Understanding exact space-times

By Jiří Podolský 

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)

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

By Sabine Hossenfelder

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. Continue reading

Some features of the Cosmos in Loop Quantum Gravity

By Parampreet Singh, Louisiana State University, USA


CQG Editorial Board member, Param Singh, Guest Edited the Applications of loop quantum gravity to cosmology focus issue in 2016 and 2017

A successful union of Einstein’s general relativity and quantum theory is one of the most fundamental problems of theoretical physics. Though a final theory of quantum gravity is not yet available, its lessons and techniques can already be used to understand  quantization of various spacetimes. Of these, cosmological spacetimes are of special interest. They provide a simpler yet a non-trivial and a highly rich setting to explore detailed implications of quantum gravitational theories. Various conceptual and technical difficulties encountered in understanding quantum dynamics of spacetime in  quantum gravity can be bypassed in such a setting. Further, valuable lessons can be learned for the quantization of more general spacetimes.

In the last decade, progress in loop quantum gravity has provided avenues which allow us to reliably answer various interesting questions about the quantum dynamics of spacetime in the cosmological setting. Quantum gravitational dynamics of cosmological spacetimes obtained using techniques of  loop quantum gravity leads to a novel picture where singularities of Einstein’s theory of general relativity are overcome and a new window opens to test loop quantum gravity effects through astronomical observations.

The scope of the Focus Issue: Applications of loop quantum gravity to cosmology, published last year in CQG, is to provide a snapshot of some of the rigorous and novel results on this research frontier in the cosmological setting.

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How to reach infinity?

Bypassing stability conditions and curing logarithmic singularities

By Jörg Frauendiener and Jörg Hennig

Assume you want to model a general relativistic spacetime. Due to the annoying limitations of conventional computers, like finite memory and processing speed, it is tempting to focus on a finite portion of the spacetime. Then, without waiting endlessly, one can obtain an approximate description of this portion. One just has to choose a suitable numerical method and solve the field equations for the metric at some set of grid-points. While this approach is standard, it introduces unpleasant problems. Firstly, the set of equations needs to be complemented with boundary conditions at the outer edges of this finite portion, in order to obtain a complete mathematical problem. This, however, is quite unphysical as usually no information about the actual behaviour at such an artificial boundary is available. Consequently, spurious gravitational radiation enters the numerical domain. Secondly, if one is interested in accurately describing gravitational waves, one should recall that these are only well-defined at infinity. Hence it is desirable to extend the simulation up to infinity.


Jörg Frauendiener and Jörg Hennig trapped at infinity.

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Constructing Spacetimes

By Steven Carlip and Samuel Loomis

Imagine you are given a bucket of points and asked to assemble them into a spacetime. What kind of “glue” would you need?

In causal set theory, the only added ingredient is the set of causal relations, the knowledge of which points are to the past and future of which. In particular, suppose your points were taken at random from a real spacetime, at some typical length scale ℓ. Then on scales large compared to ℓ, the causal diamonds – the sets formed by intersecting the past of one point with the future of another – determine the topology; the causal relations determine the metric up to a scale factor; and the remaining scale factor is just a local volume, which can be obtained by counting points. As the slogan of Rafael Sorkin, the founder of the field, goes, “Order + Number = Geometry.”

Carlip photo

Samuel Loomis and Steven Carlip with their causal set.

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de Sitter meets Planck

by Adriana V. Araujo, Diego F. López and José G. Pereira

The Quest for Consistency in Spacetime Kinematics

Newton’s inception of the theory for the gravitational interaction in 1687 was a landmark for modern physics. In addition to explaining all known gravitational phenomena of that time, Newton’s gravitational theory was consistent with the kinematic rules of the Galilei group, known as Galilei relativity. Such consistency provided an atmosphere of intellectual comfort, which lasted for more than two centuries.


From left to right, José, Adriana and Diego. Click here to see the authors taking advantage of all dimensions of a space section of the universe.

By the mid nineteenth century, most secrets of the electric and  magnetic fields were already unveiled. Those advancements culminated with the publication by Maxwell of a comprehensive treatise on the unification of electricity and magnetism, which became known as Maxwell’s theory. This theory brought to the scene the first inconsistency of our tale. In fact, it became immediately clear that the electromagnetic theory was inconsistent with the Galilei relativity: electromagnetism was claiming for a new relativity. In response to this claim, and with contributions from Lorentz and Poincaré, Einstein published in 1905 the basics of what is know today as Einstein special relativity. According to this theory, for velocities near the velocity of light, spacetime kinematics would no longer be ruled by Galilei, but by the Poincaré group. Most importantly, electromagnetism was consistent with Einstein special relativity! Mission accomplished? Not quite! Continue reading

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.

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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