Bifurcate Killing Horizon Research Articles

Linearized second law for higher curvature gravity and nonminimally coupled vector fields

Expanding the work of Wall [], we show that black holes obey a second law for linear perturbations to bifurcate Killing horizons, in any covariant higher curvature gravity coupled to scalar and vector fields. The vector fields do not need to be gauged, and (like the scalars) can have arbitrary nonminimal couplings to the metric. The increasing entropy has a natural expression in covariant phase space language, which makes it manifestly invariant under Jacobson-Kang-Myers ambiguities. An explicit entropy formula is given for f(Riemann) gravity coupled to vectors, where at most one derivative acts on each vector. Besides the previously known curvature terms, there are three extra terms involving differentiating the Lagrangian by the symmetric vector derivative (which therefore vanish for gauge fields). Published by the American Physical Society 2024

Decoherence from horizons: General formulation and rotating black holes

Recent work by Danielson, Satishchandran, and Wald (DSW) has shown that black holes—and, in fact, Killing horizons more generally—impart a fundamental rate of decoherence on all nearby quantum superpositions. The effect can be understood from measurement and causality: An observer (Bob) in the black hole should be able to disturb outside quantum superpositions by measuring their superposed gravitational fields, but since his actions cannot (by causality) have this effect, the superpositions must automatically disturb themselves. DSW calculated the rate of decoherence up to an unknown numerical factor for distant observers in Schwarzschild spacetime, Rindler observers in flat spacetime, and static observers in de Sitter spacetime. Working in electromagnetic and Klein-Gordon analogs, we flesh out and generalize their calculation to derive a general formula for the precise decoherence rate for Killing observers near bifurcate Killing horizons. We evaluate the rate in closed form for an observer at an arbitrary location on the symmetry axis of a Kerr black hole. This fixes the numerical factor in the distant-observer Schwarzschild result, while allowing new exploration of near-horizon and/or near-extremal behavior. In the electromagnetic case we find that the decoherence vanishes entirely in the extremal limit, due to the “black hole Meissner effect” screening the Coulomb field from entering the black hole. This supports the causality picture: Since Bob is unable to measure the field of the outside superposition, no decoherence is necessary—and indeed none occurs. Published by the American Physical Society 2024

Local Poincaré algebra from quantum chaos

The local two-dimensional Poincaré algebra near the horizon of an eternal AdS black hole, or in proximity to any bifurcate Killing horizon, is generated by the Killing flow and outward null translations on the horizon. In holography, this local Poincaré algebra is reflected as a pair of unitary flows in the boundary Hilbert space whose generators under modular flow grow and decay exponentially with a maximal Lyapunov exponent. This is a universal feature of many geometric vacua of quantum gravity. To explain this universality, we show that a two-dimensional Poincaré algebra emerges in any quantum system that has von Neumann subalgebras associated with half-infinite modular time intervals (modular future and past subalgebras) in a limit analogous to the near-horizon limit. In ergodic theory, quantum dynamical systems with future or past algebras are called quantum K-systems. The surprising statement is that modular K-systems are always maximally chaotic.Interacting quantum systems in the thermodynamic limit and large N theories above the Hawking-Page phase transition are examples of physical theories with future/past subalgebras. We prove that the existence of (modular) future/past von Neumann subalgebras also implies a second law of (modular) thermodynamics and the exponential decay of (modular) correlators. We generalize our results from the modular flow to any dynamical flow with a positive generator and interpret the positivity condition as quantum detailed balance.

On staticity of bifurcate Killing horizons*

We show that bifurcate Killing horizons with closed torsion form, in spacetimes of arbitrary dimension, and satisfying a Ricci-structure condition arise from static Killing vectors. The result applies in particular to Λ-vacuum spacetimes.

Nonthermal acceleration radiation of atoms near a black hole in presence of dark energy

We investigate how dark energy affects atom-field interaction. To this end, we consider acceleration radiation of a freely falling atom close to a Schwarzschild black hole (BH) in the presence of dark energy characterized by a positive cosmological constant $\mathrm{\ensuremath{\Lambda}}$. The resulting spacetime is endowed with a BH and a cosmological (or de Sitter) horizon. Our consideration is a nonextremal ($1+1$)-dimensional geometry with horizons far apart, giving rise to a flat Minkowski-like region in between the two horizons. Assuming a scalar (spin-0) field in a Boulware-like vacuum state, and by using a basic quantum optics approach, we numerically achieve excitation probabilities for the atom to detect a photon as it falls toward the BH horizon. It turns out that the nature of the emitted radiation deeply drives its origin from the magnitude of $\mathrm{\ensuremath{\Lambda}}$. In particular, radiation emission is enhanced due to dilation of the BH horizon by dark energy. Also, we report an oscillatory nonthermal spectrum in the presence of $\mathrm{\ensuremath{\Lambda}}$, and these oscillations, in a varying degree, also depend on BH mass and atomic excitation frequency. We conjecture that such a hoedown may be a natural consequence of a constrained motion due to the bifurcate Killing horizon of the given spacetime. The situation is akin to the Parikh-Wilzcek tunneling approach to Hawking radiation where the presence of extra contributions to the Boltzmann factor deforms the thermality of flux. It apparently hints at field satisfying a modified energy-momentum dispersion relation within classical regime of general relativity arising as an effective low energy consequence of an underlying quantum gravity theory. Our findings may signal new ways of conceiving the subtleties surrounding the physics of dark energy.

Physical process first law and the entropy change of Rindler horizons

The physical process version of the first law can be obtained for bifurcate Killing horizons with certain assumptions. Especially, one has to restrict to the situations where the horizon evolution is quasi-stationary, under perturbations. We revisit the analysis of this assumption considering the horizon perturbations of Rindler horizon by a spherically symmetric object. We demonstrate that even if the quasi-stationary assumption holds, the change in entropy, in four space–time dimensions, diverges when considered between asymptotic cross-sections. However, these divergences do not appear in higher dimensions. We also analyze these features in the presence of a positive cosmological constant. In the process, we prescribe a recipe to establish the physical process first law in such ill-behaved scenarios.

Hartle-Hawking state in the real-time formalism

We study self-interacting massive scalar field theory in static spacetimes with a bifurcate Killing horizon and a wedge reflection. In this theory the Hartle-Hawking state is defined to have the $N$-point correlation functions obtained by analytically continuing those in the Euclidean theory, whereas the double Kubo-Martin-Schwinger (KMS) state is the pure state invariant under the Killing flow and the wedge reflection which is regular on the bifurcate Killing horizon and reduces to the thermal state at the Hawking temperature in each of the two static regions. We demonstrate in the Schwinger-Keldysh operator formalism of perturbation theory the equivalence between the Hartle-Hawking state and the double KMS state with the Hawking temperature, which was shown before by Jacobson in the path-integral framework.

The Hartle–Hawking–Israel state on spacetimes with stationary bifurcate Killing horizons

We consider a free massive quantized Klein–Gordon field in a spacetime [Formula: see text] containing a stationary bifurcate Killing horizon, i.e. a bifurcate Killing horizon whose Killing vector field is globally time-like in the right wedge [Formula: see text] associated to the horizon. We prove the existence of the Hartle–Hawking–Israel (HHI) vacuum state, which is a pure state on the whole spacetime whose restriction to [Formula: see text] is a thermal state [Formula: see text] for the time-like Killing field at Hawking temperature [Formula: see text], where [Formula: see text] is the surface gravity of the horizon. We show that the HHI state is a Hadamard state and is the unique Hadamard state which is equal to the double [Formula: see text]-KMS state in the double wedge [Formula: see text]. We construct the HHI state by Wick rotation in Killing time coordinates, using the notion of the Calderón projector for elliptic boundary value problems.

QFT and Topology in Two Dimensions: $$\mathrm{SL}(2, {{\mathbb {R}}})$$-Symmetry and the de Sitter Universe

We study bosonic quantum field theory on the double covering \(\widetilde{dS}_{2}\) of the two-dimensional de Sitter universe, identified to a coset space of the group \(\mathrm{SL}(2, {{\mathbb {R}}})\). The latter acts effectively on \(\widetilde{dS}_{2}\) and can be interpreted as it relativity group. The manifold is locally identical to the standard the Sitter spacetime \({dS}_2\); it is globally hyperbolic, geodesically complete and an inertial observer sees exactly the same bifurcate Killing horizons as in the standard one-sheeted case. The different global Lorentzian structure causes, however, drastic differences between the two models. We classify all the \(\mathrm{SL}(2, {{\mathbb {R}}})\)-invariant two-point functions and show that: (1) there is no Hawking–Gibbons temperature; (2) there is no covariant field theory solving the Klein–Gordon equation with mass less than 1/2R , i.e., the complementary fields go away.

Exact renormalization group, entanglement entropy, and black hole entropy

The study of black hole physics revealed a fundamental connection between thermodynamics, quantum mechanics, and gravity. Today, it is known that black holes are thermodynamical objects with well-defined temperature and entropy. Although black hole radiance gives us the mechanism from which we can associate a well-defined temperature to the black hole, the origin of its entropy remains a mystery. Here we investigate how the quantum fluctuations from the fields that render the black hole its temperature contribute to its entropy. By using the exact renormalization group equation for a self-interacting real scalar field in a spacetime possessing a bifurcate Killing horizon, we find the renormalization group flow of the total gravitational entropy. We show that throughout the flow one can split the quantum field contribution to the entropy into a part coming from the entanglement between field degrees of freedom inside and outside the horizon and a part due to the quantum corrections to the Wald entropy coming from the Noether charge. The renormalized black hole entropy is shown to be constant throughout the flow while the balance between the effective black hole entropy at low energies and the infra-red entanglement entropy changes. A similar conclusion is valid for the Wald entropy part of the total entropy. Additionally, our calculations show that there is no mismatch between the renormalization of the coupling constants coming from the effective action or the gravitational entropy, solving an apparent "puzzle" that appeared to exist for interacting fields.

Heating up an environment around black holes and inside de Sitter space

We study quantum fields on spacetimes having a bifurcate Killing horizon by allowing the possibility that left- and right- (in-going and out-going) modes have different temperatures. We consider in particular the Rindler for both massless and massive fields, the static de Sitter and Schwarzschild black hole backgrounds for massive fields. We find that in all three cases, when any of the temperatures is different from the canonical one (Unruh, Hawking and Gibbons--Hawking, correspondingly) the correlation functions have extra singularities at the horizon.

Mass formulae and staticity condition for dark matter charged black holes

The Arnowitt–Deser–Misner formalism is used to derive variations of mass, angular momentum and canonical energy for Einstein–Maxwell dark matter gravity in which the auxiliary gauge field coupled via kinetic mixing term to the ordinary Maxwell one, which mimics properties of hidden sector. Inspection of the initial data for the manifold with an interior boundary, having topology of S^2, enables us to find the generalised first law of black hole thermodynamics in the aforementioned theory. It has been revealed that the stationary black hole solution being subject to the condition of encompassing a bifurcate Killing horizon with a bifurcation sphere, which is non-rotating, must be static and has vanishing magnetic Maxwell and dark matter sector fields, on static slices of the spacetime under consideration.

Light cone thermodynamics

We show that null surfaces defined by the outgoing and infalling wave fronts emanating from and arriving at a sphere in Minkowski spacetime have thermodynamical properties that are in strict formal correspondence with those of black hole horizons in curved spacetimes. Such null surfaces, made of pieces of light cones, are bifurcate conformal Killing horizons for suitable conformally stationary observers. They can be extremal and non-extremal depending on the radius of the shining sphere. Such conformal Killing horizons have a constant light cone (conformal) temperature, given by the standard expression in terms of the generalisation of surface gravity for conformal Killing horizons. Exchanges of conformally invariant energy across the horizon are described by a first law where entropy changes are given by $1/(4\ell_p^2)$ of the changes of a geometric quantity with the meaning of horizon area in a suitable conformal frame. These conformal horizons satisfy the zeroth to the third laws of thermodynamics in an appropriate way. In the extremal case they become light cones associated with a single event; these have vanishing temperature as well as vanishing entropy.

Entropy in Born-Infeld gravity

There is a class of higher derivative gravity theories that are in some sense natural extensions of cosmological Einstein's gravity with a unique maximally symmetric classical vacuum and only a massless spin-2 excitation about the vacuum and no other perturbative modes. These theories are of the Born-Infeld determinantal form. We show that the macroscopic dynamical entropy as defined by Wald for bifurcate Killing horizons in these theories are equivalent to the geometric Bekenstein-Hawking entropy (or more properly Gibbons-Hawking entropy for the case of de Sitter spacetime) but given with an effective gravitational constant which encodes all the information about the background spacetime and the underlying theory. We also show that the higher curvature terms increase the entropy. We carry out the computations in generic n-dimensions including the particularly interesting limits of three, four and infinite number of dimensions. We also give a preliminary discussion about the black hole entropy in generic dimensions for the BI theories.

The quantum null energy condition in curved space

The quantum null energy condition (QNEC) is a conjectured bound on components of the stress tensor along a null vector ka at a point p in terms of a second k-derivative of the von Neumann entropy S on one side of a null congruence N through p generated by ka. The conjecture has been established for super-renormalizeable field theories at points p that lie on a bifurcate Killing horizon with null tangent ka and for large-N holographic theories on flat space. While the Koeller–Leichenauer holographic argument clearly yields an inequality for general , more conditions are generally required for this inequality to be a useful QNEC. For , for arbitrary backgroud metric we show that the QNEC is naturally finite and independent of renormalization scheme when the expansion θ of N at the point p vanishes. This is consistent with the original QNEC conjecture which required θ and the shear to satisfy , . But for more conditions than even these are required. In particular, we also require the vanishing of additional derivatives and a dominant energy condition. In the above cases the holographic argument does indeed yield a finite QNEC, though for we argue these properties to fail even for weakly isolated horizons (where all derivatives of vanish) that also satisfy a dominant energy condition. On the positive side, a corrollary to our work is that, when coupled to Einstein–Hilbert gravity, holographic theories at large N satisfy the generalized second law (GSL) of thermodynamics at leading order in Newton’s constant G. This is the first GSL proof which does not require the quantum fields to be perturbations to a Killing horizon.

The first law of black hole mechanics for fields with internal gauge freedom

We derive the first law of black hole mechanics for physical theories based on a local, covariant and gauge-invariant Lagrangian where the dynamical fields transform non-trivially under the action of some internal gauge transformations. The theories of interest include General Relativity formulated in terms of tetrads, Einstein–Yang–Mills theory and Einstein–Dirac theory. Since the dynamical fields of these theories have some internal gauge freedom, we argue that there is no natural group action of diffeomorphisms of spacetime on such dynamical fields. In general, such fields cannot even be represented as smooth, globally well-defined tensor fields on spacetime. Consequently the derivation of the first law by Iyer and Wald cannot be used directly. Nevertheless, we show how such theories can be formulated on a principal bundle and that there is a natural action of automorphisms of the bundle on the fields. These bundle automorphisms encode both spacetime diffeomorphisms and internal gauge transformations. Using this reformulation we define the Noether charge associated to an infinitesimal automorphism and the corresponding notion of stationarity and axisymmetry of the dynamical fields. We first show that we can define certain potentials and charges at the horizon of a black hole so that the potentials are constant on the bifurcate Killing horizon, giving a generalised zeroth law for bifurcate Killing horizons. We further identify the gravitational potential and perturbed charge as the temperature and perturbed entropy of the black hole which gives an explicit formula for the perturbed entropy analogous to the Wald entropy formula. We then obtain a general first law of black hole mechanics for such theories. The first law relates the perturbed Hamiltonians at spatial infinity and the horizon, and the horizon contributions take the form of a ‘potential times perturbed charge’ term. We also comment on the ambiguities in defining a prescription for the total entropy for black holes.

On the Construction of Hartle–Hawking–Israel States Across a Static Bifurcate Killing Horizon

We consider a linear scalar quantum field propagating in a spacetime of dimension d ≥ 2 with a static bifurcate Killing horizon and a wedge reflection. Under suitable conditions (e.g. positive mass), we prove the existence of a pure Hadamard state which is quasi-free, invariant under the Killing flow and restricts to a double β H -KMS state on the union of the exterior wedge regions, where β H is the inverse Hawking temperature. The existence of such a state was first conjectured by Hartle and Hawking (Phys Rev D 13:2188–2203, 1976) and by Israel (Phys Lett 57:107–110, 1976), in the more general case of a stationary black hole spacetime. Jacobson (Phys Rev D 50:R6031–R6032, 1994) has conjectured a similar state to exist even for interacting fields in spacetimes with a static bifurcate Killing horizon. The state can serve as a ground state on the entire spacetime and the resulting situation generalises that of the Unruh effect in Minkowski spacetime. Our result complements a well-known uniqueness result of Kay and Wald (Phys Rep 207:49–136, 1991) and Kay (J Math Phys 34:4519–4539, 1993), who considered a general bifurcate Killing horizon and proved that a certain (large) subalgebra of the free field admits at most one Hadamard state which is invariant under the Killing flow. This state is pure and quasi-free and in the presence of a wedge reflection it restricts to a β H -KMS state on the smaller subalgebra associated to one of the exterior wedge regions. Our result establishes the existence of such a state on the full algebra, but only in the static case. Our proof follows the arguments of Sewell (Ann Phys 141: 201–224, 1982) and Jacobson (Phys Rev D 50:R6031–R6032, 1994), who exploited a Wick rotation in the Killing time coordinate to construct a corresponding Euclidean theory. In particular, we show that for the linear scalar field we can recover a Lorentzian theory by Wick rotating back. Because the Killing time coordinate is ill-defined on the bifurcation surface, we systematically replace it by a Gaussian normal coordinate. A crucial part of our proof is to establish that the Euclidean ground state satisfies the necessary analogues of analyticity and reflection positivity with respect to this coordinate.

First law of black hole mechanics as a condition for stationarity

In earlier work [arXiv:1302.1237], we provided a Hilbert manifold structure for the phase space for the Einstein-Yang-Mills equations, and used this to prove a condition for initial data to be stationary. Here we use the same phase space to consider the evolution of initial data exterior to some closed 2-surface boundary, and establish a condition for stationarity in this case. It is shown that the differential relationship given in the first law of black hole mechanics is exactly the condition required for the initial data to be stationary; this was first argued non-rigorously by Sudarsky and Wald in 1992. Furthermore, we give evidence to suggest that if this differential relationship holds then the boundary surface is the bifurcation surface of a bifurcate Killing horizon.

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.

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.