How the Tiger got its Stripes

By Nicolas Yunes, Stephon Alexander, and Kent Yagi

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Nature is sometimes lazy and messy, like a 4 year-old that likes to play with all of the toys and put none of them away, always increasing the degree of disorder. In physics, we quantify disorder through the concept of entropy. The tendency of systems to always increase their entropy is encoded in the second law of thermodynamics. Aside from a kid’s disorganised bedroom, another place in the universe with a tremendous amount of entropy is a black hole. Bekenstein and Hawking [1-4] proved that the entropy of a black hole in General Relativity is proportional to its area. For astrophysical black holes of stellar-mass, this yields a value of entropy of about 10⁵⁶ Joules per Kelvin. The reason for this large value is their incredibly small temperature, about 10^-9 Kelvin for a stellar-mass black hole, which in fact cannot be any smaller for an object of its size due to Heisenberg’s uncertainty principle.

Yunes is an Associate Professor of Physics at Montana State University and co-founder of the eXtreme Gravity Institute.

Alexander is a Professor of Physics at Brown University.

Yagi is an Assistant Professor of Physics at the University of Virginia.

The entropy area scaling is surprising because the entropy of extensive systems in statistical mechanics is proportional to their volume, not their area. In fact, Bekenstein’s and Hawking’s observation was a strong motivator for the concept of holography. The typical rationalisation of the area result is that black holes are special objects because, by definition, they possess a horizon. The argument is then that the entropy is proportional to the area because somehow the internal degrees of freedom of a black hole are imprinted on the surface area associated with its horizon. The consequence of this argument is that the only objects in nature whose entropy scales with their surface area are those with event horizons. In particular, the entropy of stars, be them compact or not, should scale with their volume and not their area.

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

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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.

Space Is the Place*

How to master spatial average properties of the Universe?

by Thomas Buchert, Pierre Mourier & Xavier Roy.

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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.

Constructing AdS-like spacetimes

By Diego A. Carranza and Juan A. Valiente Kroon

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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.

Diego A. Carranza is a PhD student at Queen Mary University of London, working under the supervision of Juan A. Valiente Kroon on conformal analysis of AdS-like spacetimes.

Juan A. Valiente Kroon is Reader in Applied Mathematics at Queen Mary University of London and author of a monograph on the use of conformal methods in General Relativity.

Building modified theories of gravity from models of quantum spacetime

Hints from non-commutative geometry

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By Marco de Cesare, Mairi Sakellariadou, and Patrizia Vitale 

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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.

Mairi Sakellariadou

Marco de Cesare

Patrizia Vitale

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

By Rodolfo Gambini and Jorge Pullin

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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.

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 

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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.

Gravitational waves measure colour of black holes

By Enrico Barausse, Richard Pires Brito, Vitor Cardoso, Irina Dvorkin, and Paolo Pani

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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.

Enrico Barausse

Richard Pires Brito

Professor Vitor Cardoso

Irina Dvorkin

Paolo Pani

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].

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References:
[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

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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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Understanding exact space-times

By Jiří Podolský 

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General relativity is a unique gem, Einstein’s most brilliant idea, and his greatest gift to humankind. Conceived in 1915, it still remains the best theory of gravity. I’m sure Einstein himself would be surprised how remarkably well it describes reality, even in the most violent and dynamical situations. Just recall its recent spectacular vindication by the first direct detection of gravitational waves from binary black hole mergers at cosmological distances. What an achievement! Gravitational waves, black holes, cosmology – all three main ingredients and predictions of Einstein’s theory combined together.

Exact space-times

As we all know, Einstein’s equations determine the space-time geometry, which is the gravitational field. And we must take all their predictions seriously. Exact solutions to Einstein’s field equations include the mathematical truth about the physical reality. Unfortunately, it is often obscured, usually very deeply hidden. To dig out the physically measurable invariant quantities and consequences, is a painful mining process involving various techniques and methods. It is the real art of science.

It is essential to be well-equipped for the investigation of exact space-times. Nevertheless, here we are preparing to descend old silver mines in Kutná Hora, the source of great wealth of the Kingdom of Bohemia in the Middle Ages. (Jerry Griffiths and Jiří Podolský, April 2006)

If space-time has defects, how could we find out?

By Sabine Hossenfelder

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Whether space and time come in discrete chunks is one of the central questions of quantum gravity, the still missing unification of quantum theory with gravity. Discretization is a powerful method to tame infinities exactly like the ones that appear when we try to quantize gravity. It is thus not surprising that many approaches to quantum gravity rely on some discrete structure, may that be condensed matter analogies, triangulations, or approaches based on networks.

Sabine Hossenfelder

Many researchers in the field hope that besides taming the infinities that appear in the quantization of gravity, discretization will also prevent the formation of singularities that general relativity predicts, for example at the big bang and inside black holes.  If space-time was fundamentally made of finite-sized chunks, then the singularities would merely be mathematical artefacts, just like singularities in hydrodynamics are merely mathematical artefacts of using the fluid-approximation on distances when we should instead use atomic physics.