### Some Negativity About Negative Tension

Well, I'm going to take a break from going on about science under attack and return to some interesting physics, for today at least.

Yesterday we had an interesting seminar by Paul Smyth, who is a graduate student at Imperial College, London, studying with Kelly Stelle. Paul’s talk was titled “The Stability of Harava-Witten Space-times”. He was concerned with the implications of the negative tension branes required by some extra-dimensional models, both those arising from string theory and those appearing in some more phenomenological models. In Paul’s seminar, he discussed a detailed analysis of the energy of such braneworld space-times in 5 dimensions, and demonstrated that they are stable, using a non-perturbative positive energy theorem.

This, and related questions have interested me for a while, and Don Marolf and I wrote a paper that I really liked about a related physical setup. I thought it might be fun to recount the general ideas here.

An interesting possibility, that has seen new life in the last seven years or so, is that our three familiar spatial dimensions may exist as a submanifold of a higher dimensional space-time. Common to most of the modern incarnations of this idea is the condition that all interactions other than gravity be confined to our submanifold, or brane. Since departures from 3+1 dimensional gravity are relatively difficult to constrain compared to those of other forces, this permits significant freedom to modify the gravitational interactions in the extra-dimensional space. It is in this modification of gravity, and of the extra dimensions themselves, that recent approaches differ from one another.

In many of the proposed models, the hierarchy problem of particle physics – how is the large difference between the weak scale (100 GeV) and the Planck scale (10

One specific hurdle that extra-dimensional theories must clear is that the brane-bulk system should be a consistent, stable solution to Einstein gravity. For those constructions that include negative-tension branes, this can pose a particular problem, although perturbative dynamical objections can be overcome by placing the offending brane at a point of special symmetry in the extra-dimensional space, known to aficionados as an “orbifold fixed point”.

Nevertheless, Don and I were interested in the possibility that there might be nonperturbative instabilities of such a space-time. We though there might be because of a simple (and, it turns out, naïve) thermodynamical argument.

This comes from the generalized second law of thermodynamics, which states that the total entropy in matter and black holes does not decrease. The entropy of a black hole is proportional to the area of the event horizon. Therefore, any process that leaves the entropy in matter fixed and decreases the total area of black hole horizons, leads to a violation of the generalized second law.

It is important to know that the area of a black hole event horizon increases when positive energy crosses the horizon, and decreases when negative energy crosses the horizon. Don and I were intrigued by the question: what happens if a black hole, initially far away from a negative tension brane, falls towards the brane and captures some of the brane within its horizon? One might expect that the generalized second law could be violated in this way, since the part of the brane that is swallowed by the black hole carries negative energy across the horizon.

Because the central issue can be muddied by questions involving gravitational radiation (and, frankly, because to go further than this seemed much harder) we considered a lower-dimensional system - a negative tension 1-brane in a 2+1 dimensional space-time – because gravitational radiation does not arise in lower dimensions. What we discovered in analyzing this system was something more dramatic (at least on the face of it) than a violation of the generalized second law.

In 2+1 dimensions, the only black holes are the so-called BTZ black holes, named after their original discoverers Maximo Banados, Claudio Teitelboim and Jorge Zanelli. These black holes arise only in space-times with a negative cosmological constant, the so-called Anti de-Sitter (AdS) space-times. This turned out to be particularly appropriate, because the kinds of brane-world constructions that people have worked on typically require an AdS space-time, for other reasons.

In the main part of our work (this is the bit where the cartoon caption reads "and then a miracle happens"), we made use of a certain amount of cute mathematics, involving the fact that both a BTZ space-time and a negative tension brane space-time can be constructed as quotient spaces of AdS. I certainly won’t go into this here. Rather, I’m going to hope that it suffices to say that this enabled us to construct a general exact solution describing the collision of a black hole and a negative tension brane at one of these points of special symmetry

What we found was catastrophic. The endpoint of this evolution is not, in fact, an equilibrium configuration, but instead is a space-like singularity (similar to the `big crunch' of closed cosmological models with only ordinary matter content). What we discovered, therefore, was a non-linear dynamical instability of gravitating negative tension branes at orbifolds – in effect, the entire space-time outside the black hole collapses when the brane is located at an orbifold fixed plane. Because BTZ black holes can form dynamically from the collision of matter, we also expect non-linear instabilities with negative tension branes in the presence of matter fields, even if no black holes are initially present.

The obvious question to ask is whether the same sort of singularity that we found in 2+1 dimensions also occurs in higher dimensional cases (which are of real interest). For a number of reasons, examples such as this are not sufficient to argue convincingly for a dynamical instability in higher dimensions. That's for future work, and I think I may even have interested Paul in the idea, at least a little.

It's worth mentioning that the ways in which orientifold constructions in string theory obviate this proof is a very interesting story in its own right. If you have the fortitude for it you can read about that in a nice paper by Don Marolf and Simon Ross.

So what is the bottom line of our work? It’s that an orbifold boundary condition may not, by itself, be sufficient to render negative tension branes stable. As a result, if use is to be made of negative tension branes in various models, it is necessary to show that the particular branes being used are immune from these effects. Within string theory this is probably not a problem, but phenomenological models might not be immune.

Well, that was a little more technical than usual, but since I've been thinking back over these ideas recently, I thought I'd provide a short overview. Come to think of it, there's a bunch of less technical and very fun things I'm itching to tell you about extra-dimensional models. I'll get into them soon.

Yesterday we had an interesting seminar by Paul Smyth, who is a graduate student at Imperial College, London, studying with Kelly Stelle. Paul’s talk was titled “The Stability of Harava-Witten Space-times”. He was concerned with the implications of the negative tension branes required by some extra-dimensional models, both those arising from string theory and those appearing in some more phenomenological models. In Paul’s seminar, he discussed a detailed analysis of the energy of such braneworld space-times in 5 dimensions, and demonstrated that they are stable, using a non-perturbative positive energy theorem.

This, and related questions have interested me for a while, and Don Marolf and I wrote a paper that I really liked about a related physical setup. I thought it might be fun to recount the general ideas here.

An interesting possibility, that has seen new life in the last seven years or so, is that our three familiar spatial dimensions may exist as a submanifold of a higher dimensional space-time. Common to most of the modern incarnations of this idea is the condition that all interactions other than gravity be confined to our submanifold, or brane. Since departures from 3+1 dimensional gravity are relatively difficult to constrain compared to those of other forces, this permits significant freedom to modify the gravitational interactions in the extra-dimensional space. It is in this modification of gravity, and of the extra dimensions themselves, that recent approaches differ from one another.

In many of the proposed models, the hierarchy problem of particle physics – how is the large difference between the weak scale (100 GeV) and the Planck scale (10

^{19}GeV) protected from the destabilizing effect of quantum corrections - is recast by bringing the fundamental mass scale of physics down to the weak scale. The large Planck mass observed on our brane is then a derived quantity, the size of which arises from the relatively large volume of the extra-dimensional manifold.One specific hurdle that extra-dimensional theories must clear is that the brane-bulk system should be a consistent, stable solution to Einstein gravity. For those constructions that include negative-tension branes, this can pose a particular problem, although perturbative dynamical objections can be overcome by placing the offending brane at a point of special symmetry in the extra-dimensional space, known to aficionados as an “orbifold fixed point”.

Nevertheless, Don and I were interested in the possibility that there might be nonperturbative instabilities of such a space-time. We though there might be because of a simple (and, it turns out, naïve) thermodynamical argument.

This comes from the generalized second law of thermodynamics, which states that the total entropy in matter and black holes does not decrease. The entropy of a black hole is proportional to the area of the event horizon. Therefore, any process that leaves the entropy in matter fixed and decreases the total area of black hole horizons, leads to a violation of the generalized second law.

It is important to know that the area of a black hole event horizon increases when positive energy crosses the horizon, and decreases when negative energy crosses the horizon. Don and I were intrigued by the question: what happens if a black hole, initially far away from a negative tension brane, falls towards the brane and captures some of the brane within its horizon? One might expect that the generalized second law could be violated in this way, since the part of the brane that is swallowed by the black hole carries negative energy across the horizon.

Because the central issue can be muddied by questions involving gravitational radiation (and, frankly, because to go further than this seemed much harder) we considered a lower-dimensional system - a negative tension 1-brane in a 2+1 dimensional space-time – because gravitational radiation does not arise in lower dimensions. What we discovered in analyzing this system was something more dramatic (at least on the face of it) than a violation of the generalized second law.

In 2+1 dimensions, the only black holes are the so-called BTZ black holes, named after their original discoverers Maximo Banados, Claudio Teitelboim and Jorge Zanelli. These black holes arise only in space-times with a negative cosmological constant, the so-called Anti de-Sitter (AdS) space-times. This turned out to be particularly appropriate, because the kinds of brane-world constructions that people have worked on typically require an AdS space-time, for other reasons.

In the main part of our work (this is the bit where the cartoon caption reads "and then a miracle happens"), we made use of a certain amount of cute mathematics, involving the fact that both a BTZ space-time and a negative tension brane space-time can be constructed as quotient spaces of AdS. I certainly won’t go into this here. Rather, I’m going to hope that it suffices to say that this enabled us to construct a general exact solution describing the collision of a black hole and a negative tension brane at one of these points of special symmetry

What we found was catastrophic. The endpoint of this evolution is not, in fact, an equilibrium configuration, but instead is a space-like singularity (similar to the `big crunch' of closed cosmological models with only ordinary matter content). What we discovered, therefore, was a non-linear dynamical instability of gravitating negative tension branes at orbifolds – in effect, the entire space-time outside the black hole collapses when the brane is located at an orbifold fixed plane. Because BTZ black holes can form dynamically from the collision of matter, we also expect non-linear instabilities with negative tension branes in the presence of matter fields, even if no black holes are initially present.

The obvious question to ask is whether the same sort of singularity that we found in 2+1 dimensions also occurs in higher dimensional cases (which are of real interest). For a number of reasons, examples such as this are not sufficient to argue convincingly for a dynamical instability in higher dimensions. That's for future work, and I think I may even have interested Paul in the idea, at least a little.

It's worth mentioning that the ways in which orientifold constructions in string theory obviate this proof is a very interesting story in its own right. If you have the fortitude for it you can read about that in a nice paper by Don Marolf and Simon Ross.

So what is the bottom line of our work? It’s that an orbifold boundary condition may not, by itself, be sufficient to render negative tension branes stable. As a result, if use is to be made of negative tension branes in various models, it is necessary to show that the particular branes being used are immune from these effects. Within string theory this is probably not a problem, but phenomenological models might not be immune.

Well, that was a little more technical than usual, but since I've been thinking back over these ideas recently, I thought I'd provide a short overview. Come to think of it, there's a bunch of less technical and very fun things I'm itching to tell you about extra-dimensional models. I'll get into them soon.

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