodynamical first laws of the extremal black holes are modified by a one form associated with the vector field. In particular, due to the existence of the non-minimal coupling, the vector forms thermodynamic conjugates with the graviton mode and partly contributes to the one form modifying the first laws. For a minimally coupled theory with $\Lambda_0\neq 0$, we also obtain one class of asymptotically flat extremal black hole solutions in general dimensions. This is possible because the parameters $(m^2,\gamma_4)$ take certain values such that $V=0$. In particular, we find that the vector also forms thermodynamic conjugates with the graviton mode and contributes to the corresponding first laws, although the non-minimal coupling has been turned off. Thus all the extremal black hole solutions that we obtain provide highly non-trivial examples how the first law of thermodynamics can be modified by a either minimally or non-minimally coupled vector field. We also study Gauss-Bonnet gravity non-minimally coupled to a vector and obtain asymptotically flat black holes and Lifshitz black holes.

Near horizon symmetries of the non-extremal black hole solutions of Generalized Minimal Massive Gravity

We consider the Generalized Minimal Massive Gravity (GMMG) model in the first order formalism. We show that all the solutions of the Einstein gravity with negative cosmological constants solve the equations of motion of considered model. Then we find an expression for the off-shell conserved charges of this model. By considering the near horizon geometry of a three dimensional black hole in the Gaussian null coordinates, we find near horizon conserved charges and their algebra. The obtained algebra is centrally extended. By writing the algebra of conserved charges in terms of Fourier modes and considering the BTZ black hole solution as an example, one can see that the charge associated with rotations along $\mathcal{Y}_{0}$ coincides exactly with the angular momentum, and he charge associated with time translations $\mathcal{T}_{0}$ is the product of the black hole entropy and its temperature. As we expect, in the limit when the GMMG tends to the Einstein gravity, all the result we obtain in this paper reduce to the result of the paper \cite{5}.

Superradiantly stable non-extremal Reissner–Nordström black holes

The superradiant stability is investigated for non-extremal Reissner-Nordstrom black hole. We use an algebraic method to demonstrate that all non-extremal Reissner-Nordstrom black holes are superradiantly stable against a charged massive scalar perturbation. This improves the results obtained before for non-extremal Reissner-Nordstrom black holes.

Logarithmic Local Energy Decay for Scalar Waves on a General Class of Asymptotically Flat Spacetimes

This paper establishes that on the domain of outer communications of a general class of stationary and asymptotically flat Lorentzian manifolds of dimension $$d+1$$ , $$d\ge 3$$ , the local energy of solutions to the scalar wave equation $$\square _{g}{\uppsi }=0$$ decays at least with an inverse logarithmic rate. This class of Lorentzian manifolds includes (non-extremal) black hole spacetimes with no restriction on the nature of the trapped set. Spacetimes in this class are moreover allowed to have a small ergoregion, but are required to satisfy an energy boundedness statement. Without making further assumptions, this logarithmic decay rate is shown to be sharp. Our results can be viewed as a generalisation of a result of Burq, dealing with the case of the wave equation on flat space outside compact obstacles, and results of Rodnianski–Tao for asymptotically conic product Lorentzian manifolds. The proof will bridge ideas from Rodnianski and Tao (see [58]) with techniques developed in the black hole setting by Dafermos and Rodnianski (see [21, 22]). As a soft corollary of our results, we will infer an asymptotic completeness statement for the wave equation on the spacetimes considered in the case where no ergoregion is present.

Kerr black holes as accelerators of spinning test particles

It has been shown that ultraenergetic collisions can occur near the horizon of an extremal Kerr black hole. Previous studies mainly focused on geodesic motions of particles. In this paper, we consider spinning test particles whose orbits are non-geodesic. By employing the Mathisson-Papapetrou-Dixon equation, we find the critical angular momentum satisfies $J=2E$ for extremal Kerr black holes. Although the conserved angular momentum $J$ and energy $E$ have been redefined in the presence of spin, the critical condition remains the same form. If a particle with this angular momentum collides with another particle arbitrarily close to the horizon of the black hole, the center-of-mass energy can be arbitrarily high. We also prove that arbitrarily high energies cannot be obtained for spinning particles near the horizons of non-extremal Kerr black holes.

The hot attractor mechanism: decoupling without deep throats

Non-extremal black holes in $\mathcal{N}=2$ supergravity have two horizons, the geometric mean of whose areas recovers the horizon area of the extremal black hole obtained from taking a smooth zero temperature limit. In prior work (arxiv:1410.3478), using the attractor mechanism, we deduced the existence of several moduli independent invariant quantities obtained from averaging over a decoupled inter-horizon region. We establish that non-extremal geometries at the Reissner--Nordstr\"om point, where the scalar moduli are held fixed, can be lifted to solutions in supergravity with a near-horizon AdS$_3\times$S$^2$. These solutions have the same entropy and temperature as the original black hole and therefore allow an interpretation of the underlying gravitational degrees of freedom in terms of CFT$_2$. Symmetries of the moduli space enable us to explicate the origin of entropy in the extremal limit.

Extremal higher spin black holes

The gauge sector of three-dimensional higher spin gravities can be formulated as a Chern-Simons theory. In this context, a higher spin black hole corresponds to a flat connection with suitable holonomy (smoothness) conditions which are consistent with the properties of a generalized thermal ensemble. Building on these ideas, we discuss a definition of black hole extremality which is appropriate to the topological character of 3d higher spin theories. Our definition can be phrased in terms of the Jordan class of the holonomy around a non-contractible (angular) cycle, and we show that it is compatible with the zero-temperature limit of smooth black hole solutions. While this notion of extremality does not require supersymmetry, we exemplify its consequences in the context of sl(3|2) + sl(3|2) Chern-Simons theory and show that, as usual, not all extremal solutions preserve supersymmetries. Remarkably, we find in addition that the higher spin setup allows for non-extremal supersymmetric black hole solutions. Furthermore, we discuss our results from the perspective of the holographic duality between sl(3|2) + sl(3|2) Chern-Simons theory and two-dimensional CFTs with W_{(3|2)} symmetry, the simplest higher spin extension of the N=2 super-Virasoro algebra. In particular, we compute W_{(3|2)} BPS bounds at the full quantum level, and relate their semiclassical limit to extremal black hole or conical defect solutions in the 3d bulk. Along the way, we discuss the role of the spectral flow automorphism and provide a conjecture for the form of the semiclassical BPS bounds in general N=2 two-dimensional CFTs with extended symmetry algebras.

Instabilities of microstate geometries with antibranes

One can obtain very large classes of horizonless microstate geometries corresponding to near-extremal black holes by placing probe supertubes whose action has metastable minima inside certain supersymmetric bubbling solutions. We show that these minima can lower their energy when the bubbles move in certain directions in the moduli space, which implies that these near-extremal microstates are in fact unstable once one considers the dynamics of all their degrees of freedom. The decay of these solutions corresponds to Hawking radiation, and we compare the emission rate and frequency to those of the corresponding black hole. Our analysis supports the expectation that generic non-extremal black holes microstate geometries should be unstable. It also establishes the existence of a new type of instabilities for antibranes in highly-warped regions with charge dissolved in fluxes.

Rotating black hole and quintessence

We discuss spherically symmetric exact solutions of the Einstein equations for quintessential matter surrounding a black hole, which has an additional parameter ($\omega$) due to the quintessential matter, apart from the mass ($M$). In turn, we employ the Newman\(-\)Janis complex transformation to this spherical quintessence black hole solution and present a rotating counterpart that is identified, for $\alpha=-e^2 \neq 0$ and $\omega=1/3$, exactly as the Kerr\(-\)Newman black hole, and as the Kerr black hole when $\alpha=0$. Interestingly, for a given value of parameter $\omega$, there exists a critical rotation parameter ($a=a_{E}$), which corresponds to an extremal black hole with degenerate horizons, while for $a<a_{E}$, it describes a non-extremal black hole with Cauchy and event horizons, and no black hole for $a>a_{E}$. We find that the extremal value $a_E$ is also influenced by the parameter $\omega$ and so is the ergoregion.

Scalar waves from a star orbiting a BTZ black hole

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.

Supertranslations and Superrotations at the Black Hole Horizon.

We show that the asymptotic symmetries close to nonextremal black hole horizons are generated by an extension of supertranslations. This group is generated by a semidirect sum of Virasoro and Abelian currents. The charges associated with the asymptotic Killing symmetries satisfy the same algebra. When considering the special case of a stationary black hole, the zero mode charges correspond to the angular momentum and the entropy at the horizon.

Near-horizon geometry and warped conformal symmetry

We provide boundary conditions for three-dimensional gravity including boosted Rindler spacetimes, representing the near-horizon geometry of non-extremal black holes or flat space cosmologies. These boundary conditions force us to make some unusual choices, like integrating the canonical boundary currents over retarded time and periodically identifying the latter. The asymptotic symmetry algebra turns out to be a Witt algebra plus a twisted u(1) current algebra with vanishing level, corresponding to a twisted warped CFT that is qualitatively different from the ones studied so far in the literature. We show that this symmetry algebra is related to BMS by a twisted Sugawara construction and exhibit relevant features of our theory, including matching micro- and macroscopic calculations of the entropy of zero-mode solutions. We confirm this match in a generalization to boosted Rindler-AdS. Finally, we show how Rindler entropy emerges in a suitable limit.