Killing Horizon Research Articles

Uniqueness of the de Sitter spacetime among static vacua with positive cosmological constant

We prove that, among all (n + 1)-dimensional spin static vacua with positive cosmological constant, the de Sitter spacetime is characterized by the fact that its spatial Killing hori-zons have minimal modes for the Dirac operator. As a consequence, the de Sitter spacetime is the only vacuum of this type for which the induced metric tensor on some of its Killing horizons is at least equal to that of a round (n -- 1)-sphere. This extends unique-ness theorems shown by Boucher-Gibbons-Horowitz and Chruciel to more general horizon metrics and to the non-single horizon case.

Three-dimensional black holes and descendants

We determine the most general three-dimensional vacuum spacetime with a negative cosmological constant containing a non-singular Killing horizon. We show that the general solution with a spatially compact horizon possesses a second commuting Killing field and deduce that it must be related to the BTZ black hole (or its near-horizon geometry) by a diffeomorphism. We show there is a general class of asymptotically AdS3 extreme black holes with arbitrary charges with respect to one of the asymptotic-symmetry Virasoro algebras and vanishing charges with respect to the other. We interpret these as descendants of the extreme BTZ black hole.

Causal structures in Gauss-Bonnet gravity

We analyze causal structures in Gauss-Bonnet gravity. It is known that Gauss-Bonnet gravity potentially has superluminal propagation of gravitons due to its noncanonical kinetic terms. In a theory with superluminal modes, an analysis of causality based on null curves makes no sense, and thus, we need to analyze them in a different way. In this paper, using the method of the characteristics, we analyze the causal structure in Gauss-Bonnet gravity. We have the result that, on a Killing horizon, gravitons can propagate in the null direction tangent to the Killing horizon. Therefore, a Killing horizon can be a causal edge as in the case of general relativity; i.e. a Killing horizon is the ``event horizon'' in the sense of causality. We also analyze causal structures on nonstationary solutions with $(D\ensuremath{-}2)$-dimensional maximal symmetry, including spherically symmetric and flat spaces. If the geometrical null energy condition, ${R}_{AB}{N}^{A}{N}^{B}\ensuremath{\ge}0$ for any null vector ${N}^{A}$, is satisfied, the radial velocity of gravitons must be less than or equal to that of light. However, if the geometrical null energy condition is violated, gravitons can propagate faster than light. Hence, on an evaporating black hole where the geometrical null energy condition is expected not to hold, classical gravitons can escape from the ``black hole'' defined with null curves. That is, the causal structures become nontrivial. It may be one of the possible solutions for the information loss paradox of evaporating black holes.

On quantization of AdS3 gravity I: semi-classical analysis

In this work we explore ideas in quantizing AdS$_3$ Einstein gravity. We start with the most general solution to the 3d gravity theory which respects Brown-Henneaux boundary conditions. These solutions are specified by two holomorphic functions and satisfy simple superposition rule. These geometries generically have a bifurcate Killing horizon (with a noncompact bifurcation curve) which is not an event horizon and are hence not black holes. Nonetheless, there are superpositions of these geometries which have event horizon. We propose to view these geometries as "semiclassical fuzzball microstates" of BTZ black holes appearing as superposition of these geometries. The details of quantization of these semiclassical microstates will be discussed in an upcoming work.

Killing horizons and spinors

We study the near horizon geometry of generic Killing horizons constructing suitable coordinates and taking the appropriate scaling limit. We are able to show that the geometry will always show an enhancement of symmetries, and, in the extremal case, will develop a causally disconnected "throat" as expected. We analyze the implications of this to the Kerr/CFT conjecture and the attractor mechanism. We are also able to construct a set of special (pure) spinors associated with the horizon structure using their interpretation as maximally isotropic planes. The structure generalizes the usual reduced holonomy manifold in an interesting way and may be fruitful to the search of new types of compactification backgrounds.

Anomalous effective action, Noether current, Virasoro algebra and Horizon entropy

Several investigations show that in a very small length scale there exist corrections to the entropy of black hole horizon. Due to fluctuations of the background metric and the external fields the action incorporates corrections. In the low energy regime, the one-loop effective action in four dimensions leads to trace anomaly. We start from the Noether current corresponding to the Einstein–Hilbert plus the one-loop effective action to calculate the charge for the diffeomorphisms which preserve the Killing horizon structure. Then a bracket for the charges is calculated. We show that the Fourier modes of the bracket are exactly similar to the Virasoro algebra. Then using the Cardy formula the entropy is evaluated. Finally, the explicit terms of the entropy expression is calculated for a classical background. It turns out that the usual expression for the entropy; i.e. the Bekenstein–Hawking form, is not modified.

Ray tracing Einstein-Æther black holes: Universal versus Killing horizons

Violating Lorentz invariance, and so implicitly permitting some form of super-luminal communication, necessarily alters the notion of a black hole. Nevertheless, in both Einstein-\ae{}ther gravity and Ho\ifmmode \check{r}\else \v{r}\fi{}ava-Lifshitz gravity, there is still a causally disconnected region in black-hole solutions, now being bounded by a ``universal horizon,'' which traps excitations of arbitrarily high velocities. To better understand the nature of these black holes, and their universal horizons, we study ray trajectories in these spacetimes. We find evidence that Hawking radiation is associated with the universal horizon, while the ``lingering'' of ray trajectories near the Killing horizon hints at reprocessing there. In doing this we solve an apparent discrepancy between the surface gravity of the universal horizon and the associated temperature derived by the tunneling method. These results advance the understanding of these exotic horizons and provide hints for a full understanding of black-hole thermodynamics in Lorentz-violating theories.

Revisiting Vaidya Horizons

In this study, we located and compared different types of horizons in the spherically symmetric Vaidya solution. The horizons we found were trapping horizons, which can be null, timelike, or spacelike, null surfaces with constant area change and also conformal Killing horizons. The conformal Killing horizons only exist for certain choices of the mass function. Under a conformal transformation, the conformal Killing horizons can be mapped into true Killing horizons. This allows conclusions drawn in the dynamical Vaidya spacetime to be related to known properties of static spacetimes. We found the conformal factor that performs this transformation and wrote the new metric in explicitly static coordinates. Using this construction we found that the tunneling argument for Hawking radiation does not umabiguously support Hawking radiation being associated with the trapping horizon. We also used this transformation to derive the form of the surface gravity for a class of physical observers in Vaidya spacetimes.

Entropy from near-horizon geometries of Killing horizons

We derive black hole entropy based on the near-horizon symmetries of black hole space-times. To derive these symmetries we make use of an $(R,T)$-plane close to a Killing horizon. We identify a set of vector fields that preserves this plane and forms a Witt algebra. The corresponding algebra of Hamiltonians is shown to have a non-trivial central extension. Using the Cardy formula and the central charge we obtain the Bekenstein-Hawking entropy.

Stationary black holes as holographs II

Within the generic null characteristic initial value problem a reduced set of the evolution equations are deduced from the coupled Newman–Penrose and Maxwell equations for smooth four-dimensional electrovacuum spacetimes allowing a non-zero cosmological constant. It is shown that these reduced equations make up a first-order symmetric hyperbolic system of evolution equations, and also that the solutions to this reduced system are also solutions to the full set of the Newman–Penrose and Maxwell equations provided that the inner equations hold on the initial data surfaces. The derived generic results are applied in carrying out the investigation of electrovacuum spacetimes distinguished by the existence of a pair of null hypersurfaces, and , generated by expansion and shear-free geodesically complete null congruences such that they intersect on a two-dimensional spacelike surface, . Besides the existence of this pair of null hypersurfaces, no assumption concerning the asymptotic structure is made. It is shown that both the spacetime geometry and the electromagnetic field are uniquely determined, in the domain of dependence of once a complex vector field ξA (determining the metric induced on ), the τ spin coefficient and the ϕ1 electromagnetic potential are specified on . The existence of a Killing vector field—with respect to which the null hypersurfaces and comprise a bifurcate-type Killing horizon—is also justified in the domain of dependence of . Since, in general, the freely specifiable data on do not have any sort of symmetry, the corresponding spacetimes do not possess any symmetry in addition to the horizon Killing vector field. Thereby, they comprise the class of generic ‘stationary’ distorted electrovacuum black hole spacetimes which for the case of a positive cosmological constant may also (or, in certain cases, only) contain a distorted de Sitter-type cosmological horizon to which our results equally apply. It is also shown that there are stationary distorted electrovacuum black hole configurations such that parallelly propagated curvature blow-up occurs both to the future and to the past ends of some of the null generators of their bifurcate Killing horizon, and also that this behavior is universal. In particular, it is shown that in the space of vacuum solutions to Einstein’s equations, in an arbitrarily small neighborhood of the Schwarzschild solution this type of distorted vacuum black hole configurations always exist. A short discussion on the relation of these results and some of the recent claims on the instability of extremal black holes is also given.

Static spherically symmetric black holes with scalar field

Static spherically symmetric black holes and particle like solutions with self interacting minimally coupled scalar field $$\varphi $$ are analyzed. They are asymptotically flat or anti-de Sitter (AdS). We express them in terms of a single function $$\rho $$ which undergoes simple conditions. If $$\varphi $$ is nontrivial the ADM mass $$M$$ has to be positive. No-hair theorems are generalized to the AdS asymptotic. For both asymptotics the Killing horizon is nondegenerate and its radius cannot be bigger than $$2M$$ . Derivatives of $$\rho $$ at singularity determine properties of admissible potentials $$V(\varphi )$$ as regularity, boundedness and behavior for maximal values of $$\varphi $$ . Several classes of solutions with singular or nonsingular potentials are obtained. Their examples are presented in a form of plots.

KIDs like cones

We analyze vacuum Killing Initial Data on characteristic Cauchy surfaces. A general theorem on existence of Killing vectors in the domain of dependence is proved, and some special cases are analyzed in detail, including the case of bifurcate Killing horizons.

“Tunnelling” Black-Hole Radiation with ϕ 3 Self-Interaction: One-Loop Computation for Rindler Killing Horizons

Tunnelling processes through black hole horizons have recently been investigated in the framework of WKB theory discovering interesting interplay with the Hawking radiation. A more precise and general account of that phenomenon has been subsequently given within the framework of QFT in curved spacetime by two of the authors of the present paper. In particular, it has been shown that, in the limit of sharp localization on opposite sides of a Killing horizon, the quantum correlation functions of a scalar field appear to have thermal nature, and the tunnelling probability is proportional to exp{−βHawkingE}. This local result is valid in every spacetime including a local Killing horizon, no field equation is necessary, while a suitable choice for the quantum state is relevant. Indeed, the two-point function has to verify a short-distance condition weaker than the Hadamard one. In this paper we consider a massive scalar quantum field with a ϕ3 self-interaction and we investigate the issue whether or not the black hole radiation can be handled at perturbative level, including the renormalisation contributions. We prove that, for the simplest model of the Killing horizon generated by the boost in Minkowski spacetime, and referring to Minkowski vacuum, the tunnelling probability in the limit of sharp localization on opposite sides of the horizon preserves the thermal form proportional to exp{−βHE} even taking the one-loop renormalisation corrections into account. A similar result is expected to hold for the Unruh state in the Kruskal manifold, since that state is Hadamard and looks like Minkowski vacuum close to the horizon.

Marginally outer trapped surfaces in higher dimensions

We review the basic setup of Kaluza–Klein theory, namely a five-dimensional vacuum with a cyclic isometry, which corresponds to Einstein–Maxwell-dilaton theory in four-dimensional spacetime. We first recall the behaviour of Killing horizons under bundle lift and projection. We then show that the property of compact surfaces of being (stably) marginally trapped is preserved under lift and projection provided the appropriate (‘Pauli’) conformal scaling is used for the spacetime metric. We also discuss and compare recently proven area inequalities for stable axially symmetric two-dimensional and three-dimensional marginally outer trapped surfaces.

Classification of Near-Horizon Geometries of Extremal Black Holes.

Any spacetime containing a degenerate Killing horizon, such as an extremal black hole, possesses a well-defined notion of a near-horizon geometry. We review such near-horizon geometry solutions in a variety of dimensions and theories in a unified manner. We discuss various general results including horizon topology and near-horizon symmetry enhancement. We also discuss the status of the classification of near-horizon geometries in theories ranging from vacuum gravity to Einstein-Maxwell theory and supergravity theories. Finally, we discuss applications to the classification of extremal black holes and various related topics. Several new results are presented and open problems are highlighted throughout.

Thermodynamics of a black hole with moon

For a Kerr black hole perturbed by a particle on the ``corotating'' circular orbit (angular velocity equal to that of the unperturbed event horizon), the black hole remains in equilibrium in the sense that the perturbed event horizon is a Killing horizon of the helical Killing field. The associated surface gravity is constant over the horizon and should correspond to the physical Hawking temperature. We calculate the perturbation in surface gravity/temperature, finding it negative: the moon has a cooling effect on the black hole. We also compute the change in horizon angular frequency, which is positive, and the change in surface area/entropy, which vanishes.

Generalized second law at linear order for actions that are functions of Lovelock densities

In this article we consider the second law of black holes (and other causal horizons) in theories where the gravitational action is an arbitrary function of the Lovelock densities. We show that there exists an entropy which increases locally, for linearized perturbations to regular Killing horizons. In addition to a classical increase theorem, we also prove a generalized second law for semiclassical, minimally-coupled matter fields.

Self-energy anomaly of an electric pointlike dipole in three-dimensional static spacetimes

We calculate the self-energy anomaly of a pointlike electric dipole located in a static ($2+1$)-dimensional curved spacetime. The energy functional for this problem is invariant under an infinite-dimensional (gauge) group of transformations parametrized by one scalar function of two variables. We demonstrate that the problem of the calculation of the self-energy anomaly for a pointlike dipole can be reduced to the calculation of quantum fluctuations of an effective two-dimensional Euclidean quantum field theory. We reduced the problem in question to the calculation of the conformal anomaly of an effective scalar field in two dimensions and obtained an explicit expression for the self-energy anomaly of an electric dipole in an asymptotically flat, regular ($2+1$)-dimensional spacetime, which may have electrically neutral black-hole-like metrics with regular Killing horizon.

CFTs in rotating black hole backgrounds

We use AdS/CFT to construct the gravitational dual of a 5D CFT in the background of a non-extremal rotating black hole. Our boundary conditions are such that the vacuum state of the dual CFT corresponds to the Unruh state. We extract the expectation value of the stress tensor of the dual CFT using holographic renormalization and show that it is stationary and regular on both the future and the past event horizons. The energy density of the CFT is found to be negative everywhere in our domain and we argue that this can be understood as a vacuum polarization effect. We construct the solutions by numerically solving the elliptic Einstein–DeTurck equation for stationary Lorentzian spacetimes with Killing horizons. Communicated by H Reall

Stationary Holographic Plasma Quenches and Numerical Methods for Non-Killing Horizons

We explore use of the harmonic Einstein equations to numerically find stationary black holes where the problem is posed on an ingoing slice that extends into the interior of the black hole. Requiring no boundary conditions at the horizon beyond smoothness of the metric, this method may be applied for horizons that are not Killing. As a nontrivial illustration we find black holes which, via AdS-CFT, describe a time-independent CFT plasma flowing through a static spacetime which asymptotes to Minkowski in the flow's past and future, with a varying spatial geometry in between. These are the first nonperturbative examples of stationary black holes which do not have Killing horizons. When the CFT spacetime slowly varies, the CFT stress tensor derived from gravity is well described by viscous hydrodynamics. For fast variation it is not, and the solutions are stationary analogs of dynamical quenches, with the plasma being suddenly driven out of equilibrium. We find evidence these flows become unstable for sufficiently strong quenches, and speculate the instability may be turbulent.