Quantum imprints of a black hole’s shape

Can quantum fields tell us about the curvature of a black hole event horizon?

By Tom Morley,  Peter Taylor, Elizabeth Winstanley

The event horizon of a black hole completely surrounds a singularity. It seems obvious that the event horizon takes the form of a (possibly distorted) sphere, a surface with positive curvature. If the space-time far from the black hole is flat, this must be the case. Suppose instead that the space-time in which the black hole is situated itself has negative curvature (this is known as anti-de Sitter space-time and arises in string theory). Then the event horizon does not have to have positive curvature; it can have zero or negative curvature.

How do these different horizon shapes affect black hole physics? If we look at our reflection in a flat mirror, it is undistorted, but a mirror with positive or negative curvature distorts our reflection, as might be experienced in a “hall of mirrors” at a fairground (see image below for some similar effects).


Tom Morley is a PhD student at the University of Sheffield.


Peter Taylor is Assistant Professor of Mathematical Sciences at the Centre for Astrophysics and Relativity, Dublin City University.


Elizabeth Winstanley is Professor of Mathematical Physics at the University of Sheffield.

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Space Is the Place*

How to master spatial average properties of the Universe?

by Thomas Buchert, Pierre Mourier & Xavier Roy.

The question of how to define a cosmological model within General Relativity without symmetry assumptions or approximations can be approached by spatially averaging the scalar parts of Einstein’s equations. This yields general balance equations for average properties of the Universe.  One open issue that we address here is whether the form and solutions of these equations depend on the way we split spacetime into spatial sections and a global cosmological time. We also discuss whether we can at all achieve this – given the generality of possible spacetime splits.

Our CQG Letter explores the general setting with a surprisingly simple answer.

Currently most researchers in cosmology build model universes with a simplifying principle that is almost as old as General Relativity itself.  One selects solutions that are isotropic about every point, so that no properties of the model universe depend on direction. This local assumption restricts one to homogeneous geometries that define the cosmological model globally, up to the topology that is specified by initial conditions. Spacetime is foliated into hypersurfaces of constant spatial curvature, labelled by a global cosmological time-parameter. The homogeneous fluid content of these model universes is assumed to define a congruence of fundamental observers moving in time along the normal to these hypersurfaces. Einstein’s equations reduce, in this flow-orthogonal foliation, to the equations of Friedmann and Lemaître. The only gravitational degree of freedom is encoded in a time-dependent scale factor, which measures the expansion of space. Continue reading

Constructing AdS-like spacetimes

By Diego A. Carranza and Juan A. Valiente Kroon

Maldacena’s AdS-CFT correspondence has brought the study of properties of anti de Sitter-like spacetimes (AdS spacetimes for short) to the centre of attention of a wide community of researchers. This class of spacetimes is characterised by a time-like conformal boundary similar to that of the anti-de Sitter spacetime. Maldacena’s correspondence relates AdS spacetimes to dual conformal field theories defined on the boundary of the spacetime. In particular, it allows to obtain information otherwise not easily accessible about the conformal field theories through the numerical computation of the dual spacetime. Thus, numerical simulations of these spacetimes have received a substantial amount of attention in recent years.  The existence of the time-like conformal boundary in these spacetimes also has implications of interest to mathematicians studying general properties of solutions to the Einstein equations. AdS spacetimes are examples of non-globally hyperbolic solutions to the Einstein field equations. Accordingly, if one wants to formulate a well-posed initial value problem for an AdS spacetime, in addition to the initial data, it is necessary to provide some information on the boundary. The prescription of boundary data is linked to the question of stability of this kind of solutions to the Einstein equations as, during the last years, numerical evidence has showed that under certain boundary conditions the anti-de Sitter spacetime is unstable under non-linear perturbations.

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A Successful Eccentric

by Blake Moore

Humankind has been obsessed with circles for a long time. It comes as no surprise then that the modeling of gravitational waves had focused until recently on those emitted by black holes or neutron stars in circular orbits around each other. But in the case of gravitational wave modeling, there is good reason for this obsession. Gravitational waves remove energy and angular momentum from a binary, forcing the eccentricity to decay and the orbit to circularize rapidly. Since the 1960s, the expectation has then been that the gravitational waves that ground-based detectors would observe would correspond to circular binaries.

Travis, Blake and David at Yellowstone

Travis Robson (right), Blake Moore (center), and David Anderson (left) are members of the eXtreme Gravity Institute at Montana State University. Here they are at nearby Yellowstone National Park.

But as with most things in physics, Nature adores the complex if one looks closely enough. Several astrophysical studies have recently shown that binaries may form with moderate eccentricities at orbital separations at which they would be emitting gravitational waves that ground-based detectors could observe very soon. These binaries would form near a supermassive black hole or in globular clusters, where three- or many-body interactions may source eccentricity. And if they are detected, they could shed light on their true population and formation scenario.

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Building modified theories of gravity from models of quantum spacetime

Hints from non-commutative geometry

By Marco de Cesare, Mairi Sakellariadou, and Patrizia Vitale 

It is often argued that modifications of general relativity can potentially explain the properties of the gravitational field on large scales without the need to postulate a (so far unobserved) dark sector. However, the theory space seems to be virtually unconstrained. One may then legitimately ask whether there is any guiding principle —such as symmetry— that can be invoked to build such a modified gravity theory and ground it in fundamental physics. We also know that the classical picture of spacetime as a Riemannian manifold must be abandoned at the Planck scale. The question then arises as to what kind of geometric structures may replace it, and if there are any novel gravitational degrees of freedom that they bring along. Importantly, one may ask whether there are any potentially observable effects away from the experimentally inaccessible Planck regime. These questions are crucial both from the point of view of quantum gravity and for model building in cosmology; trying to answer them will help us in the attempt to bridge the gap between the two fields, and could have far-reaching implications for our understanding of the quantum structure of spacetime.

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Gravitation in terms of observables: breathing new life into a bold proposal of Mandelstam

By Rodolfo Gambini and Jorge Pullin


Rodolfo Gambini and Jorge Pullin have been collaborating for 27 years

In the 1960’s Stanley Mandelstam set out to reformulate gravity and gauge theories in terms of observable quantities. The quantities he chose are curves, but specified intrinsically. The simplest way of understanding what does “specified intrinsically” means is to think how the trajectory of a car is specified by a GPS unit. The unit will give commands “turn right”, “advance a certain amount”, “turn left”. In this context “right” and “left” are not with respect to an external coordinate system, but with respect to your car. The list of commands would remain the same whatever external coordinate system one chooses (in the case of a car it could be a road marked in kilometres or miles, for instance). The resulting theories are therefore automatically invariant under coordinate transformations (invariant under diffeomorphisms). They can therefore constitute a point of departure for the quantization of gravity radically different from other ones. For instance, they would share in common with loop quantum gravity that both are loop-based approaches. However, in loop quantum gravity one has to implement the symmetry of the theory under diffeomorphisms. Intrinsically defined loops, on the other hand, are space-time diffeomorphism invariant, therefore such a symmetry is already implemented. It is well known that in loop quantum gravity diffeomorphism invariance is key in selecting in almost unique way the inner product of the theory and therefore on determining the theory’s Hilbert space. Intrinsically defined loops are likely to be endowed with a very different inner product and Hilbert space structure. In fact, since the loops in the Mandelstam approach are space-time ones it lends itself naturally to an algebraic space-time covariant form of quantization. Continue reading

What are the fundamental gauge symmetries of the gravitational field?

Uncovering the gauge symmetries of general relativity via Noether’s theorem.

By Merced Montesinos, Diego Gonzalez, and Mariano Celada 

Symmetries are the cornerstone of modern physics. They are present in almost all its subfields and have become the language in which the underlying laws of the universe are expressed. Indeed, in the standard model of particle physics, our best understanding of nature down to the subatomic world, the interactions among fundamental particles are dictated by internal gauge symmetries.

Although the four fundamental interactions can be fitted within the framework of gauge theories, gravity still remains as the weird family member. While gravity can be conceived as a gauge theory on its own, it seems to be one that differs from those describing the non-gravitational interactions. Indeed, the latter are embedded within the so-called Yang-Mills theories, but gravity is something else.


Merced Montesinos (centre) is a theoretical physicist at Departamento de Física, Cinvestav, Mexico.
Diego Gonzalez (left) is a postdoctoral researcher at Instituto de Ciencias Nucleares, UNAM, Mexico.
Mariano Celada (right) is a postdoctoral researcher at Departamento de Física, UAM-I, Mexico.

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Gravitational waves measure colour of black holes

By Enrico BarausseRichard Pires BritoVitor Cardoso, Irina Dvorkin, and Paolo Pani

Black holes predicted by Einstein are, well, black. In classical physics, nothing can escape their event horizon, not light, not matter, and neither gravitational waves.

There is a deep reason for this absolute blackness. If the event horizon were not a perfectly absorbing surface, but rather a partially reflective one, spinning black holes would become unstable and would shed most of their rotational energy into gravitational waves. This process is known as superradiant instability [1], and is tightly linked to the presence of an ergoregion, a region of spacetime just outside the event horizon, where modes of negative energy are allowed to exist. Negative energy-modes can form in the ergoregion of normal humdrum black holes, but are eventually doomed to fall in the event horizon.

If (what look like) black objects had a surface, such modes would be partially reflected by it, and they would bounce back and forth between the horizon and the boundary of the ergoregion (which they cannot cross, since negative energy modes cannot travel to infinity). Each time they reach the ergoregion boundary, they come out as positive energy-modes, thus inside the ergoregion they would keep growing in amplitude (i.e. their energy would keep decreasing and becoming more negative) eventually producing an instability. Indeed, these ‘bounces’ produce ‘echoes’ in the gravitational wave signal [2] from the remnant black hole forming from binary mergers, and there are claims [3] (albeit controversial [4]) that they may have been seen in the LIGO data.

In this paper we do not look at the black holes that form from binary mergers, but rather at isolated ones. These black holes can have a wide range of masses (from stellar masses for stellar-origin black holes up to millions or billions solar masses for supermassive black holes) and a variety of spins (on which we have some knowledge thanks to electromagnetic observations). Normally, isolated black holes do not emit gravitational waves, but if their event horizon had some reflectivity (that is, if these objects were not totally black), they would turn into black-hole bombs due to superradiance, and they would shed almost all their angular momentum in gravitational waves. These gravitational wave signals would be too weak to be detected singularly, but because there are in general many more black holes in isolation than in binaries, they can produce a very large stochastic background. Indeed, this background would be orders of magnitude larger than the current upper bounds from LIGO/Virgo. Similar results also apply to supermassive black holes, in the yet-unexplored LISA band.

So in conclusion, the existing stochastic background constraints from LIGO and Virgo show that black holes are very black, although some shades of grey may still be allowed. Indeed, while 100% reflection from the horizon is ruled out, smaller reflection coefficients may still be possible depending on the spin of the object [5].

[1] W. H. Press and S. A. Teukolsky, “Floating Orbits, Superradiant Scattering and the Black-hole Bomb“. Nature. 238 (5361): 211-212 (1972);
Brito, Cardoso, Pani; “Superradiance“, Springer (2015)
[2] Cardoso, Franzin, Pani, “Is the gravitational-wave ringdown a probe of the event horizon?“, Phys. Rev. Lett. 116, 171101 (2016)
[3] Abedi & Afshordi, “Echoes from the Abyss: Tentative evidence for Planck-scale structure at black hole horizons“, Phys. Rev. D 96, 082004 (2017)
[4] Ashton+ https://arxiv.org/abs/1612.05625; Abedi, Dykaar, Afshordi, https://arxiv.org/abs/1701.03485 and https://arxiv.org/abs/1803.08565;
Westerweck+, “Low significance of evidence for black hole echoes in gravitational wave data“, Phys. Rev. D 97, 124037 (2018)
[5] Maggio, Pani, Ferrari “Exotic Compact Objects and How to Quench their Ergoregion Instability“, Phys. Rev. D 96, 104047 (2017); Maggio, Cardoso, Dolan, Pani, http://arxiv.org/abs/arXiv:1807.08840

Read the full article in Classical and Quantum Gravity:
The stochastic gravitational-wave background in the absence of horizons
Enrico Barausse et al 2018 Class. Quantum Grav. 35 20LT01

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Winner of the 2018 IOP Gravitational Physics Group (GPG) thesis prize: Dr Viraj A A Sanghai

Can you tell us a little bit about the work in your thesis?

One of the challenges of modern cosmology is to interpret observations in a consistent and model-independent way. There are several assumptions in interpreting cosmological/astrophysical data. For example, it is often assumed that Einstein’s theory of gravity is the correct theory of gravity. Furthermore, fundamental to cosmology is the assumption that the universe is homogeneous and isotropic on the largest scales and hence, this is the correct starting point to interpret cosmological data. To test these assumptions, approaches are needed, which work in a model-independent way. Broadly speaking, my thesis addresses these questions.

Thinking back, what was the most interesting thing that happened during your PhD?


Dr Viraj A A Sanghai is a postdoctoral fellow working on theoretical cosmology at Dalhousie University in Halifax, Canada

The most interesting thing that happened during my PhD was the discovery of gravitational waves by LIGO, due to the merging of two black holes. This opened up a new avenue into testing Einstein’s theory of gravity and started the new field of gravitational wave astronomy. Before this, all our astronomical observations relied on electromagnetic radiation. This discovery is helping us to have a deeper understanding of our Universe. Consequently, the Nobel Prize in Physics was awarded for this discovery. Continue reading

Soft hair and you

by Josh Kirklin

Awaken, quantum relativist.

Have breakfast, and notice that a black hole has found its way into your laboratory. You measure its mass M, electric charge Q and angular momentum J, double-check the statement of the no hair theorem, and tell yourself that you can learn no more about this particular black hole.

But the quantum mechanic inside of you objects. As a devout believer in unitarity, you are convinced that the black hole must contain a complete description of the matter involved in its formation. So, you think, the no hair theorem must not apply. Finding no logical inconsistency in the mathematical steps involved in its proof, you decide that something must be wrong with its initial assumptions.

One such assumption is that two black holes related by a gauge transformation are physically equivalent. But this cannot be correct, since if we do a gauge transformation whose action does not vanish sufficiently quickly at infinity, there are observable consequences. The quickest way to become convinced of this fact is to find the Poisson brackets appropriate for a description of gravity and electromagnetism, and to use them to compute the actions generated by M, Q and J. The action of each is a gauge transformation that is non-trivial at infinity, and it is a basic fact of Hamiltonian mechanics that if a quantity is observable (M, Q and J certainly are), then so too must be the action that it generates.


About the author: Josh Kirklin is a PhD student in the Relativity and Gravitation group at the Department of Applied Mathematics and Theoretical Physics of the University of Cambridge. He studies black hole thermodynamics and the role of information in quantum gravity.

You deduce that you have at least three new gauge-dependent observable parameters (one for each of M, Q and J) to describe your black hole. In fact, you have infinitely many more, since the dipole, quadrupole and higher order moments of M, Q and J also generate gauge transformations that are non-trivial at infinity. The no hair theorem guarantees that the higher order moments of M, Q and J themselves must vanish, but it does not claim the same for the resultant new gauge-dependent variables.

So you conclude that your stationary black hole is actually described by infinitely many degrees of freedom. These are known as soft hairs, and the quantities which generate their transformations are known as soft charges.

During lunch, you manage to devise a clever experiment which makes use of the gravitational and electromagnetic memory effects to measure the soft hairdo of your black hole. Partially out of respect for the event horizon, but mostly out of fear, you decide to stay far enough away from the black hole that you must treat yourself as an idealised observer at infinity in your calculations. So, carrying out your experiment, you obtain a set of numbers describing how the soft hair appears to an observer at infinity.

But the black hole is an isolated body in spacetime, whose properties should be intrinisic to it. What you really desire is a description of the soft hair that is local to the black hole – one that would reflect what a more courageous observer, who was more willing to closely approach the event horizon, might see. You wonder what the best way would be to deduce such a local description from your observations at infinity, and, absent-mindedly leafing through the latest issue of CQG, you stumble upon a paper that takes you part of the way towards the answer. Following its advice, you are able to write the soft charges in terms of fields close to the black hole.

At dinner, you decide that you would like to justify your scientific credentials by predicting the future of the black hole. Before the discovery of soft hair, you would have only been able to treat the black hole as a single thermodynamical system, supporting an entropy and emitting Hawking radiation. This was necessary because you only had a macroscopic description for your black hole, being ignorant of its microscopic physics. But now you feel that you can improve on this, since you have a candidate for the black hole’s microscopic degrees of freedom – its soft hair. To make some predictions, all you need is a theory that governs the dynamics of the soft hair, and that same paper seems to again provide some assistance. It argues that a softly hairy black hole should actually be treated as an infinity of thermodynamic systems, all in thermal contact with each other. This thermal contact manifests as a heat current on the event horizon.

You use these results to make your predictions, and climb into bed after an exhausting day of hands-on relativity. Drifting off to sleep, you wonder whether the discovery of soft hair will be enough to solve one of the biggest mysteries of black hole physics – the information paradox. To have any hope of this being the case, you ought to be able to use the existence of soft hair to derive the black hole area entropy relation. You have heard rumors that this result is close at hand, and that the paper announcing it will bear Stephen Hawking’s name (making it his last published work).

Until then, you can only dream of tomorrow’s meals and measurements…

Read the full article in Classical and Quantum Gravity:                                                     Localisation of soft charges, and thermodynamics of softly hairy black holes
Josh Kirklin,
2018 Class. Quantum Grav. 35 175010

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CQG papers are selected for promotion based on the content of the referee reports. The papers you read about on CQG+ have been rated ‘high quality’ by your peers.

This work is licensed under a Creative Commons Attribution 3.0 Unported License.