Anti de Sitter space happens to be unstable

Circle Limit III by M.C. Escher, a pictorical bidimensional representation of an AdS-like hyperbolic space.| Credit: Wikimedia Commons

Quantum field theory has been the paradigm of modern physics since it was formulated in the first half of the last century. It is based on the idea that everything in physics corresponds to the interaction of some fundamental fields that permeate all space and the excitations of which represent the particles we see in our experiments and we are made of. However, as everybody that have read at least a couple of popular science books knows, there is a fundamental force that cannot be quantized and therefore treated on a par with the rest of interactions in a well-formulated quantum field theory. That force is gravity and we describe it instead by using the Theory of General Relativity that Albert Einstein formulated around 1915, which states that both space and time behave as a fabric that can curve under the influence of energy and mass, being this curvature, which pushes little massive bodies towards very massive ones, the responsible for the attractive force we call gravity. Therefore, quantum field theory only studies the rest of physics, leaving gravity aside and thus working only in flat space (the one which is not curved by the influence of gravity), which is a very good approximation for the surface of the Earth and its surroundings.

However, gravity can be actually added to a quantum field theory in a partial way if we introduce it differently from the rest of fields. Since it cannot be quantized, it cannot be a dynamical field of the theory (dynamics will lead to the need of a quantum description of the force in some moment) and it is introduced instead as a background, id est, General Relativity just sets, by saying how the spacetime is curved, the scenario in which the rest of fields will interact. Obviously, this is not an easy thing to do and part of the difficulty comes from the question of which ones of the large amount of classical solutions of Einstein equations can be used as backgrounds. Are all of them allowed? If not, which is the criteria that determines whether they are or they are not? The answer to this question is that only stable spacetimes are allowed to be backgrounds to define a field theory over them. But this leads to another question, namely what a stable spacetime is.

The simplest answer to this question comes from considering little perturbations of the gravity field around the spacetime of interest. If we study the dynamics of such perturbations and they happen to grow and produce such a curvature that is comparable to the background one, then we conclude that the spacetime is unstable, since the adding of any field or any matter would perturb it enough to be destroyed. On the other hand, if we can find perturbations that are little enough to never destroy the background geometry, the spacetime is said to be stable. This is the method that O. Dias et al have explored recently1 in order to check if a spacetime of special interest in modern physics, called Anti de Sitter (AdS), is stable or not. Even if we had known such spacetime since the very first decades of the last century, to date we only had some conjectures and not a real proof about wether it was stable or not. The result of Dias et al, relays in a very special fact of AdS, namely that it has a well defined boundary. In other words, AdS is a box. To understand why this is so relevant, it is interesting to compare with the case of flat space.

Image of a two dimensional AdS space embebbed into a three dimensional space.| Credit: Wikimedia Commons

When we study perturbations of gravity in a flat space, we can think of them as gravitational waves that propagate from different sources and at some point they collide with each other. When this happens, if the waves were very big, they would produce a black hole and therefore change the background geometry from the previous flat one to the one that surrounds the hole. However, since we are free to choose perturbations as little as we want, we can always select them not to form the black hole but to cross each other without interacting and continue propagating infinitely. What Dias and his fellows computed is that when we tried to reproduce this scheme in the interior of AdS, things are radically different because of the boundaries of the spacetime.

In this case, we can again think of the perturbations as gravitational waves that collide and we are still able to choose them little enough not to form a black hole when crossing each other but, contrary to the flat space example, and since AdS has boundaries, they do not travel infinitely but instead they are reflected from the boundaries, allowing them to cross again and again after every reflection. This fact together with the result of the equations of motion for these perturbations, which say that their intensity grows with time while they propagate, do the trick. Even if we choose them to be little enough not to form the black hole in their first crossing, they could do it in their second or third one and, eventually, they will always do it, destroying the background geometry.

The relevance of this results comes from the fact that this instability only shows when the perturbations are chosen to be non-linear, a thing that physicists usually do not do when studying perturbations for good reasons (mostly because computing outside of the linear regime is a really pain in the neck and sometimes even impossible). This is, maybe, the reason why this instability has been hidden till now.

However, there are still things to be studied after this very first result. As the authors themselves say, their result is strictly classic so it still has to be reviewed in the light of quantum theory. Their argument is that in the quantum regime, the black hole could not form, but things are still not clear. Also, another con of this computation is that it is done in 2+1 dimensions instead of the 3+1 ones of our (observable) universe. However, Dias et al state in their conclusions that they have found similar results in that scenario, but a real proof is not published yet.

Even with those little inconveniences, the result achieved by O. Dias, G. Horowitz and J. Santos is extremely interesting and I am pretty sure that it will produce a lot of responses from the scientific community, especially in the light of AdS/CFT duality, one of the hottest topics of our days, which relates gravity on AdS spaces to field theories in a very interesting way. But this is another story and shall be told another time.

More info:

Slides of the talk by O. Dias in the XVIII Xmas Workshop of the IFT UAM-CSIC (Madrid)


  1. “Gravitational turbulent instability of anti-de Sitter space”, Óscar J. C. Dias et al, Class. Quantum Grav. 29, 194002 (2012)

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