Boundaries, Corners and Creases

 by Joseph Samuel.

Cricketing nations have a very good idea what a boundary is, it’s good for a cool four runs, without the bother of running! Corners are tense moments in a football (soccer to some) match when a well struck ball can curve into the goal. The crease is what a batsman lunges for when the wicket keeper ….. wait! this is not a sports column, but CQG+! Let’s back up and explain what our paper really is about.

In a path integral approach to quantum gravity, one has to divide up spacetime into pieces and focus on the action within each piece. In the elementary case of particle mechanics, this “skeletonisation” converts the action expressed as a Riemann integral into a discrete sum. A desirable property of the action is that it should be additive when we glue the pieces back together. This is achieved only when one properly takes into account the boundaries of the pieces.  The boundaries can be spacelike, timelike or null. Much work has focused on the first two cases. The Einstein–Hilbert Action principle for spacetime regions with null boundaries has only recently attracted attention (look up the Arxiv for papers by E. Poisson et al and Parattu et al; references would not be consistent with the chatty, informal style of  CQG+). These papers deal with the appropriate boundary terms that appear in all boundary signatures.

Our paper (authors: Ian Jubb, Joseph Samuel, Rafael Sorkin and Sumati Surya) gives a unified approach to all boundary signatures using Cartan’s tetrad formalism. An unexpected feature of the boundary term required here is that it is not gauge invariant under local Lorentz transformations (although its variation is). As the tetrad formalism may not be familiar to some readers of CQG, we also give a treatment in terms of metrics. When the boundary has corners the action has to also contain corner terms. Cartan’s tetrad formalism gives a simple way to arrive at the corner terms, exploiting the gauge non invariance of the boundary terms.

Spacetime boundaries can be null. A classic example is the region exterior to a black hole, whose boundary is a frozen wavefront, the event horizon. Horizons can have creases where the null normal is discontinuous, as happens when new null generators enter (or leave) the horizon. Another example of a null boundary is that which appears in a Causal diamond. A causal diamond is the intersection of a past set with a future set and looks (when it is drawn on a blackboard) much like the diamond in a baseball game. Yet another example of a null boundary is Scri, asymptotic null infinity. Null boundaries deserve special attention since their normals are also their tangents. Our unified treatment paints all kinds of boundaries with the same brush.

A photograph taken on the terrace of the library building during the workshop around December 2015.

This work, involving a collaboration across continents had its genesis in a series of workshops organised at the Raman Research Institute in the last few years by Sumati Surya. Come December, when the skies are grey in the northern latitudes, some of our colleagues, like migrating birds, wing their way south, to the Raman Research Institue (RRI) in Bangalore, India. The photo above shows some of them with friends and families. Visible in the photograph are the four authors of our paper. The photograph is taken on the terrace of the library building, where the talks took place. The talks and discussions revolve around general relativity (mostly from the Causet point of view championed by Sorkin), Quantum Measure theory, entanglement entropy, the cosmological constant and topology change. Sometime in December 2014, the discussions around the meeting raised the question of null boundaries. This question was partly answered and then revived at the subsequent meetings and culminated in the paper. Do take a look at it.

A few words about the RRI campus may be in order here: it is wooded and distinctly cooler than the surrounding areas because of the foliage. The trees are home to a variety of bird, animal and insect life. Common birds are sunbirds, bulbuls, koels and barbets, whose calls you can listen to as a welcome distraction from work. There is a dwindling population of slender lorises and a thriving population of lazy cats. During November / December / January the skies are clear blue (though dotted with soaring kites) and the wooded campus attracts a seasonal feathered visitor, the paradise flycatcher. Each year these birds stop at the Institute campus for about two weeks before going further south to their destination in the Nilgiri hills. What attracts the birds here is probably the insect life, which is also pretty diverse (Arachnophobes are advised to desist from clicking on this link.).

The Raman Institute has groups doing research in four select areas of physics: Astrophysics, Theoretical Physics, Soft Condensed Matter and Light and Matter Physics. There is also research in chemistry and a substantial thrust in instrumentation related to Astronomical Observations at telescopes in India and around the world. For more information on this look at Facebook or Twitter. The theoretical physics group has interest mainly in General Relativity and Non equilibrium Statistical Physics. Apart from the permanent faculty at RRI, we have postdocs, PhD students and a vibrant Visiting Student program at the Bachelor’s and Master’s level. We also have an outreach programme to interface with schools and colleges. Check out our homepage for more details.

About the Author: I am a theoretical physicist at the Raman Research Institute. My interests include general relativity, optics, the geometric phase in quantum mechanics, DNA elasticity and science popularisation. I keep moderately fit by raising and lowering indices. I enjoy gardening and relax by cooking exquisitely textured lacy appams for my friends.

Read the full article in Classical and Quantum Gravity:
Boundary and corner terms in the action for general relativity
Ian Jubb et al 2017 Class. Quantum Grav. 34 065006

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Going NUTs

By Paul I. Jefremov and Volker Perlick.

Among all known solutions to Einstein’s vacuum field equation the (Taub-)NUT metric is a particularly intriguing one. It is that metric that owing to its counter-intuitive features was once called by Charles Misner “a counter-example to almost anything”. In what follows we give a brief introduction to the NUT black holes, discuss what makes them interesting for a researcher and speculate on how they could be detected should they exist in nature.

paul jefremov-and-volker

Volker Perlick and Pavel (Paul) Ionovič Jefremov from the Gravitational Theory group at the University of Bremen in Germany. Volker is a Privatdozent and his research interests are in classical relativity, (standard and non-standard) electrodynamics and Finsler geometry. He is an amateur astronomer and plays the piano with great enthusiasm and poor skills. Paul got his diploma in Physics at the National Research Nuclear University MEPhI in Moscow, 2014. Now he is a PhD Student in the Erasmus Mundus Joint Doctorate IRAP Programme at the University of Bremen. Beyond the scientific topics in physics his interests include philosophy in general, philosophy of science, Eastern and ancient philosophy, religion, political and social theories and last but not the least organic farming.

The NUT (Newman–Unti–Tamburino) metric was obtained by Newman, Unti and Tamburino (hence its name) in 1963. It describes a black hole which, in addition to the mass parameter (gravito-electric charge) known from the Schwarzschild solution, depends on a “gravito-magnetic charge”, also known as NUT parameter. If the NUT metric is analytically extended, on the other side of the horizon it becomes isometric to a vacuum solution of Einstein’s field equations found by Abraham Taub already in 1951. However, for an observer who is prudent enough to stay outside the black hole, the Taub part is irrelevant.

At first sight, the existence of the NUT metric seems to violate the uniqueness (“no-hair”) theorem of black holes according to which a non-spinning uncharged black hole is uniquely characterised by its mass. Actually, there is no contradiction because Continue reading

Black holes without special relativity

By Jishnu Bhattacharyya, Mattia Colombo and Thomas Sotiriou.

Black holes are perhaps the most fascinating predictions of General Relativity (GR). Yet, their very existence (conventionally) hinges on Special Relativity (SR), or more precisely on local Lorentz symmetry. This symmetry is the local manifestation of the causal structure of GR and it dictates that the speed of light is finite and the maximal speed attainable. Accepting also that light gravitates, one can then intuitively arrive at the conclusion that black holes should exist — as John Michell already did in 1783!

One can reverse the argument: does accepting that black holes exist, as astronomical observations and the recent gravitational wave direct detections strongly suggest, imply that Lorentz symmetry is an exact symmetry of nature? In other words, is this ground breaking prediction of GR the ultimate vindication of SR?

Jishnu Bhattacharyya, Mattia Colombo and Thomas Sotiriou from the School of Mathematical Sciences, University of Nottingham.

Jishnu Bhattacharyya, Mattia Colombo and Thomas Sotiriou from the School of Mathematical Sciences, University of Nottingham.

These questions might seem ill-posed if one sees GR simply as a generalisation of SR to non-inertial observers. On the same footing, one might consider questioning Lorentz symmetry as a step backwards altogether. Yet, there is an alternative perspective. GR taught us that our theories should be expressible in a covariant language and that there is a dynamical metric that is responsible for the gravitational interaction. Universality of free fall implies that Continue reading

Issue of the Beginning: Initial Conditions for Cosmological Perturbations

by Abhay Ashtekar and Brajesh Gupt.


Abhay Ashtekar holds the Eberly Chair in Physics and the Director of the Institute for Gravitation and the Cosmos at the Pennsylvania State University. Currently, he is a Visiting Professor at the CNRS Centre de Physique Théorique at Aix-Marseille Université.

Although our universe has an interesting and intricate large-scale structure now, observations show that it was extraordinarily simple at the surface of last scattering. From a theoretical perspective, this simplicity is surprising. Is there a principle to weed out the plethora of initial conditions which would have led to a much more complicated behavior also at early times?

In the late 1970s Penrose proposed such a principle through his Weyl curvature hypothesis (WCH) [1,2]: in spite of the strong curvature singularity, Big Bang is very special in that the Weyl curvature vanishes there. This hypothesis is attractive especially because it is purely geometric and completely general; it is not tied to a specific early universe scenario such as inflation.

However, the WCH is tied to general relativity and its Big Bang where classical physics comes to an abrupt halt. It is generally believed that quantum gravity effects would intervene and resolve the big bang singularity. The question then is Continue reading

Pants on fire!

One of the authors, Ian Jubb, discussing a pair of trousers with his colleagues at Imperial College London. Ian Jubb is currently the PhD student of Fay Dowker at Imperial College London.

One of the authors, Ian Jubb, discussing a pair of trousers with his colleagues at Imperial College London. Ian Jubb is currently the PhD student of Fay Dowker in the Theoretical Physics group at Imperial College London.

by Ian Jubb and Michel Buck.

Did you know that Quantum Gravity literally sets pants on fire?

Your pants are not just a nifty garment, they are also a perfect example of a space undergoing a process known as topology change. Take a space that initially consists of two separate circles. If they were to meet and merge into a single circle, the topology of the space would have changed. The trousers allow us to visualise each stage of this process, with cross sections higher up the trouser leg corresponding to later times in the process (if we hold the trousers upside-down, we get the reverse process, corresponding to a single circle splitting into two circles). Instead of viewing this process as the space changing in time, Einstein would tell us to view the trousers in their entirety, as one whole spacetime — the trousers spacetime.

But why should we care about spaces that can ‘split’ and ’attach’ like this? It turns out that there are good reasons to believe that Continue reading

Pulsed Gravitational Waves


Timothy J. Walton occupies some quantum state between a physicist and a mathematician, having obtained his PhD from the physics department at Lancaster University in 2008 but now masquerading as a lecturer in mathematics at the University of Bolton.

by Timothy J. Walton.

Applying techniques from classical electrodynamics to generate new gravitational wave perturbations

I must begin with a confession: I don’t view myself as a gravitational physicist. Despite my PhD at Lancaster University involving a formulation of relativistic elasticity and an awful lot of differential geometry, my research thus far has been within the realm of classical and quantum electrodynamics. But it was precisely within that domain, along one particular avenue of investigation, where the first seeds of an idea were sown. Following my earlier work on a class of exact finite energy, spatially compact solutions to the vacuum source-free Maxwell equations – pulsed electromagnetic waves – describing single cycle pulses of laser light [1], together with Shin Goto at Kyoto University in Japan and my former PhD supervisor Robin Tucker at Lancaster University, a new question arose: “do pulsed gravitational waves exist?’’

As I recall, this question was posed and began to take root during one of the regular meetings I have with Robin. Within my institution, I am fortunate enough Continue reading

Want to crush a singularity? First make it strong and then …

by Parampreet Singh.

Parampreet Singh

Parampreet Singh with a young student who often asks him the most difficult and so far unanswerable questions on the resolution of singularities. Dr Parampreet Singh is Associate Professor at Department of Physics and Astronomy at Louisiana State University.

Einstein’s theory of classical general relativity breaks down when spacetime curvature
becomes extremely large near the singularities. To answer the fundamental questions
about the origin of our Universe or what happens at the central singularity of the black holes thus lies beyond the validity of Einstein’s theory. Our research deals with discovering the framework which guarantees resolution of singularities.

It has been long expected that quantum gravitational effects tame the classical singularities leading to insights on the above questions. A final theory of quantum gravity is not yet there but the underlying techniques can be used to understand whether quantum gravitational effects resolve cosmological and black hole singularities. Our goal is Continue reading

Propagation in the absence of classical spacetime

Written by Madhavan Varadarajan


The author’s research group busy at work. Madhavan Varadarajan is a Professor at the Raman Research Institute in Bangalore, India.

At the Planck scale of 10−33cm, where the very notion of classical spacetime ceases to exist due to large quantum fluctuations of spacetime geometry, can meaning be given to the notion of “causality”? We are interested in this question in the context of Loop Quantum Gravity (LQG).

The basic quantum states of LQG are labelled by graphs. Each such state describes discrete one dimensional excitations of spatial geometry along the edges of its graph label. These ‘graphical’ states provide the Continue reading

CQG+ Insight: More Classical Charges for Black Holes

Written by Geoffrey Compère, a Research Associate at the Université Libre de Bruxelles. He has contributed to the theory of asymptotic symmetries, the techniques of solution generation in supergravity, the Kerr/CFT correspondence and is generally interested in gravity, black hole physics and string theory.

Why mass and angular momentum might not be enough to characterize a stationary black hole

Geoffrey Compère

Geoffrey Compère having family time in the park Le
Cinquantenaire in Brussels.

“Black hole have no hair.” This famous quote originates from John Wheeler in the sixties. In other words, a stationary black hole in general relativity is only characterized by its mass and angular momentum. This is because multipole moments of the gravitational field are sources for gravitational waves which radiate the multipoles away and only the last two conserved quantities, mass and angular momentum, remain. That’s the standard story.

Now, besides gravitational waves, general relativity contains another physical phenomenon which does not exist in Newtonian theory: the memory effect. It was discovered beyond the Iron Curtain by Zeldovich and Polnarev in the seventies and rediscovered in the western world and further extended by Christodoulou in the nineties. While gravitational waves lead to spacetime oscillations, the memory effect leads to a finite permanent displacement of test observers in spacetime. The effect exists for any value of the cosmological constant but in asymptotically flat spacetimes, it can be understood in terms of an asymptotic diffeomorphism known as a BMS supertranslation.

In order to understand that, let’s go back to the sixties where the radiative properties of sources were explored in general relativity; it was found by Bondi, van der Burg, Mezner and Sachs that there is a fundamental ambiguity in the coordinate frame at null infinity. Most expected that Continue reading

CQG+ Insight: Spacetime near an extreme black hole

Written by James Lucietti, a Lecturer in Mathematical Physics in the School of Mathematics at the University of Edinburgh; and Carmen Li, previously a graduate student in the School of Mathematics at the University of Edinburgh and now a postdoc in the Institute of Theoretical Physics at the University of Warsaw.

How many extreme black holes are there with a given throat geometry?


James Lucietti, University of Edinburgh

The classification of equilibrium black hole states is a major open problem in higher dimensional general relativity. Besides being of intrinsic interest, it has numerous applications in modern approaches to quantum gravity and high energy physics. Two key questions to be answered are: What are the possible topologies and symmetries of a black hole spacetime? What is the ‘moduli’ space of black hole solutions with a given topology and symmetry? For vacuum gravity in four spacetime dimensions, these questions are answered by the celebrated no-hair theorem which reveals a surprisingly simple answer: the Kerr solution is the only possibility. However, since Emparan and Reall’s discovery of the black ring — an asymptotically flat five dimensional black hole with ‘doughnut’ topology — it has become clear that there is a far richer set of black hole solutions to the higher dimensional Einstein equations.


Carmen Li, University of Warsaw, at the top of Ben Nevis in the UK.

Over the last decade, a number of general results have been derived which Continue reading