Golden Wedding of Black Holes and Thermodynamics

An Online Celebration

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Book of Abstracts

Invited Speakers

Horizon Symmetries and Black Hole Entropy

Steven Carlip (University of California, Davis — USA)

One of the most mysterious properties of black hole entropy is its universality. The area law, with the same fixed coefficient, occurs for a huge variety of “black” objects, and is reproduced by a great many microscopic approaches, even though they seem to describe vastly different underlying states. I describe an approach to this problem that appeals to a possible underlying symmetry of a generic horizon, and discuss its implication for the microscopic description of black hole entropy.

Hadamard States and Black Holes: A story written at infinity

Claudio Dappiaggi (University of Pavia — Italy)

We review the cornerstones of a longstanding programme aimed at constructing Hadamard states for free quantum fields on asymptotically flat spacetimes by exploiting invariance under the action of the BMS asymptotic symmetry group. In particular we outline how this procedure has been used to give a rigorous construction of the Unruh state for a massless, conformally coupled, real scalar field on Schwarzschild spacetime and we survey its application to higher spin, free fields. To conclude we examine the future perspectives on the interplay between Hadamard states, algebraic quantum field theory and black hole physics.

Measurement in Quantum Field Theory: measurement schemes, state updates, impossible measurements, and all that

Christopher J. Fewster (University of York — United Kingdom)

Many presentations of quantum mechanics include a postulate that the state of a system undergoes an instantaneous change following a measurement. This is clearly incompatible with special and general relativity and raises questions concerning the description of measurement in quantum field theory (QFT). Attempts to extend measurement postulates to QFT by hand have produced pathologies, such as the “impossible measurements” described long ago by Sorkin. I will present a recent operational approach to these questions, which models measurement of one quantum field (the system) by coupling it to another (the probe). This is all accomplished in a model-independent way within algebraic quantum field theory (AQFT). The resulting framework provides a description of measurement in QFT that is causal, covariant and consistent, and includes state update rules that are derived from the formalism, and works equally well in flat or curved spacetimes. No prior knowledge of AQFT will be assumed.

On the derivation of Hawking radiation

Klaus Fredenhagen (Universität Hamburg — Germany)

About 50 years after Hawking’s prediction of radiation of black holes I want to review the status of its derivation in the framework of quantum field theory on curved spacetimes. Besides indirect arguments based on an assumed merger between thermodynamics and general relativity the analysis was mainly concentrated on an analysis of classical wave equations on stationary spacetimes with black holes, with boundary conditions mimicking the situation in a gravitational collapse. Not much was done for the interacting case, and also for free fields the results for generic states on spacetimes describing collapsing stars are rare. I will present an ansatz which might be useful for further progress and is based on older ideas on the use of the conformal anomaly.

Alice falls into a black hole: entanglement, clocks and quantum technologies in Schwarzschild spacetime

Ivette Fuentes (University of Southampton and Keble College, Oxford — United Kingdom)

The covariance matrix formalism, commonly used in quantum optics, has enabled the application of methods in quantum information and quantum metrology to quantum field theory in curved spacetime. In this talk, I will explain how these techniques can be used to study the entanglement between modes of quantum fields in a Schwarzschild spacetime. The states analysed include global bosonic modes, propagating light wave-packets and gaussian states of fields contained within cavities hovering at a fixed distance from a black hole horizon. We show theoretically that space-based experiments are now capable of testing the effects of curvature on quantum communications, including quantum teleportation and cryptography. We have proposed schemes, using quantum metrology, to estimate with high precision the spacetime parameters of the Earth. Finally, I will discuss how table-top experiments can be used to study the effects of relativity on quantum clocks.

2D Moving-Mirror Models Revisited: Equivalence Principle, Conformal Invariance, and Their Limitations

Stephen A. Fulling (Texas A&M University — USA)

Because all two-dimensional space-times are conformally flat and most massless field equations are conformally invariant, exact solutions are easy to find, even for reflection boundary conditions on arbitrary timelike curves (“mirror trajectories”). Such models played an important role in establishing the paradigm of quantum field theory in curved space-time in the 1970s, and they have recently reattracted attention in connection with radiation from black holes or their near environs. The critical issues (not yet completely solved even for classical fields) are the correct formal definition and degree of physical reality of “radiation” when source and detector are in relative acceleration. One arrives at apparently paradoxical conclusions such as that a stationary mirror radiates from the perspective of an accelerated observer. Our conclusion is that such statements are correct (and necessary to satisfy principles of equivalence and general covariance), but one must be careful about combining them too literally with our flat-space and classical intuitions.

My principal collaborators in this work have been Paul Davies in the 1970s and Justin Wilson in the 2010s; Bill Unruh and Marlan Scully have had major influences.

Black Holes in The Quantum Effective Action

Aaron Held (École Normale Supérieure — France)

In classical General Relativity, black holes are stable and key to cosmic censorship. The semi-classical limit suggests that black holes slowly evaporate due to Hawking radiation. In an attempt to go beyond the semi-classical limit, my talk addresses black holes in the effective action expected from full quantum gravity. In particular, I will present recent progress on the well-posed nonlinear evolution when including the leading quantum corrections and summarize what we know about the resulting implications for black-hole dynamics and recent observational advances in black-hole astronomy.

Black Hole Interiors

Stefan Hollands (University of Leipzig — Germany)

Kerr or Reissner-Nordström black holes have inner horizons which delineate the domain of predictability of solutions to wave equations (Klein-Gordon, EM, gravitational,…). Often, though not e.g. for certain (A)deSitter black holes, these inner horizons are classically dynamically unstable, and get turned into some sort of singularity by perturbations, and thus drastically changing the nature of the inner horizons relative to the underlying exact solutions, and relegating the problem of predictability to the real of quantum gravity (“strong cosmic censorship”). Recent work has shown that semi-classical quantum effects dominate classical ones very near the inner horizons, and are expected lead to singularity in cases where the inner horizon is stable classically. I review recent progress surrounding these issues.

Particle detectors and purification partners for the observed particles

Masahiro Hotta (Tohoku University — Japan)

In quantum field theory, the term “particle” is defined as that which is observed by a particle detector, as emphasized by Unruh and DeWitt. We present a general formula for determining a complementary purification particle corresponding to the particle observed by the detector.


  • 1503.06109: On the partner particles for moving mirror radiation and black hole evaporation
  • 1807.03467: Spatially Overlapped Partners in Quantum Field Theory
  • 1906.05009: Partner formula for an arbitrary moving mirror in $1+1$ dimensions

But what is entropy?

Bernard Kay (University of York — United Kingdom)

Bardeen Carter and Hawking’s first law of black hole mechanics, $\delta M = (\kappa/8\pi) \delta A$ + etc.\ was thought by them to be only analogous to the first law of thermodynamics — with $\kappa/8\pi$ analogous to temperature and $A$ analogous to entropy. However, as we all know now, and as anticipated by Bekenstein, and demonstrated by Hawking by considering quantum field theory on a black hole collapse spacetime, it is more than an analogy and actually (lending and borrowing a factor of 4) for a black hole, $\kappa/2\pi$ is its temperature and $A/4$ is its entropy. But what is entropy? Soon after Hawking’s discovery, some people took his work as giving us a clue, that, contrary to the standard, and still popular, `coarse-graining’ view, entropy must really be something physically real and objective. But if so, what exactly is it? In this talk I will briefly review my 1998 proposal that it is matter-gravity entanglement entropy. I will then describe some recent plausibility arguments (see arXiv: 2305.11723 for a shorter version and arXiv: 2206.07445 for a longer version) that the end state of the evaporation of a (large) black hole will consist of a pure entangled state of (mainly) photons and gravitons with a photon-graviton entanglement entropy greater than $A/4$ where $A$ is the surface area of the initial black hole. If true, this would be consistent with the second law and thus resolve (one version of) the information loss puzzle.

A review on the questions of spin and spin quantum correlations in the relativistic regime

Shrobona Bagchi (Korea Institute of Science and Technology — South Korea)

Quantum correlations are one of the most important aspects of the modern day quan- tum information and computation theory. However, the majority of understanding of the quantum correlations is in the field of non-relativistic quantum mechanics. To develop the quantum information and computation tasks fully, one must inevitably take into account the relativistic effects. In this regard, the spin is one of the central tools to implement these qubit operations in almost all quantum information processing tasks. For this purpose, it is of paramount importance to understand and characterize fully the theory of spin in relativistic quantum mechanics and relativistic quantum information theory where the spin states act as qubit. This area is still far from being resolved as a current state of art. As a result, in this talk we will explore the recent studies of the concepts of the spin and spin quantum correlations in inertial frames and some apparent paradoxes regarding this concept. We will mainly focus on the problem of characterizing the concept of spin, reduced spin density matrices and consequently spin quantum correlations in inertial reference frames and the apparent paradoxes involved therein, yet to be verified experimentally. Another important aspect is the use of tools of quantum field theory to extend concepts in non- relativistic domain to relativistic one. In this regard, we will analyze the development of the theory of relativistic secret sharing and a correlation measure namely the entanglement of purification. We will also explore how these developments may be mapped to quantum information processing task and discuss the future promises.

Generalizations of Classical Relativity Theorems

Eleni-Alexandra Kontou (King’s College London — United Kingdom)

Several classical relativity theorems, including the famous singularity theorems, have in their assumptions pointwise energy conditions. Those conditions bound the energy density (or similar quantities) on every spacetime point and are easily violated by quantum fields. One way to examine the applicability of those theorems in semiclassical gravity is to replace them with an averaged version, where the energy density is bounded on a segment of a causal geodesic. The index form method, used instead of the Raychaudhuri equation, provides a direct way of using those weakened conditions. In this talk I will apply this method to the Penrose singularity theorem and the Hawking area theorem, and present progress and challenges for both. Finally, I will discuss the connection between the area theorem and black hole thermodynamics.

Black Hole Chemistry

David Kubiznak (Charles University — Czechia, University of Waterloo and Perimeter Institute — Canada)

The mass of a black hole has traditionally been identified with its energy. We describe a new perspective on black hole thermodynamics, one that identifies the mass of a black hole with chemical enthalpy, and the cosmological constant as thermodynamic pressure. This leads to an understanding of black holes from the viewpoint of chemistry, in terms of concepts such as Van der Waals fluids, reentrant phase transitions, and triple points. Both charged and rotating black holes exhibit novel chemical-type phase behaviour, hitherto unseen

Quantum Communication in Curved Spacetimes

André G. S. Landulfo (Federal University of ABC — Brazil)

How much classical or quantum information can be reliably transmitted between two (or more) parts? This is one of the fundamental questions addressed by quantum information theory and within such scenarios relativity can play a major role. A related and equally relevant inquiry is: how much energy is needed to convey the information? This has important consequences not only for practical proposes (such as engineering communication networks) but because it may also shed light in one of the major open problems in semiclassical/quantum gravity nowadays, namely, what is the fate of the information which falls into a black hole.

In this seminar, I will describe the communication of classical and quantum information between two (or three) arbitrary observers in asymptotically flat spacetimes (possibly containing black holes) and investigate what is the energy cost for such information transmission. In addition to computing the achievable communication rates, It will be shown that the change in the expectation value of the energy of the system during the communication process can be separated in: (i) a contribution coming from the particle creation due to the change of the spacetime, (ii) a contribution associated with the energy needed to switch-on/off each qubit, and (iii) a term which comes from the communication process itself. For the quantum channel considered here, we show that the extra energy cost needed for communication vanishes. As a result, if one has already created a system of qubits for some specific task (e.g., quantum computation) one can also reliably convey information between its parts with no extra energy cost.

Information flows, entanglement and communication in the presence of black holes

Eduardo Martín-Martínez (University of Waterloo and Perimeter Institute — Canada)

In this talk we will review several aspects of how information and correlations flow through quantum fields in the presence of black holes. We will focus on entanglement harvesting (the extraction of entanglement from a quantum field vacuum) and quantum communication in regions of strong curvature. We will use these results to gain some insight on how information is stored in quantum fields in the presence of strong gravity and horizons and what can relativistic quantum information teach us about them.

Temperature, Entropy and Energy from the QFT Point of View

Ko Sanders (FAU Erlangen-Nürnberg — Germany)

Black hole thermodynamics is an analogy (or more?) between the laws of black hole physics and the laws of thermodynamics. This analogy identifies properties of black holes (surface gravity, horizon area and mass) with general thermodynamic quantities (temperature, entropy and energy). The discovery of Hawking radiation suggests, that a proper justification of these identifications should involve quantum theory. This leads us to the investigation of thermodynamic quantities of quantum field theories (QFTs) in curved spacetimes. In this talk I will review some mathematical aspects about results on temperature, entropy and energy from the QFT point of view.

Black holes Beyond General Relativity

Frank Saueressig (Radboud Universiteit Nijmegen — The Netherlands)

Black holes described by general relativity are extremely simple. They are completely characterized by their asymptotic mass, spin, and charge. In particular, Birkhoff’s theorem ensures that spherically symmetric black holes, described by the Schwarzschild metric, come with a single free parameter. Following up on Bardeen’s interest in black holes, this talk will discuss the corrections to the Schwarzschild geometry arising from including the Goroff-Sagnotti counterterm in the gravitational dynamics. Specifically, I will discuss the corresponding ``no hair’’ theorem and the impact of the corrections on black hole thermodynamics and shadow imaging.

Quantum Switch on a Superposition of Spherical Shells

Nelson Yokomizo (Federal University of Minas Gerais — Brazil)

A characteristic generally expected to hold in a theory of quantum gravity is the existence of states describing superpositions of distinct geometries. Phenomenological models have recently been explored for the analysis of possible effects due to a superposition of geometries, including the occurrence of processes with indefinite order. In particular, in a gravitational quantum switch, the order of operations applied by two agents on a target system is entangled with the state of the geometry. We consider a model describing the superposition of geometries produced by distinct arrangements of spherical mass shells, and show that a protocol for the implementation of a gravitational quantum switch can be formulated in such a system. The geometries in superposition are identical in an exterior region outside a given radius, and differ within such a radius. The exterior region provides a classical frame with a definite geometry from which the superposition of geometries in the interior region can be probed. One of the agents crosses the interior region and becomes entangled with the geometry, which is explored as a resource for the implementation of the quantum switch.

Contributed Talks

Nonperturbative Renormalization Group Flow for a Particle Detector

Níckolas de Aguiar Alves (Federal University of ABC — Brazil)

The functional renormalization group is a powerful tool to study nonperturbative physics, but it has not been much explored within quantum field theory in curved spacetime yet. On the other hand, particle detectors are an omnipresent tool within quantum field theory in curved spacetime and relativistic quantum information but are often treated only perturbatively. In this paper, we present the first computation of the functional renormalization group flow for a particle detector. The chosen model is an inertial Unruh–DeWitt detector in Minkowski spacetime, for simplicity. A new development in heat kernel techniques—the Taylor trick—is necessary to perform the calculations and it is important to carefully choose cutoffs that diverge at the ultraviolet limit in order to keep the beta functions finite. We compare our results with the MS-bar results at one-loop and find that both computations agree qualitatively, as expected.

Based on arXiv: 2305.17453 [gr-qc] and work to appear.

Entanglement from rotating black holes in thermal baths

Ivan Agullo (Louisiana State University — USA)

The question of how much entanglement is produced during Hawking’s evaporation process has primarily been investigated mostly for spherically symmetric black holes in isolation. However, most black holes in our universe are spinning, and exist within a thermal environment—the cosmic microwave background. In this presentation, I will describe how black hole spin and the thermal environment impact the generation of entanglement. The predictions are suitable for experimental testing in the laboratory using analog platforms.

Horizons as Eavesdroppers: Black Holes Decohere Quantum Superpositions

Daine L. Danielson (University of Chicago — USA)

We show that if a massive (or charged) body is put in a quantum superposition of spatially separated states in the vicinity of a black hole or cosmological horizon, the mere presence of the horizon will eventually destroy the coherence of the superposition. This occurs because, in effect, the long-range fields sourced by the superposition register on the horizon which forces the emission of entangling “soft gravitons/photons” through the horizon. This enables the horizon to harvest “which path” information about the superposition. We provide estimates of the decoherence time for such quantum superpositions in the presence of a black hole and cosmological horizon. We further sharpen and generalize this mechanism by recasting the gedankenexperiment in the language of (approximate) quantum error correction. This yields a complementary picture where the decoherence is due to an “eavesdropper” inside the black hole attempting to obtain “which path” information by measuring the long-range fields of the superposed body. We compute the quantum fidelity to determine the amount of information such an interior observer can obtain, and give a direct relationship between the information content of the interior and the decoherence of the superposition in the exterior. In particular, we show that the decoherence of the superposition corresponds to the “optimal” measurement performable in the black hole interior.

Grand Canonical ensemble of a d-dimensional Reissner-Nordström Black Hole Space in a Cavity

Tiago Vasques Fernandes (GRIT, CENTRA, Instituto Superior Técnico — Portugal)

The grand canonical ensemble of a d-dimensional Reissner-Nordström black hole space in a cavity is analyzed using the Euclidean path integral approach. The partition function of the ensemble, which describes the thermodynamics of the black hole, is given by the Euclidean path integral depending on the Euclidean action with fixed temperature and fixed electric potential at the boundary of the cavity. One performs the zero loop approximation of the path integral, i.e., only the contribution of the stationary points of the action are considered. One finds two solutions for the stationary points of the action corresponding to two possible values of the horizon radius of the charged black hole, $r_{+1}$ and $r_{+2}$, where $r_{+2}$ is the largest radius solution. The stability is analyzed by considering perturbations of the reduced action, yielding instability of $r_{+1}$ and stability of $r_{+2}$. By the correspondence between the partition function and the thermodynamic grand potential, one derives the mean energy, mean charge, the entropy and the thermodynamic pressure of the system. Regarding phase transitions, a spontaneous process in the grand canonical ensemble can never increase the grand canonical potential $W$, or else the second law of thermodynamics would be violated. Therefore, for a cavity with fixed size, temperature, and electric potential, it is of interest to compare the value of $W$ for the black hole $r_{+2}$ solution with the value of $W$ of a nongravitating charged shell, which serves as a model for charged hot flat space. This enables one to study possible phase transitions between black holes and hot flat space.

The Entropy of Dynamical Black Holes

Victor Zhang (University of Chicago — USA)

We propose a new formula for the entropy of a dynamical black hole—valid to leading order for perturbations off of a stationary black hole background—in an arbitrary classical diffeomorphism covariant Lagrangian theory of gravity in $n$ dimensions. In stationary eras, this formula agrees with the usual Noether charge formula, but in nonstationary eras, we obtain a nontrivial correction term. In particular, in general relativity, our formula gives the entropy of a dynamical black hole as its area minus an integral involving the expansion of the null generators of the horizon, which we show is equal to the area of the apparent horizon to leading order. Our formula for entropy in a general theory of gravity is obtained from the requirement that a “local physical process version” of the first law of black hole thermodynamics hold for perturbations of a stationary black hole. It follows immediately that for first order perturbations sourced by external matter that satisfies the null energy condition, our entropy obeys the second law of black hole thermodynamics. For vacuum perturbations, the leading order change in entropy occurs at second order in perturbation theory, and the second law is obeyed at leading order if and only if the “modified canonical energy flux” is positive (as is the case in general relativity but presumably would not hold in more general theories of gravity). Our formula for the entropy of a dynamical black hole differs from the formula proposed independently by Dong and by Wall (which are both equal to the Bekenstein–Hawking entropy in general relativity), and we obtain the relationship between their formula and ours, thereby generalizing their results to our class of Lagrangians. We finally consider the generalized second law in semiclassical gravity for first order perturbations of a stationary black hole, and its relation (via the quantum null energy condition, or QNEC) to our formula for the entropy of a dynamical black hole and the one given by Dong and Wall. In this talk, I will focus the discussion on our formula in general relativity, and will show that it satisfies a classical second law and a generalized second law.

Quantum signatures of black hole mass superpositions

Joshua Foo (Stevens Institute of Technology — USA)

We present a new operational framework for studying “superpositions of spacetimes,” which are of fundamental interest in the development of a theory of quantum gravity. Our approach capitalizes on nonlocal correlations in curved spacetime quantum field theory, allowing us to formulate a metric for spacetime superpositions as well as characterizing the coupling of particle detectors to a quantum field. We apply our approach to analyze the dynamics of a detector (using the Unruh-deWitt model) in a spacetime generated by a Banados-Teitelboim-Zanelli black hole in a superposition of masses. We find that the detector exhibits signatures of quantum-gravitational effects corroborating and extending Bekenstein’s seminal conjecture concerning the quantized mass spectrum of black holes in quantum gravity. Crucially, this result follows directly from our approach, without any additional assumptions about the black hole mass properties.

The four laws of spacetime thermodynamics

Arno Keppens (Space Pole, Brussels — Belgium)

Bardeen, Carter, and Hawking presented the four laws of black hole thermodynamics fifty years ago, in 1973. Only 45 years later, in 2018, Wang and Braunstein rigorously demonstrated that surfaces away from horizons are not thermodynamic, unless they are spherically symmetric. This entails that the laws of black hole thermodynamics can only be generalized to surfaces that are concentric with black hole horizons. Assuming the Schwarzschild metric, this generalization is achieved with a twist: By equating two temperature definitions that should simultaneously hold for an equilibrium thermodynamics of spacetime’s degrees of freedom, it is immediately found that the latter only applies to surfaces with constant Newtonian gravitational potential. The laws of thermodynamics can correspondingly be rephrased in terms of spacetime’s state variables at these surfaces. Note: This work elaborates on an essay written for the Gravity Research Foundation 2023 Awards for Essays on Gravitation, which was awarded an Honorable Mention.

Path Integrals on Curved Spacetime: the case for second order geometry

Folkert Kuipers (LMU Munich — Germany)

The search for a theory that consistently combines quantum theory with general relativity forces us to consider geometrical frameworks beyond standard (first order) differential geometry. One candidate for such a generalized geometrical framework is second order (stochastic) differential geometry, which incorporates a violation of Leibniz rule, that is characteristic to the Wiener integral, into the geometry. This feature makes the framework an ideal tool in the study of covariant path integrals on curved spacetimes.

In this talk, I will review the basic ingredients of this framework in both a Euclidean and Lorentzian signature. Here, I will pay particular attention to the deformations of spacetime symmetries that arise due to the coupling of quantum fluctuations to the affine connection. Finally, I will discuss some connections to non-commutative geometry.

Energy Conditions for a Minimally Coupled Scalar Field

Hideki Maeda (Hokkai-Gakuen University — Japan, and Albert-Einstein-Institut — Germany)

Energy conditions are a set of inequalities imposed on an energy-momentum tensor that ensure that the corresponding matter field is physically reasonable. In fact, some of the standard energy conditions are imposed as assumptions in the proofs of various strong theorems in general relativity. It is expected that fundamental matter fields satisfy the energy conditions, but it is not obvious. In this talk, without imposing any spacetime symmetry, we derive simple equivalent representations of the standard energy conditions for a minimally coupled scalar field with potential.

Extended thermodyanmics of the bumblebee black hole

Zhan-Feng Mai (Kavli Institute for Astronomy and Astrophysics — China)

The bumblebee gravity is a potential theory to test Lorentz symmetry violation. Recently, a new class of numerical spherical black holes in the bumblebee theory was constructed. In this paper, we investigate the associated local thermodynamic properties. By introducing a pair of conjugated thermodynamic quantities X and Y , which can be interpreted as an extension of electric potential and charge of the Reissner-Nordström black holes, we numerically construct a new first law of thermodynamics for bumblebee black holes. We then study the constant-Y processes in the entropy-charge parameter space. For the constant-Y processes, we also calculate the heat capacity to study the local thermodynamic stability of the bumblebee black holes

Do classical-limit Schwarzschild Black Holes transcend the Laws of Thermodynamics?

Kiril Maltsev (University of Heidelberg/HITS — Germany)

In 1971 Geroch presented an ingenious gedanken experiment of a thermodynamic cycle in which a classical Schwarzschild Black Hole (CSBH) acts as a perfect heat sink that converts radiation energy, which is poured onto the event horizon out of a perfectly insulating box that is lowered from infinity, into mechanical work at 100% efficiency. His argument was that therefore, CSBHs are to be assigned absolute zero Kelvin temperature. We review the gedanken experiment and conclude that, for reversibility of the thermodynamic cycle, a minimal distance needs to be maintained when pouring out radiation onto the event horizon. This implies the assignment of only asymptotically zero Kelvin temperature: a classical Schwarzschild horizon is a mechanical thermostat operated at asymptotically unit efficiency. We then make the case that although a CSBH never enters into thermal equilibrium with its environment, the Zeroth Law is not prima facie transcended by the refrigerator analogy. Despite that horizon surface gravitation constitutes a frictionless Perpetuum Mobile of Third Kind (motion at no power supply), we argue that the First Law is not transcended, because no gravitational energy can be extracted from a quasi-stationary CSBH. Bekenstein in 1973 proved that the Generalized Second Law (GSL) cannot be violated in the Geroch cycle, if the Bekenstein entropy bound is assumed. Then, the horizon surface area increase will be at least as great as the entropy loss behind the horizon, and we show that the GSL persists to hold in the classical-limit case. Since the Bekenstein-Hawking entropy diverges in the classical limit, the Planck statement of the Third Law is violated. However, the Nernst statement is not, since the entropy change of the divergent CSBH entropy approaches zero in the asymptotically zero classical limit of Hawking temperature. The overall conclusion is that the Laws of Thermodynamics are not transcended by classical-limit Schwarzschild Black Holes, despite their extremal thermodynamic properties.

The talk is based on

Quantum energy inequalities in integrable models

Jan Mandrysch (FAU Erlangen — Germany)

Many results in general relativity rely crucially on classical energy conditions. This includes for instance singularity theorems a la Hawking and Penrose which predict the formation of black holes. Quantum matter, however, violates these conditions since the energy density can fluctuate and in particular become arbitrarily negative at a point. A reminiscent notion of stability can be captured by weaker conditions often referred to as quantum (weak) energy inequalities (QEIs). Such inequalities have been proven in many free QFT models and in some cases lead to similar singularity theorems as for classical matter.

However, there exist only few results in interacting theories. A first step in this direction is to find QEI bounds in QFT without gravity. Here we will focus on a certain class of two-dimensional interacting QFTs known as “integrable models”. In particular, we present results on the O(N)-nonlinear-sigma and sinh-Gordon model at one- and two-particle level.

The talk is partly based on 2302.00063.

Why 5? Semi-classical analysis of a Kerr-AdS black hole in five dimensions

Alessandro Monteverdi (University of Sheffield — UK)

Some of the quantum properties of rotating black holes in (3+1)-dimensional spacetimes remain unresolved. However, studying higher-dimensional cases may provide insight into their behaviour. In this talk, we investigate the behaviour of a massive scalar field in a Kerr-AdS (4+1)-dimensional spacetime. Specifically, we focus on the existence of a Hartle-Hawking state, which is a vacuum state with important properties, such as being a thermal and Hadamard state, which means that it is well-behaved and has a finite energy density. While the Hartle-Hawking state has been studied in various types of black holes, our focus is on rotating black holes where we have observed a correlation between the presence of the light surface and the existence of the Hartle-Hawking state. It should be noted that the Hartle-Hawking state does not exist in Kerr black holes, but it does exist in Kerr-AdS black holes. In four dimensions, the analysis of this state is very challenging, but in five dimensions, the enhanced symmetry of the system simplifies the analysis. Finally, by using the Hartle-Hawking state, we also present a method for evaluating observables, starting with the vacuum polarization.

Asymptotic quasinormal modes of string-theoretical d-dimensional black holes

Filipe Moura (Instituto Universitário de Lisboa — Portugal)

We compute the quasinormal frequencies of d-dimensional large spherically symmetric black holes with Gauss-Bonnet corrections in the highly damped regime. We solve perturbatively the master differential equation and we compute the monodromies of the master perturbation variable (analytically continued to the complex plane) in different contours, in order to obtain the quasinormal mode spectra. We consider tensorial, vectorial and scalar gravitational perturbations, obtaining the same frequencies for the three cases like in Einstein gravity.

Black-hole Thermodynamics in Higher-Curvature Gravities

Ángel Murcia (INFN, Sezione di Padova — Italy)

In this talk, I will present a class of higher-curvature gravities admitting non-hairy generalizations of the Schwarzschild solution whose thermodynamics can be computed analytically. This class of theories, called Generalized Quasitopological Gravities, turn out to provide a perturbative basis for the space of effective theories of gravity, so the study of their thermodynamic properties is of relevance. Among other things, I will review some intriguing novel features of black-hole thermodynamics in these theories and present other interesting phenomena regarding the thermodynamics of charged black holes when non-minimal couplings to a U(1) gauge field are included.

Kinematic and Thermodynamic Properties of Dynamical Regular Black Holes

Sebastian Murk (Quantum Gravity Unit, Okinawa Institute of Science and Technology — Japan)

Quantum gravity candidate theories are generally expected to cure the singularities inherent to the mathematical black holes predicted by general relativity. In the absence of such a theory, singularity-free models of so-called regular black holes have become a popular alternative to avoid the nontrivial causal structures typically associated with mathematical black holes. Based on the assumption that semiclassical physics remains valid in the vicinity of their horizons, we derive the generalized dynamical version of the first law of black hole mechanics and demonstrate that the need for corrections to the conventional form of the first law is ineluctably linked to the introduction of a minimal length scale and can therefore be seen as a direct consequence of the spacetime regularization. We analyze the behavior of the null energy condition and find that it is violated in the vicinity of the outer horizon and satisfied in the vicinity of the inner horizon, which implies that the trapped spacetime region (as determined from the behavior of null geodesic congruences) is effectively separated into an NEC-violating and an NEC-non-violating domain. Moreover, we show that massive observers and particles can cross the inner and outer horizon on an ingoing geodesic, and thus entering and exiting the supposedly trapped spacetime region is possible. We outline the physical implications of these results for the information loss problem and black hole thermodynamics.

Unpacking Black Hole Complementarity

Siddharth Muthukrishnan (University of Pittsburgh — USA)

To what extent does the black hole information paradox lead to violations of quantum mechanics? I explain how black hole complementarity provides a framework to articulate how quantum characterizations of black holes can remain consistent despite the information paradox. I point out that there are two ways to cash out the notion of consistency in play here: an operational notion and a descriptive notion. These two ways of thinking about consistency lead to (at least) two principles of black hole complementarity: an operational principle and a descriptive principle. Our background philosophy of science regarding realism/instrumentalism might initially lead us to prefer one principle over the other. However, the recent physics literature, which applies tools from quantum information theory and quantum computational complexity theory to various thought experiments involving quantum systems in or around black holes, implies that the operational principle is successful where the descriptive principle is not. This then lets us see that for operationalists the black hole information paradox might no longer be pressing.

Fisher Information of Thermal Black Hole Parameter

Everett Patterson (Institute for Quantum Computing, University of Waterloo — Canada)

Relativistic quantum metrology is a framework that not only accounts for both relativistic and quantum effects when performing measurements and estimations, but further improves upon classical estimation protocols by exploiting quantum relativistic properties of a given system. Here I present recent developments in the Fisher information analysis associated with black hole spacetimes. I review recent work in relativistic quantum metrology that examined Fisher information for estimating thermal parameters in (2+1)-dimensional AdS and the static BTZ black hole spacetimes. Treating Unruh-DeWitt detectors coupled to a massless scalar field as probes in an open quantum systems framework, I extend these recent results to the (2+1)-dimensional rotating black hole spacetime. We find that varying the mass and angular momentum of the BTZ black hole leads to dramatic change in the Fisher information provided the appropriate black hole and detector parameters.

Extracting The Temperature of Quantum Fields Through Local Measurements: general features of the thermalization of particle detectors.

T. Rick Perche (University of Waterloo, Perimeter Institute and the Institute of Quantum Computing — Canada)

Realistic particle detectors can be used to accurately probe the temperature of quantum fields in general backgrounds. In this talk we will show that if the field is in a state that satisfies the KMS condition with inverse temperature β with respect to the detector’s local notion of time evolution, reasonable assumptions ensure that the probe thermalizes to the temperature 1/β in the limit of long interaction times. Our method also imposes bounds on the size of the system with respect to its proper acceleration and spacetime curvature in order to accurately probe the KMS temperature of the field. We then comment on applications to the case of detectors probing the Unruh and Hawking temperatures.

Noncommutative correction to black hole entropy

Filip Pozar (Institute Ruder Boskovic — Croatia)

Noncommutative geometry is an established candidate for modeling aspects of quantum gravity. We will outline the calculation of the correction to black hole entropy due to noncommutative geometry in the Brick wall approach and find logarithmic corrections, providing evidence to the claim that noncommutative spacetime encapsulates quantum effects of the spacetime.

Light-cone thermodynamics: purification of the Minkowski vacuum

Salvatore Ribisi (CPT–Marseille — France)

We explicitly express the Minkowski vacuum of a massless scalar field in terms of the particle notion associated with suitable spherical conformal killing fields. These fields are orthogonal to the light wavefronts originating from a sphere with a radius of rH in flat spacetime: a bifurcate conformal killing horizon that exhibits semiclassical features similar to those of black hole horizons and Cauchy horizons of non-extremal spherically symmetric black holes. Our result highlights the quantum aspects of this analogy and extends the well-known decomposition of the Minkowski vacuum in terms of Rindler modes, which are associated with the boost Killing field normal to a pair of null planes in Minkowski spacetime (the basis of the Unruh effect). While some features of our result have been established by Kay and Wald’s theorems in the 90s—on quantum field theory in stationary spacetimes with bifurcate Killing horizons—the added value we provide here lies in the explicit expression of the vacuum.

Modular Localization and The Problem of Localizability of Relativistic Particles

Ivan Romualdo (University of Sao Paulo — Brazil/ FAU-Erlangen-Nuremberg — Germany)

It is a longstanding open problem to define a notion of localizability of states for relativistic quantum systems, that is, to construct probability measures that predict the likelihood of detecting a particle in a given region of space or spacetime. The first attempt was due to Newton and Wigner (Newton, Wigner, 1949), later improved by Wightman (Wightman, 1962), where the referred probability measures were constructed in terms of a projection-valued-measure defined on the Borel sets of a Cauchy surface. Despite its many accomplishments, the Newton-Wigner localizability is not entirely satisfactory. Localized states, in this framework, can instantaneously spread to non-local regions. In this work, we show how the formalism of Modular Localization (Brunetti, Guido, Longo, 2002) can contribute to this problem in a way that is fully satisfactory with relativistic principles. In addition, we show that the probability measures defined in this new approach approximate the ones constructed by Newton-Wigner when the regions of spacetime are separated by distances larger than the Compton wavelength of the particle.

The Idealization Paradox in Hawking Radiation

Dominic Ryder (London School of Economics — United Kingdom)

In this talk I argue that three mainstream derivations of Hawking radiation all contain an unresolved paradox. These are Hawking’s original derivation (1975), Fredenhagen and Haag’s mathematically “watertight” derivation (1990), and the algebraic approach (Dimock and Kay, 1987). The paradox is due to the fact that the derivations are carried out in a spacetime which does not model black hole evaporation whereas, given the existence of Hawking radiation, black holes are expected to evaporate. Therefore, one should be able to deidealize the derivations and carry them out on spacetimes which model black hole evaporation. However, assumptions essential for the derivations breakdown in evaporation spacetimes, and so it appears the idealization of non-evaporation is essential for these derivations.

I first present and defend the existence of the idealization paradox by showing that for all three approaches, assumptions necessary for them to be carried out breakdown in evaporation spacetimes. Hawking himself recognized this problem and proposed an approximation regime to respond to it. Therefore, I next describe this proposal and argue that for each derivation it is non-trivial to show that Hawking radiation can be recovered in the approximation regime. For Hawking’s derivation the prospects of success for the approximation regime are very limited; for Fredenhagen and Haag’s the prospects are better; and for the algebraic approach the prospects are good.

Finally, canvas possible resolutions to the idealization paradox and argue that, given a resolution, we may learn about: the operation of idealizations in physics, how global structure encodes local structure in general relativity, the nature of Hawking radiation, and how science uses models to make progress.

Holographic Einstein Rings of a Black Hole in Massive Gravity

Rabia Saleem (COMSATS University Islamabad, Lahore Campus — Pakistan)

In the context of holography, the Einstein ring of an AdS black hole (BH) in massive gravity (MG) is depicted. An oscillating Gaussian source on one side of the AdS boundary propagates in bulk, and we impose a response function to explain it. Using a wave optics imaging system, we obtain the optical appearance of the Einstein ring. Our research reveals that the ring can change into a luminosity-deformed ring or light spots depending on the variation of parameters and observational positions. When observers are positioned at the north pole, the holographic profiles always appear as a ring with concentric stripe surroundings, and a bright ring appears at the location of the photon sphere of the BH. The amplitude of the lensed response function $|\langle O\rangle|$ increases with the increasing values of the graviton parameter $m$, for the fixed value of the horizon $u_{e}$. On the other hand, the amplitude of the lensed response function $|\langle O\rangle|$ decreases with the increasing values of the horizon $u_{e}$, for the fixed value of graviton parameter $m$. These differences are also reflected in the Einstein ring, where the intensities and the locations of the Einstein ring significantly vary according to the numerical values of the involved parameters. These findings are also observed in the brightness profiles and the best fit comparison between the results obtained by wave optics and geometric optics for different values of graviton parameter $m$.

QFT in Non-smooth Spacetimes

Yafet Erasmo Sanchez Sanchez (University of Genoa — Italy)

Quantum Field Theory (QFT) offers a robust framework for understanding the universe at a microscopic level. However, its traditional formulation relies on the assumption of smooth spacetime—an idealization that fails in the context of certain astrophysical models and self-gravitating fluids.

In this talk, I will explore the extension of QFT to accommodate non-smooth spacetimes. I will outline the primary concepts involved in the algebraic quantization of the free scalar field. The construction of observables will draw from the findings presented in arXiv:1910.13789, while the discussion of states will reference arXiv:2207.01429.

AdS/BCFT correspondence and BTZ black hole entropy within Horndeski gravity

Fabiano Francisco dos Santos (Federal University of Maranhão — Brazil)

This talk presents some results results of the BTZ black hole thermodynamics. These were performed by gravity duals of conformal field theories with boundaries$-$ known as AdS/BCFT correspondence. Through holographic renormalization, the free energy is obtained and with it, the other thermodynamic quantities are derived. Beyond these results, the Hawking-Page phase transition was computed to show the stable and unstable phases throughout the plane of free energy versus temperature.

Quantum Corrected Anti-de Sitter Spacetime: Backreaction from the RSET

Jacob Thompson (University of Sheffield — UK)

We consider (3+1)-dimensional anti-de Sitter spacetime and take the renormalised stress energy tensor as computed by Allen, Folacci and Gibbons in 1987. This talk will focus on the effects of the renormalisation upon the spacetime geometry by solving the backreaction problem, manifest in the semi-classical Einstein equations as a set of differential equations for the metric on the LHS, using the RSET as a source term. This metric is then what is known as the quantum corrected metric, and this talk will explain how it is solved for and discuss some interesting properties it exhibits, as well as discuss ‘how far away from’ pure-adS the quantum corrected metric is.

Holographic Dual of Extended Black Hole Thermodynamics

Manus Visser (University of Cambridge — United Kingdom)

We establish a novel exact holographic dictionary between the laws of extended black hole thermodynamics in Anti-de Sitter Space and the thermodynamic laws of the dual conformal field theory (CFT). We formulate the CFT first law that is exactly dual to the first law of extended black hole thermodynamics with variable cosmological constant but fixed Newton’s constant. Moreover, we find the holographic dual of the Smarr relation for AdS black holes, coined the ‘holographic Euler relation’. On the field theory side we include two independent pairs of thermodynamic conjugate variables: the central charge-chemical potential term and the pressure-volume term. In this setting we uncover various new phase transitions and critical behaviour in the CFT.


The Impact of Gravitational Lensing in The Reconstruction og Stellar Oobits Around SGR A*

Silvia Pietroni(INAF — Italy)

After the amazing discoveries by the GRAVITY collaboration in the last few years on the star S2 orbiting the black hole Sgr A* in the center of the Milky Way, we present a detailed investigation of the impact of gravitational lensing on the reconstruction of stellar orbits around this massive black hole.

One Lorentz Violating Model and Black hole thermodynamics

Jan Novák(Czech Technical University — Czech Republic)

We review the basic problems why it is notoriously difficult to do considerations in thermodynamics, when we deal with the Lorentz breaking theories. Concretely, we formulated a new approach to quantum gravity, which is called the ring paradigm. Graviton is a phonon on a dynamical grid, which would ultimately mean that we need to break the Lorentz invariance. It is clear from the computations in the toy model with graviton-phonons that we must define some temperature T_g pertaining to the longitudinal vibrations of the rings, which mediates gravity. But the crystal made of rings was present at the instant of the Big Bang, before the occurrence of any particles. Therefore T_g > 0, even when the absolute thermodynamic scale was not defined. It looks like we need to introduce an artificial temperature scale concerning just the vibrations of the rings. This is a completely new phenomenon in thermodynamics.

Yang-Mills Casimir Wormholes in D=2+1

Alana Carolina Lima dos Santos(Federal University of Ceara — Brazil)

The Casimir effect is associated with quantum vacuum fluctuations when we impose boundary conditions on quantum fields. The first configuration analyzed, two neutral parallel conducting plates separated by a very short distance, produces an attractive force. However, the nature of this force is strongly dependent on the geometry and dimension in question and can be repulsive, for example. This manifestation of quantum effects on macroscopic scales was confirmed in 1958 and has been subjected to increasingly precise experiments. On the other hand, a distinct characteristic of the Casimir energy density is the violation of energy conditions due to its negative value for some configurations. This fact promotes it as one of the scarce energy sources capable of sustaining traversable wormholes - solutions of Einstein’s equations analogous to a hypothetical tunnel in spacetime between two distant points in the universe, whose traversability requires exotic matter. Then, this work presents new three-dimensional traversable wormhole solutions sourced by the Casimir energy density and pressures related to the quantum vacuum fluctuations in the Yang-Mills theory. Initially, we analyze the noninteracting Casimir source with an arbitrary state parameter ω and determine a simple constant wormhole shape function. Next, we introduce a new methodology for deforming the state parameter to find well-behaved redshift functions. The wormhole can be interpreted as a true Casimir wormhole with an expected average state parameter of ω = 2. Then, we investigate the wormhole curvature properties, energy conditions, and stability of the solutions. Furthermore, we discover a novel family of traversable wormhole solutions sourced by the quantum vacuum fluctuations of interacting (confined) Yang-Mills fields with a more complex shape function. Deforming the effective state parameter similarly, we obtain well-behaved redshift functions and traversable wormhole solutions. Finally, we also examine the energy conditions and stability of solutions in the interacting scenario compared to the noninteracting case, with instability near the wormhole throat becoming more stable towards the ω = 2 Casimir configuration.