## The Conformal Structure of Space-Time | SpringerLink

Quantum Grav. The Arrow of Time in the Landscape, R. Vaas ed. The talk was based on Paper 65, above.

The results in Paper 75 are interestingâ€¦.. Because they are based on some very complicated calculations involving a certain black hole, and this black hole is used to mimic the motion of the QGP in the immediate aftermath of a peripheral [off-centre] collision of two lead nuclei at the LHC in Geneva.

But it does this very badly! Roughly speaking, the idea is this: when a black hole rotates, it tries to make everything outside it rotate too [even things with zero angular momentum! This differential rotation effect can [in Anti-de Sitter spacetimes ] even persist at infinity. Now peripheral collisions do produce a QGP with a differential velocity profile, dependent on distance from the axis of the collision. So the idea is to use the differentially rotating black hole geometry to model that internal motion of the spinning QGP.

All this was in Paper The problem is that in that paper I had to use a specific black hole, the one discovered long ago by Klemm et al. Question: if you replaced that black hole by one which does have a realistic velocity profile at infinity, might that make the angular momentum cutoff go away? The straightforward way to answer this question is as follows.

In reality, however, this is utterly beyond reach! So we have a problem: reproduce the results of Paper 75, starting with the general shape of the profile, and without knowing the black hole metric. This sounds completely impossible, but that is exactly what is done in Paper 76 [to appear in Nuclear Physics B]. To cut a [very] long story short, I use some fancy global differential geometry to reduce the problem to a special case. Normally we tell our first-year students that this is hopeless, and they have to use computers.

But in this case [for some reason] I contradicted this advice and tried to solve it. Amazingly, I found that I could! Very satisfying to use such extremely abstract mathematics to solve such a concrete problem. When lead ions collide, at extremely high energies, at the LHC in Geneva, the collisions are not always head-on. By the. We know [from my work, see below] that AdS black holes with topologically spherical event horizons are stable no matter how rapidly they spin. This is not as dramatic as it seems, however, because it takes a tiny but non-zero amount of time for the instability to developâ€¦.

And the QGP only exists for an extremely short time anyway. The rate at which the instability develops can be studied by working out how long it takes for something to fall into the black hole, a relatively simple calculation. All this is explained in Paper So there is every reason to fear that a rotating black hole might likewise become unstable.

In our Universe, that is indeed true, as a beautiful theorem due to Stephen Hawking shows.

And in fact, strange as it seems, black holes in AdS -like spacetimes can have non-spherical topologies and geometries! In the early Universe, when temperatures were so high that a quark -gluon plasma can form, or in the core of a neutron star, the spacetime geometry is deformed away from being exactly flat, and one has to ask what happens then.

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Well, why should anything special happen? This may have important consequences for the detailed shape of the quark matter phase diagram [see below]. In particular, the location of the triple point may be different in situations where the curvature is weak [ eg in a particle accelerator here on Earth] and in cases where it is strong [ eg inside a neutron star]. This is the theme of paper 72, due to appear soon in Nuclear Physics B.

## Hermann Weyl

Under normal circumstances you can never see a quark they are said to be confined. This can also happen at unimaginably high pressures. Here T is the temperature and you can think of the horizontal axis as pressure. This is the state of matter at the highest temperatures. So what is that temperature? In paper 71 above I tried to answer that question. The idea is to use the dual description of the plasma in terms of electrically charged black holes in Anti-de Sitter spacetime [see below].

These black holes have a very peculiar and intricate differential geometry which tends to make the black hole become unstable if it gets too cold. After a long and complicated argument I find that this temperature is a chilly 70 MeV , or around billion degrees Kelvin. That may seem a bit extreme, but higher temperatures than that can be achieved in particle accelerator collisions, for example such temperatures are being reached at the RHIC experiment [see picture below] and at the Large Hadron Collider.

The theory of cosmic inflation is primarily an attempt to explain why our Universe is so big. That may sound strange, but think of it this way: near to the Big Bang, the dominant physics was particle physics: the Universe was a soup made of quarks and gluons and radiation and so on, all interacting furiously and very rapidly over very short length scales. So how did we get something so huge and so long-lived out of a system in which typical distances are tiny fractions of a millimetre , not billions of light-years?

The answer is that, very early on, the Universe expanded at a fantastic rate and became really huge in a very short time. This theory explains a lot, but it has one problem: it needs a certain field [the inflaton ] to be in a very special state initially.

This goes back to the Arrow of Time problem.

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But even if we have a solution to that problem, we are still in trouble because Inflation is not supposed to start right away there is a period during which the Universe needs to reach the right size [after expanding relatively slowly]. But how do we keep the inflaton in its special state during that waiting period?

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Hyperbolic space, as shown, is infinite:. But the picture [in which the demons or cows are all really the same size] suggests that it might be possible to break it up into an infinite collection of identical finite pieces. That can indeed be done, and recently Gabai and co-workers have found the way to do this so that the pieces have the smallest possible volume. Compactifying means that you just declare that all of the identical pieces are the same, so you get a finite space.

When rays of light move outward from a point in such a compactified space, they are forced to return to a neighbourhood of where they started, so everything gets mixed up. This mixing can keep the inflaton in its special state.

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My paper 70 above is about finding some numerical bounds that have to be satisfied if the idea is to be made to work. It looks like it can. And who cares, you ask. Well, there is a special kind of black hole that can [apparently] be made to reach absolute zero!

Now this spacetime is nothing like our own spacetime , but it is nowadays a subject of intense interest in String Theory. String theory works exceptionally well in AdS spacetimes. So you can think of us living on the boundary in the picture, but using the five-dimensional interior to understand physics in our world. What sort of physics? Well, one branch of physics which is not yet fully understood is the theory of the strong interaction, the force that holds quarks together inside protons, neutrons, etc. This may be what happens in the core of a neutron star, which is what is left over after many supernovae [see the other picture].

It is not really understood what happens to quarks under such pressures. But AdS spacetimes give us a way to study this. You just have to put a black hole into the five-dimensional space and charge it up. But that means, as we said, making it very cold. This is a fascinating example of mathematics being developed for one purpose finding a use in a totally different, completely unexpected direction.

My work shows that in fact if you try to make this particular kind of black hole too cold, it just disintegrates. So there is a limit to how cold it can be. See paper 69 above. When things fall into black holes, the information they contain is apparently lost.