Degenerate Horizons Research Articles

Anti-evaporation and evaporation of an n -dimensional Reissner-Nordström black hole

We generalize $f(R)$-theory of anti-evaporation/evaporation for a Reissner-Nordstr\"om black hole in $n$-dimensional space-time. We consider non-linear conformally invariant Maxwell field. By perturbing the fields over Nariai-like space-time associated with degenerate horizon, we describe dynamical behavior of horizon. We show that $f(R)$-gravity can offer both anti-evaporation and evaporation in $n$-dimensional Reissner-Nordstr\"om black hole depending on the dimension $n$ and the functional form of $f(R)$. Furthermore, we argue that, in one class of non-oscillatory solution, stable and unstable anti-evaporation/evaporation exist. In the other class of oscillatory solution anti-evaporation/evaporation exists only with instability. The first class of solution may explain a long-lived black hole.

Charged rotating black holes coupled with nonlinear electrodynamics Maxwell field in the mimetic gravity

In mimetic gravity, we derive D-dimension charged black hole solutions having flat or cylindrical horizons with zero curvature boundary. The asymptotic behaviours of these black holes behave as (A)dS. We study both linear and nonlinear forms of the Maxwell field equations in two separate contexts. For the nonlinear case, we derive a new solution having a metric with monopole, dipole and quadrupole terms. The most interesting feature of this black hole is that its dipole and quadruple terms are related by a constant. However, the solution reduces to the linear case of the Maxwell field equations when this constant acquires a null value. Also, we apply a coordinate transformation and derive rotating black hole solutions (for both linear and nonlinear cases). We show that the nonlinear black hole has stronger curvature singularities than the corresponding known black hole solutions in general relativity. We show that the obtained solutions could have at most two horizons. We determine the critical mass of the degenerate horizon at which the two horizons coincide. We study the thermodynamical stability of the solutions. We note that the nonlinear electrodynamics contributes to process a second-order phase transition whereas the heat capacity has an infinite discontinuity.

Multiple Killing horizons and near horizon geometries

Near horizon geometries with multiply degenerate Killing horizons are considered, and their degenerate Killing vector fields identified. We prove that they all arise from hypersurface-orthogonal Killing vectors of any cut of with the inherited metric—cuts are spacelike co-dimension two submanifolds contained in . For each of these Killing vectors on a given cut, there are three different possibilities for the near horizon metric which are presented explicitly. The structure of the metric for near horizon geometries with multiple Killing horizons of order is thereby completely determined, and in particular we prove that the cuts on must be warped products with maximally symmetric fibers (ergo of constant curvature). The question whether multiple degenerate Killing horizons may lead to inequivalent near horizon geometries by using different degenerate Killings is addressed, and answered on the negative: all near horizon geometries built from a given multiple degenerate Killing horizon (using different degenerate Killings) are locally isometric.

Rotating black hole in Rastall theory

Rotating black hole solutions in theories of modified gravity are important as they offer an arena to test these theories through astrophysical observation. The non-rotating black hole can be hardly tested since the black hole spin is very important in any astrophysical process. We present rotating counterpart of a recently obtained spherically symmetric exact black hole solution surrounded by perfect fluid in the context of Rastall theory, viz, rotating Rastall black hole that generalize the Kerr–Newman black hole solution. In turn, we analyze the specific cases of the Kerr–Newman black holes surrounded by matter like dust and quintessence fields. Interestingly, for a set of parameters and a chosen surrounding field, 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} with value a_E is also influenced by these parameters. We also discuss the thermodynamical quantities associated with rotating Rastall black hole, and analyze the particle motion with the behavior of effective potential.

Hairy black hole solutions in a three-dimensional Galileon model

We investigate stationary rotationally symmetric solutions of a particular truncation of Horndeski theory in three dimensions, including a non-minimal scalar kinetic coupling to the curvature. After discussing the special case of a vanishing scalar charge, which includes most of the previously known solutions, we reduce the general case to an effective mechanical model in a three-dimensional target space. We analyze the possible near-horizon behaviors, and conclude that black hole solutions with degenerate horizons and constant curvature asymptotics may exist if the minimal and non-minimal scalar coupling constants have the same sign. In a special case, we find a new analytic rotating black hole solution with scalar hair and degenerate horizon. This is geodesically and causally complete, and asymptotic to the extreme BTZ metric. We also briefly discuss soliton solutions in another special case.

Kaluza-Klein black lens in five dimensions

We obtain a supersymmetric Kaluza-Klein black lens solution in Taub-NUT space in the five-dimensional minimal ungauged supergravity. It is shown that the spacetime has a degenerate horizon with the spatial cross section of the lens space topology L(n,1) and looks like the four-dimensional Minkowski spacetime in the neighborhood of spatial infinity. In contrast to the horizon topology, from a five-dimensional point of view, the spatial infinity has the topology of S^3 rather than the lens space. For this reason, this solution has an asymptotically flat limit. We discuss several properties of such a black lens, in particular, the effect by the compactification of an extra-dimension and some physical differences from the asymptotically flat supersymmetric black lens which has recently been found.

Linear waves on constant radius limits of cosmological black hole spacetimes

In this paper we consider the Klein-Gordon equation on spherically symmetric background spacetimes with a constant area radius. The spacetimes under consideration are Nariai and Plebanski-Hacyan, and can be considered constant radius limits of Reissner-Nordstrom-de Sitter spacetimes. We prove boundedness in the case of a non-negative Klein Gordon mass and decay unless the mass is zero. In the latter case we prove decay of solutions that are supported on all harmonic modes with angular momentum l greater or equal to 1. We show that the l=0 modes of solutions to the massless Klein-Gordon equation do not decay. They are subject to conservation laws along degenerate Killing horizons. We apply the estimates in Nariai to give decay of solutions to the massive Klein-Gordon equation on an n-dimensional de Sitter background, using only the vector field method and with no restrictions on the positive Klein-Gordon mass.

Nonexistence of degenerate horizons in static vacua and black hole uniqueness

We show that in any spacetime dimension D≥4, degenerate components of the event horizon do not exist in static vacuum configurations with positive cosmological constant. We also show that without a cosmological constant asymptotically flat solutions cannot possess a degenerate horizon component. Several independent proofs are presented. One proof follows easily from differential geometry in the near-horizon limit, while others use Bakry–Émery–Ricci bounds for static Einstein manifolds.

Black hole horizons at the extremal limit in Lorentz-violating gravity

Lorentz-violating gravity theories with a preferred foliation can have instantaneous propagation. Nonetheless, it has been shown that black holes can still exist in such theories and the relevant notion of an event horizon has been dubbed `universal horizon'. In stationary spacetimes the universal horizon has to reside in a region of spacetime where the Killing vector associated with stationarity is spacelike. This raises the question of what happens to the universal horizon in the extremal limit, where no such region exists anymore. We use a decoupling limit approximation to study this problem. Our results suggest that at the extremal limit, the extremal Killing horizon appears to play the role of a degenerate universal horizon, despite being a null and not a spacelike surface, and hence not a leaf of the preferred foliation.

Horizon thermodynamics from Einstein's equation of state

By regarding the Einstein equations as equation(s) of state, we demonstrate that a full cohomogeneity horizon first law can be derived in horizon thermodynamics. In this approach both the entropy and the free energy are derived concepts, while the standard (degenerate) horizon first law is recovered by a Legendre projection from the more general one we derive. These results readily generalize to higher curvature gravities where they naturally reproduce a formula for the entropy without introducing Noether charges. Our results thus establish a way of how to formulate consistent black hole thermodynamics without conserved charges.

Kinematic restrictions on particle collisions near extremal black holes: A unified picture

In 2009, Banados, Silk and West (BSW) pointed out the possibility of having an unbounded limit of centre-of-mass collision energy for test particles in the field of an extremal Kerr black hole, if one of them has fine-tuned parameters and the collision point is approaching the horizon. The possibility of this "BSW effect" attracted much attention: it was generalised to arbitrary ("dirty") rotating black holes and an analogy was found for collisions of charged particles in the field of non-rotating charged black holes. Our work considers the unification of these two mechanisms, which have so far been studied only separately. Exploring the enlarged parameter space, we find kinematic restrictions that may prevent the fine-tuned particles from reaching the limiting collision point. These restrictions are first presented in a general form, which can be used with an arbitrary black-hole model, and then visualised for the Kerr-Newman solution by plotting the "admissible region" in the parameter space of critical particles, reproducing some known results and obtaining a number of new ones. For example, we find that (marginally) bounded critical particles with enormous values of angular momentum can, curiously enough, approach the degenerate horizon, if the charge of the black hole is very small. Such "mega-BSW" behaviour is excluded in the case of a vacuum black hole, or a black hole with large charge. It may be interesting in connection with the small "Wald charge" induced on rotating black holes in external magnetic fields.

Asymptotically flat multiblack lenses

We present an asymptotically flat and stationary multi-black lens solution with bi-axisymmetry of $U(1)\times U(1)$ as a supersymmetric solution in the five-dimensional minimal ungauged supergravity. We show that the spatial cross section of each degenerate Killing horizon admits different lens space topologies of $L(n,1)=S^3/{\mathbb Z_n}$ as well as a sphere $S^3$. Moreover, we show that in contrast to the higher dimensional Majumdar-Papapertrou multi-black hole and multi-BMPV black hole spacetimes, the metric is smooth on each horizon even if the horizon topology is spherical.

The Characteristic Gluing Problem and Conservation Laws for the Wave Equation on Null Hypersurfaces

We obtain necessary and sufficient conditions for the existence of “conservation laws” on null hypersurfaces for the wave equation on general four-dimensional Lorentzian manifolds. Examples of null hypersurfaces exhibiting such conservation laws include the standard null cones of Minkowski spacetime and the degenerate horizons of extremal black holes. Another (limiting) example of such a conservation law is that which gives rise to the well-known Newman–Penrose constants along the null infinity of asymptotically flat spacetimes. The existence of such conservation laws can be viewed as an obstruction to a certain gluing construction for characteristic initial data for the wave equation. We initiate the general study of the latter gluing problem and show that the existence of conservation laws is in fact the only obstruction. Our method relies on a novel elliptic structure associated to a foliation with 2-spheres of a null hypersurface.

The Trapping Effect on Degenerate Horizons

We show that degenerate horizons exhibit a new trapping effect. Specifically, we obtain a non-degenerate Morawetz estimate for the wave equation in the domain of outer communications of extremal Reissner-Nordstrom up to and including the future event horizon. We show that such an estimate requires 1) a higher degree of regularity for the initial data, reminiscent of the regularity loss in the high-frequency trapping estimates on the photon sphere, and 2) the vanishing of an explicit quantity that depends on the restriction of the initial data on the horizon. The latter condition demonstrates that degenerate horizons exhibit a global trapping effect (in the sense that this effect is not due to individual underlying null geodesics as in the case of the photon sphere). We moreover uncover a new stable higher-order trapping effect; we show that higher-order estimates do not hold regardless of the degree of regularity and the support of the initial data. We connect our findings to the spectrum of the stability operator in the theory of marginally outer trapped surfaces (MOTS). Our methods and results play a crucial role in our upcoming works on linear and non-linear wave equations on extremal black hole backgrounds.

Stress-energy tensor of the quantized massive scalar field in spherically symmetric, topological and lukewarm black hole configurations inD=4andD=5

In this paper, using the general Schwinger-DeWitt approach, we construct the approximate stress-energy tensor of the quantized massive scalar field with arbitrary curvature coupling in $D=4$ and $D=5$ classical spacetimes of the electrically charged static black holes with the cosmological constant. Depending on the sign of the cosmological constant we consider both spherically symmetric and topological spacetimes with the special emphasis put on the extremal and ultraextremal configurations. Moreover, we analyze the geometries of the closest vicinity of the degenerate horizons, which topologically are the product of the maximally symmetric subspaces. We solve the semiclassical Einstein field equations for such product spacetimes. Since the transformation $k\ensuremath{\rightarrow}\ensuremath{-}k$ and ${f}^{\ensuremath{'}\ensuremath{'}}\ensuremath{\rightarrow}\ensuremath{-}{f}^{\ensuremath{'}\ensuremath{'}}$ leaves the equations unchanged, where $k$ is the curvature of the ($D\ensuremath{-}2$)-dimensional subspace and ${f}^{\ensuremath{'}\ensuremath{'}}$ is the second derivative of the metric potential at the degenerate horizon, it suffices to consider only the spherically symmetric case. It is shown that in four and five dimensions there are forbidden sectors of the parameter space.

Collision of two general particles around a rotating regular Hayward’s black holes

The rotating regular Hayward's spacetime, apart from mass ($M$) and angular momentum ($a$), has an additional deviation parameter ($g$) due to the magnetic charge, which generalizes the Kerr black hole when $g\neq0$, and for $g=0$, it goes over to the Kerr black hole. We analyze how the ergoregion is affected by the parameter $g$ to show that the area of ergoregion increases with increasing values of $g$. Further, for each $g$, there exist critical $a_E$, which corresponds to a regular extremal black hole with degenerate horizons $r=r^E_H$, and $a_E$ decrease whereas $r^E_H$ increases with an increase in the parameter $g$. Ban{\~a}dos, Silk and West (BSW) demonstrated that the extremal Kerr black hole can act as a particle accelerator with arbitrarily high center-of-mass energy ($E_{CM}$) when the collision of two particles takes place near the horizon. We study the BSW process for two particles with different rest masses, $m_1$ and $m_2$, moving in the equatorial plane of extremal Hayward's black hole for different values of $g$, to show that $E_{CM}$ of two colliding particles is arbitrarily high when one of the particles takes a critical value of angular momentum. For a nonextremal case, there always exist a finite upper bound for the $E_{CM}$, which increases with the deviation parameter $g$. Our results, in the limit $g \rightarrow 0$, reduces to that of the Kerr black hole.

Horizon supertranslation and degenerate black hole solutions

In this note we first review the degenerate vacua arising from the BMS symmetries. According to the discussion in [1] one can define BMS-analogous supertranslation and superrotation for spacetime with black hole in Gaussian null coordinates. In the leading and subleading orders of near horizon approximation, the infinitely degenerate black hole solutions are derived by considering Einstein equations with or without cosmological constant, and they are related to each other by the diffeomorphism generated by horizon supertranslation. Higher order results and degenerate Rindler horizon solutions also are given in appendices.

A comparison of remnants in noncommutative Bardeen black holes

We derive the mass term of the Bardeen metric in the presence of a noncommutative geometry induced minimal length. In this setup, the proposal of a stable black hole remnant as a candidate to store information is confirmed. We consider the possibility of having an extremal configuration with one degenerate event horizon and compare different sizes of black hole remnants. As a result, once the magnetic charge $g$ of the noncommutative Bardeen solution becomes larger, both the minimal nonzero mass $M_0$ and the minimal nonzero horizon radius $r_0$ get larger. This means, subsequently, under the condition of an adequate amount of $g$, the three parameters $g$, $M_0$, and $r_0$ are in a connection with each other linearly. According to our results, a noncommutative Bardeen black hole is colder than the noncommutative Schwarzschild black hole and its remnant is bigger, so the minimum required energy for the formation of such a black hole at particle colliders will be larger. We also find a closely similar result for the Hayward solution.

Horizon structure of rotating Einstein–Born–Infeld black holes and shadow

We investigate the horizon structure of the rotating Einstein-Born-Infeld solution which goes over to the Einstein-Maxwell's Kerr-Newman solution as the Born-Infeld parameter goes to infinity ($\beta \rightarrow \infty$). We find that for a given $\beta$, mass $M$ and charge $Q$, there exist critical spinning parameter $a_{E}$ and $r_{H}^{E}$, which corresponds to an extremal Einstein-Born-Infeld black hole with degenerate horizons, and $a_{E}$ decreases and $r_{H}^{E}$ increases with increase in the Born-Infeld parameter $\beta$. While $a<a_{E}$ describe a non-extremal Einstein-Born-Infeld black hole with outer and inner horizons. Similarly, the effect of $\beta$ on infinite redshift surface and in turn on ergoregion is also included. It is well known that a black hole can cast a shadow as an optical appearance due to its strong gravitational field. We also investigate the shadow cast by the non-rotating ($a=0$) Einstein-Born-Infeld black hole and demonstrate that the null geodesic equations can be integrated that allows us to investigate the shadow cast by a black hole which is found to be a dark zone covered by a circle. Interestingly, the shadow of the Einstein-Born-Infeld black hole is slightly smaller than for the Reissner-Nordstrom black hole. F urther, the shadow is concentric circles whose radius decreases with increase in value of parameter $\beta$.

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.