# Global Structure in Semi-Classical Gravity (21-23 July 2022)

## Idea & Motivation

Semi-classical gravity (SCG)---the coupling of the dynamics of quantum fields to the classical spacetime geometry of general relativity byway of the semi-classical Einstein field equation (SCEFE)---is the framework within which black hole thermodynamics is formulated. As such, it is widely held to provide the most secure theoretical clues

to a search for a theory of quantum gravity. Considered in its own right, however, it already presents us with a plethora of foundational and conceptual problems that require investigation, even over and above the most famous one, Hawking's Information-Loss Paradox. In what sense is it legitimate to think of black holes as thermodynamical systems? What is the causal structure of an evaporating black hole spacetime? What is black hole entropy? How does the Generalized Second Law ramify into the host of other problems of fundamental physical and philosophical import that the ordinary Second Law does, from the arrow of time and the nature of spacetime to issues of probability, causality, predictability and determinism? What is the difference between geometry and matter? Are there singularities in the semi-classical regime, and does a form of cosmic censorship hold in it? Are there physically reasonable full solutions to the SCEFE? What role do classical energy conditions and quantum energy inequalities play, and how do they relate to entropy and information? How does holography relate to and constrain all this?

This will be the first major conference to address foundational issues associated with SCG, and to try to examine how they all bear on each other. Invited physicists and philosophers have been chosen to represent different approaches to SCG (quantum field theory on curved spacetime, emergent gravity, canonical gravity, holography) and to discuss different kinds of issues (locality, information loss, the nature of black hole entropy, singularities and cosmic censorship, epistemology, methodology, ontology, inter alia), in an attempt to foster a dialogue among the different fields. Contributed talks from both physicists and philosophers will be chosen in part so as to complement the perspectives of the invited speakers. Such a gathering will spur new, innovative approaches to the problem, as well as connecting and invigorating work on existing approaches. It will also provide young researchers with a comprehensive introduction to the state of the art of this central field of research, and established researchers with a comprehensive overview.

## Confirmed Speakers

- Beatrice Bonga
- Ted Jacobson
- Eleni Kontou
- JB Manchak
- Marija Tomasevic
- Chris Smeenk
- Rainer Verch
- Bob Wald
- Chris Wüthrich
- Bernard Kay
- Bryan W. Roberts

## Registration

Public-health conditions permitting, we plan to hold the conference in-person, with no video connections, although the talks and Q&A will be recorded (pending agreement by the speakers). Please send

registration requests by 26 June 2022 using the following form: https://forms.gle/rBAZakkNPxc2uxD87

We are happy to announce that thanks to the generous support of the DFG the registration fees have been waived. Only the conference dinner at Georgenhof, Friedrichstraße 1, 80801 München on Friday, July 22nd remains on dutch treat for everyone.

## Day 1 (21 July 2022) - Location: IBZ, Amalienstraße 38

Time | Event |
---|---|

09:00 - 10:00 | Registration & Coffee |

10:00 - 11:00 | Robert Wald: The Black Hole Information Loss Issue |

11:00 - 11:30 | Coffee Break |

11:30 - 12:15 | Jinzhao Wang & Renato Renner: The Black Hole Information Puzzle and the Quantum de Finetti Theorem |

12:15 - 13:00 | Erik Curiel: On Information-Loss Paradoxes, Holographic Proofs of Unitarity, Causal Structure and Non-Locality - or - Why Marolf's Argument is Excruciatingly Clever and Elegant, but Probably Wrong |

13:00 - 14:30 | Lunch Break |

14:30 - 15:30 | Chris Smeenk: Inflation: Beyond Semi-Classical Quantum Gravity |

15:30 - 16:30 | Béatrice Bonga: Foundational Issues of Gravitational Waves in Cosmological Spacetimes |

16:30 - 17:00 | Coffee Break |

17:00 - 18:00 | JB Manchak: Is the Universe As Large As It Can Be? If So, Why? |

## Day 2 (22 July 2022) - Location: LMU, Geschwister-Scholl-Platz 1, A120

Time | Event |
---|---|

09:30 - 10:00 | Registration & Coffee |

10:00 - 11:00 | Rainer Verch: Temperature and Entropy-Area Relation of Quantum Matter Near Spherically Symmetric Outer Trapping Horizons |

11:00 - 11:30 | Coffee Break |

11:30 - 12:15 | Manus Visser: Newton’s Force Law from Entanglement |

12:15 - 13:45 | Lunch Break |

13:45 - 14:45 | Bernard Kay: The Matter-Gravity Entanglement Hypothesis and the Information Loss, and Other Puzzles |

14:45 - 15:15 | Coffee Break |

15:15 - 16:00 | Tom McClain: Geometry vs. Matter: Could a Semi-Classical Approach to Gravity be Fundamental? |

16:00 - 16:45 | Ana Alonso-Serrano & Marek Liska: Quantum Phenomenological Gravitational Dynamics: A General View from Thermodynamics of Spacetime, note: REMOTE TALK VIA ZOOM |

16:45 - 17:15 | Coffee Break |

17:15 - 18:15 | Bryan Roberts: Black Holes and Geometric Thermodynamics |

19:00 | Conference Dinner @ Georgenhof, Friedrichstraße 1, 80801 München |

## Day 3 (23 July 2022) - Location: IBZ, Amalienstraße 38

Time | Event |
---|---|

09:30 - 10:00 | Registration & Coffee |

10:00 - 11:00 | Marija Tomasevic: Holography of Time Machines |

11:00 - 11:30 | Coffee Break |

11:30 - 12:30 | Chris Wüthrich: Time Travelling in Emergent Spacetime |

12:30 - 14:00 | Lunch Break |

14:00 - 15:00 | Eleni Kontou: Singularity Theorems in Semiclassical Gravity |

15:00 - 15:30 | Coffee Break |

15:30 - 16:15 | Marc Schneider: Probing Singularities with Quantum Fields in General Relativity |

16:15 - 17:00 | Daan Janssen: Constructing Quantum Fields on Evaporating Black Holes and Other Semi-globally Hyperbolic Space-Times |

17:00 - 17:30 | Coffee Break |

17:30 - 18:30 | Ted Jacobson: Diffeomorphism Invariance and the Black Hole Information Paradox, note: REMOTE TALK VIA ZOOM |

## Abstracts

### Ana Alonso-Serrano & Marek Liska: Quantum phenomenological gravitational dynamics: A general view from thermodynamics of spacetime

I present a formalism to analyze low-energy quantum gravity modifications in a completely general framework based on the thermodynamics of spacetime. I first discuss the connection between thermodynamics and gravitational dynamics in the light of the derivation of Einstein equations of motion from the proportionality of entropy with an area, and analyze the entropies involved in the process. Then, quantum gravity effects are considered via modification of entropy by an extra logarithmic term in the area. This modification is predicted by several approaches to quantum gravity, including loop quantum gravity, string theory, AdS/CFT correspondence and generalised uncertainty principle phenomenology, giving our result a general character. I show the derivation of the quantum modified gravitational dynamics from this modified entropy expression and discuss its main features. These results provide a general expression of quantum phenomenological equations of gravitational dynamics. Furthermore, I outline the application of the modified dynamics to particular models, such as cosmology, suggesting the replacement of the Big Bang singularity with a regular bounce.top

### Beatrice Bonga: Foundational issues of gravitational waves in cosmological spacetimes

There exists a solid framework to study gravitational waves in full, non-linear general relativity when the spacetime is asymptotically flat. It does not require the splitting of the metric in a background piece and a linearly perturbed part and therefore describes all non-linearities of the theory. The situation for cosmological spacetimes is different, however. Expanding spacetimes, whose expansion is decelerating such as matter- or radiation-dominated universes, share some similarities with the asymptotically flat case nonetheless. I will discuss some of the recent developments in this area.top

### Erik Curiel: On Information-Loss Paradoxes, Holographic Proofs of Unitarity, Causal Structure and Non-Locality - or - Why Marolf's Argument is Excruciatingly Clever and Elegant, but Probably Wrong

I examine Marolf's influential argument for the unitarity of black hole evaporation based on what he calls a holographic argument for boundary unitarity. I draw out a hidden assumption of the argument, that the causal structure of the interior must be nice in a particular way. I conclude that his argument thereby either begs the question or else necessitates a peculiarly strong and promiscuous global non-locality of spacetime structure, which I find hard to accept.top

### Ted Jacobson: Diffeomorphism invariance and the black hole information paradox

After briefly characterizing the paradox, I will argue that, as with other famous paradoxes of physics, its resolution is to be found in paying close attention to what is a meaningful statement within the theory. The central role of the diffeomorphism constraints will be emphasized, and recent progress (by other authors) illustrating how this works even in perturbation theory will be summarized.top

### Daan Janssen: Constructing Quantum Fields on Evaporating Black Holes and Other Semi-globally Hyperbolic Space-Times

In the semi-classical picture, that is, a classical background geometry coupled to a quantum field theory via the semi-classical Einstein equations, it is expected that when a black hole fully evaporates, the final stage of evaporation, before the black hole has fully disappeared, is a naked singularity. It is this causal defect that can be seen as the reason why an evaporating black hole space-time is not globally hyperbolic. These naked singularities are somewhat different in nature than those appearing in some classical solutions to the Einstein equations, which can often be seen as time-like singular boundaries. In this case the defect has a more discrete nature, which can be compared to the causal defects appearing in certain space-times that undergo a topology change. To capture this feature in a more general setting, I shall introduce a new causality condition, situated between stable causality and global hyperbolicity, that allows for causal defects that are, in a sense, discrete in time. This condition can be formulated in multiple equivalent ways, and relates to the existence of a cover of such a space-time by globally hyperbolic patches that can be given some natural time-ordering.

### Bernard Kay: The matter-gravity entanglement hypothesis and the information

loss, and other, puzzles

Hawking’s 1974 discovery of black hole evaporation raised a number of puzzles, including the information loss puzzle, the thermal atmosphere puzzle and the firewall puzzle. Some people also saw in Hawking’s discovery an indication that the traditional understanding of entropy and of the Second Law, in terms of a Boltzmannian coarse- grained entropy – which is partly subjective and/or vague – may need to be replaced by a new understanding in which entropy is objective and has something to do with quantum gravity. It is also unclear whether the above puzzles can be resolved within a semiclassical framework or require a quantum gravity theory.

In this talk, I recall my 1998 matter-gravity entanglement hypothesis which starts with the assumption that (low-energy) quantum gravity is a traditional Hilbert- space-based quantum theory with a unitarily time-evolving ever-pure total state and (partly motivated by the thermal atmosphere puzzle) defines the entropy of a closed system to be its matter-gravity entanglement entropy. I then outline some recent work [1] in which I argue that due to (usually neglected) photon-graviton interactions, at least if the evaporation is slowed down by putting the black hole in a slightly permeable box, the radiation remaining after a large black hole has evaporated will (be pure and) mainly consist of roughly equal numbers of photons and gravitons entangled with one another – with a (matter-gravity entanglement) entropy greater than that of the freshly formed black hole. We also give a general argument that, in the absence of such a box, the final state would be similar. If it is so, this would seem to offer a resolution to the information loss puzzle and also to improve the prospects for the resolution of the firewall puzzle since late emitted photons/gravitons would not be needed to purify early emitted photons/gravitons; instead they would purify one other.

[1] B.S. Kay, Matter-gravity entanglement entropy and the information loss puzzle. arXiv:2206.07445

### Eleni Kontou: Singularity theorems in semiclassical gravity

Classical singularity theorems predict the existence of singularities, defined using incomplete geodesics, under a set of general assumptions. One of those assumptions, namely the energy condition, is always violated by quantum fields and thus the realm of semiclassical gravity is outside the scope of these theorems. However, quantum fields do obey weaker conditions which can also be used to predict singularities. In this talk, I will present such semiclassical singularity theorems both in the timelike and the null case and discuss the challenges and open questions for each case.

### JB Manchak: Is the Universe As Large As It Can Be? If So, Why?

Leibnizian metaphysics seems to underpin the position that spacetime is inextendible -- that it is "as large as it can be" in an appropriate sense. Here, we cast doubt on this position in a variety of ways. We highlight a recent result concerning the instability of inextendibility if attention is restricted to certain collections of "physically reasonable" spacetimes.top

### Tom McClain: Geometry vs. matter: Could a semi-classical approach to gravity be fundamental?

A novel solution to the problem of quantum gravity is proposed: what if it were possible – by means of a strict correspondence between quantum fields and their classical counterparts – to source the Einstein field equations with exact, classical fields corresponding to fundamental quantum matter fields? This correspondence may be provided – possibly even required – by a geometric approach to quantum field theory founded on the twin pillars of a global version of Gunther’s polysymplectic framework for Hamilonian field theory and a simple approach to geometric quantization used in the early work of Konstant and Souriau. Through this correspondence, the classical-yet-dynamical nature of space-time could be unified with the fundamentally quantum mechanical nature of mat- ter without theoretical or philosophical tension. The theoretical and mathematical pillars of this approach are presented, and its potential merits and demerits are discussed.top

### Bryan Roberts: Black holes and geometric thermodynamics

We formulate and defend the interpretation of equilibrium thermodynamics as a Gibbs-inspired geometric theory, with models given by contact manifolds, and then reconsider the philosophical status of black hole thermodynamics from this perspective.

top

### Marc Schneider: Probing Singularities in General Relativity with Quantum Fields

The singularity theorems of Penrose and Hawking are based on geodesic incompleteness and predict the occurrence of classical singularities under rather general circumstances. In general relativity, these singularities represent absolute boundaries where space-time ends.

Physically, however, this criterion refers to the fate of point like classical test particles. We raise the question: What if one uses quantum fields instead? Intuitively, quantum probes are much more fundamental and bear a richer structure. We will begin with the prove that one can unambiguously evolve quantum fields across them in a rigorous sense. Thus when probed with quantum fields, the big bang is not an absolute boundary where physics breaks down. Additionally we will discuss the behavior of composite operators such as the expectation values of renormalized products of fields and the renormalized stress-energy tensor and show that they too remain well-defined as distributions.

For Schwarzschild black holes, we will show that a similar taming occurs for quantum probes across the gravitational singularity.

The overall conclusion is twofold: first quantum mechanical considerations provide more refined tools to probe classically singular structures, and second, singularities of classical general relativity are tamer when seen from a quantum perspective.

### Chris Smeenk: Inflation: Beyond Semi-Classical Quantum Gravity

According to inflationary cosmology, the very early universe passed through a transient phase of exponential expansion, leading to several characteristic features in the post-inflationary state, compatible with observations. This phase of exponential expansion is sourced semi-classically by the stress-energy tensor of the inflaton field, but inflation requires going beyond a semi-classical description. In particular, inflation generates Gaussian perturbations with a nearly scale-invariant spectrum via back-reaction of quantum fluctuations of the inflaton field on the gravitational field. A mean-field description sensitive only to the expectation value of the stress-energy tensor will miss these fluctuations. Standard accounts instead describe the generation of primordial fluctuations in terms of linear perturbations of the inflaton and metric field to a background spacetime. Yet there are several questions regarding the domain of applicability of these methods from the perspective of quantum gravity. Here I will focus on two distinct aspects of these debates: the robustness of the inflationary account of the generation of perturbations to assumptions regarding the initial state, and the self-consistency of the dynamical evolution.top

### Marija Tomasevic: Holography of time machines

We use holography to examine the response of interacting quantum fields to the appearance of closed timelike curves in a dynamically evolving background that initially does not contain them. For this purpose, we study a family of two- dimensional spacetimes that model very broad classes of wormhole time machines. The behavior of strongly coupled conformal theories in these spacetimes is then holographically described by three-dimensional AdS bulk geometries that we explicitly construct. The dual bulk spacetime is free from any divergences, but splits into two disconnected components, without and with CTCs, which are joined only through the boundary; then, passages across the chronology horizon are impossible for any field excitations. In dual terms, the strong self-interaction of the CFT suffices to enforce – without any gravitational backreaction – the chronology protection principle in the most explicit manner: by completely decoupling the pathological part from the rest of the spacetime. We also find that entangling the CFTs in two separate time machines connects them through a traversable bulk wormhole. Nevertheless, any entanglement-assisted chronology violations will be prevented by quantum bulk corrections, i.e., subleading 1/N effects, again without needing any gravitational backreaction of the CFT. We are led to speculate that chronology may be protected without involving Planck scale physics.top

### Rainer Verch: Temperature and entropy-area relation of quantum matter near spherically symmetric outer trapping horizons

We consider spherically symmetric spacetimes with an outer trapping horizon. Such spacetimes are generalizations of spherically symmetric black hole spacetimes where the central mass can vary with time, like in black hole collapse or black hole evaporation. These spacetimes possess in general no timelike Killing vector field, but admit a Kodama vector field which provides a replacement. Spherically symmetric spacelike cross-sections of the outer trapping horizon define in- and outgoing lightlike congruences. We investigate a scaling limit of Hadamard 2-point functions of a quantum field on the spacetime onto the ingoing lightlike congruence. The scaling limit 2-point function has a universal form and a thermal spectrum with respect to the time-parameter of the Kodama flow, where the inverse temperature is related to the surface gravity of the horizon cross-section in the same way as in the Hawking effect for an asymptotically static black hole. Similarly, the tunneling probability in the scaling limit between in- and outgoing Fourier modes with respect to the the Kodama time shows a thermal distribution with the same inverse temperature, determined by the surface gravity. This can be seen as a local counterpart of the Hawking effect for a dynamical horizon in the scaling limit. The scaling limit 2-point function as well as the 2-point functions of coherent states of the scaling-limit-theory have relative entropies behaving proportional to the cross-sectional horizon area. Thereby, we establish a local counterpart, and microscopic interpretation in the setting of quantum field theory on curved spacetimes, of the dynamical laws of outer trapping horizons, derived by Hayward and others in generalizing the laws of black hole dynamics originally shown for stationary black holes by Bardeen, Carter and Hawking.

Based on joint work with F. Kurpicz and N. Pinamonti, Letters in Mathematical Physics 111 (2021) No 110, e-Print: 2102.11547 [gr-qc]

### Manus Visser: Newton’s force law from entanglement

In this talk I derive Newton’s second law and the law of universal gravitation from the physics of quantum entanglement.

### Bob Wald: The Black Hole Information Loss Issue

It has been known for nearly 50 years that black holes radiate thermally and that, as a consequence, they should completely evaporate in a finite time. It is a clear prediction of semi-classical gravity that, in this process, an initial pure state will evolve to a final mixed state, i.e., that information will be lost into the black hole. I review these semiclassical arguments and briefly discuss various mechanisms that have been invoked to try to evade this conclusion, including fuzzballs, firewalls, tunneling via wormholes, remnants, and information emerging in a final burst. The general arguments against information loss are discussed, and it is concluded that these do not make loss of information into a black hole rise to the level of a ”paradox.

### Jinzhao Wang & Renato Renner: The black hole information puzzle and the quantum de Finetti theorem

The black hole information puzzle arises from a discrepancy between conclusions drawn from general relativity and quantum theory about the nature of the radiation emitted by a black hole. According to Hawking's original argument, the radiation is thermal and its entropy thus increases monotonically as the black hole evaporates. Conversely, due to the reversibility of time evolution according to quantum theory, the radiation entropy should start to decrease after a certain time, as predicted by the Page curve. This decrease has been confirmed by new calculations based on the replica trick, which also exhibit its geometrical origin: spacetime wormholes that form between the replicas. Here we analyse the discrepancy between these and Hawking's original conclusions from a quantum information theory viewpoint, using in particular the quantum de Finetti theorem. The theorem implies the existence of extra information, W, which is neither part of the black hole nor the radiation, but plays the role of a reference. The entropy obtained via the replica trick can then be identified to be the entropy S(R|W) of the radiation conditioned on the reference W, whereas Hawking's original result corresponds to the non-conditional entropy S(R). The entropy S(R|W), which mathematically is an ensemble average, gains an operational meaning in an experiment with N independently prepared black holes: For large N, it equals the normalised entropy of their joint radiation, S(R1⋯RN)/N. The discrepancy between this entropy and S(R) implies that the black holes are correlated. The replica wormholes may thus be interpreted as the geometrical representation of this correlation. Our results also suggest a many-black-hole extension of the widely used random unitary model, which we support with non-trivial checks.top

### Chris Wüthrich: Time travelling in emergent spacetime

Most approaches to quantum gravity suggest that relativistic spacetime is not fundamental, but instead emerges from some non-spatiotemporal structure. This talk will investigate the implications of this suggestion for the possibility of time travel in the sense of the existence of closed timelike curves in some relativistic spacetimes. In short, will quantum gravity reverse or strengthen general relativity’s verdict that time travel is possible?top

## Organizers

For information about practical matters and registration, please contact one of the organisers.

## Acknowledgement

Funded in part by the Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 312032894, CU 338/1-2. Also supported by the Lichtenberg Group for History and Philosophy of Physics at the University of Bonn, under the direction of Prof. Dr. Dennis Lehmkuhl.