I am a professor of mathematics at NUS. I like: my wife, travel, cosmology, cycling three-wheeled vehicles, black holes, my son, cars with too much horsepower, and hippopotami. Not necessarily in that order.
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Sorry, I don’t have any notes of my own on bifurcation theory, but I found the following web page extremely helpful:
How to run a business and get rich
Very nice graphing software....when in doubt, graph it!
Integrals Explained ODEs Explained Fourier explained Linear Algebra in Action!
Need Some Exercise?
Need some more exercise ? [MA1506 ]
A Very Nice Webpage With Lots of Helpful Calculators and Stuff
Click on the picture of the VORTEX to learn more about something that is even more fun than driving cars.
Preprints on Arxiv
Most Cited Paper: http:/inspirehep.net/record/568045/
The Full List:
1. Brinkman's theorem in general relativity and non-Riemannian field theories. Gen. Relativity Gravitation 12 (1980), 767.
2. [with Klotz, A. H.] On the meaning of the metric hypothesis in the nonsymmetric unified field theory. Acta Phys. Polon. B 11 (1980), 345.
3. An identification of the metric tensor of the nonsymmetric unified field theory. J. Phys. A 13 (1980), no. 12, 3657.
4. A note on Browne's "source function" in five-dimensional relativity. Gen. Relativity Gravitation 13 (1981), no. 5, 495.
5. [with Klotz, A. H.] The nonsymmetron. Acta Phys. Polon. B 13 (1982), no. 3, 181.
6. A note on the geometry of connections in gauge theories of the electroweak interaction. J. Phys. A 15 (1982), no. 11, 3623.
7. A New Approach to Symmetry Breaking, J. Phys. G, 8 (1982) 609.
8. Geometric Symmetry Breaking and the Weinberg-Salam Model, J. Phys. G, 8 (1982) 621.
9. Remarks on the geometric interpretation of superconductive flux quantisation. J. Phys. A 17 (1984), no. 16, 3101.
10. On the nonexistence of 't Hooft-Polyakov magnetic monopoles in nonsemisimple electroweak gauge theories. J. Phys. A 17 (1984), no. 17, 3287.
11. On the affine approach to Riemann-Cartan space-time geometry. Classical Quantum Gravity 1 (1984), no. 2, 115.
12. On the geometrical interpretation of "nonsymmetric" space-time field structures. Classical Quantum Gravity 1 (1984), no. 2, 105.
13. On the significance of the compatibility condition in gauge theories of the Poincare group. Classical Quantum Gravity 1 (1984), no. 1, 1.
14. Spontaneous Compactification and Singularities in Inner Space, Physics Letters B, 150 (1985), 113.
15. Global Aspects of Spontaneous Compactification, Classical and Quantum Gravity, 2 (1985), 661.
16. Spontaneous compactification and Ricci-flat manifolds with torsion. J. Math. Phys. 27 (1986), no. 8, 2029.
17. On the Splitting Problem in Higher-Dimensional Field Theories, CQG 4 (1987), 329.
18. Spontaneous splitting and internal isometries of superstring vacua. J. Math. Phys. 28 (1987), no. 11, 2564.
19. Superstrings and holonomy groups of Kaehler manifolds. Classical Quantum Gravity 5 (1988), no. 4, 561.
20. Group-theoretic aspects of the Hosotani mechanism. J. Phys. A 22 (1989), no. 13, 2309.
21. The geometry of gauge symmetry breaking in the superstring context. J. Math. Phys. 30 (1989), no. 2, 498.
22. Hosotani Breaking of E_6 to a Subgroup of Rank Five, Journal of Mathematical Physics 31 (1990), 2094.
23. Methods of Holonomy Theory for Ricci-Flat Riemannian Manifolds, JMP 32 (1991), 888.
24. Gauge Theory in Witten's Approach to the Generation Problem, Communications in Mathematical Physics, 138 (1991), 107-136.
25. The Quotient Construction for a Class of Compact Einstein Manifolds, Communications in Mathematical Physics, 154 (1993), 307.
26. Holonomy Groups of Compact Riemannian Manifolds: A Classification in Dimensions up to Ten, Journal of Mathematical Physics 34 (1993), 4273.
27. Examples of Einstein Manifolds with All Possible Holonomy Groups in Dimensions Less than Seven, Journal of Mathematical Physics 34 (1993), 4287.
28. Complex Symplectic Geometry and Compact Locally HyperKaehlerian Manifolds, Journal of Mathematical Physics 34 (1993), 4857.
29. Gauging discrete symmetries. J. Math. Phys. 36 (1995), no. 10, 5414.
30. Holonomy Groups and Holonomy Representations, Journal of Mathematical Physics 36 (1995), 4450.
31. Multiple spin structures in higher-dimensional physics. Classical Quantum Gravity 13 (1996), 3175.
32. Calabi-Yau Compactifications and the Global Structure of the Standard Group, Journal of Mathematical Physics 37 (1996), 493.
33. On Space Forms of Grassmann Manifolds, Manuscripta Mathematica 93 (1997), 205.
34. Obtaining Holonomy From Curvature, Journal of Physics A 30 (1997), 661.
35. Alice Universes, Classical and Quantum Gravity 14 (1997), 2527.
36. Disconnected forms of the standard group. J. Math. Phys. 38 (1997), no. 8, 4354-4362.
37. Existence of parallel spinors on nonsimply connected Riemannian manifolds. J. Math. Phys. 39 (1998), no. 4, 2362-2366.
38. Methods of Alice Physics, Journal of Physics A 31 (1998), 3607.
39. Metric symmetries and spin asymmetries of Ricci-flat Riemannian manifolds. J. Math. Phys. 40 (1999), no. 3, 1255-1267.
40. The Semispin Groups in String Theory, J.Math.Phys. 40 (1999) 4699, see http://arxiv.org/abs/hep-th/9906059
41. Spin Holonomy of Einstein Manifolds, Communications in Mathematical Physics 203 (1999) 349.
42. Gauge Spinors and String Duality, Nucl.Phys. B577 (2000) 439, see http://arxiv.org/abs/hep-th/9910100
43. AdS/CFT for Non-Boundary Manifolds, JHEP 0005 (2000) 025, see http://arxiv.org/abs/hep-th/0003291
44. The Topology of the AdS/CFT/Randall-Sundrum Complementarity, Nucl.Phys. B602 (2001) 132, see http://arxiv.org/abs/hep-th/0009087
45. Topologically Induced Instability in String Theory, JHEP 0103 (2001) 031, see http://arxiv.org/abs/hep-th/0101136
46. A Positive Cosmological Constant in String Theory Through AdS/CFT Wormholes, Nucl.Phys. B609 (2001) 325, see http://arxiv.org/abs/hep-th/0105151
47. Exploring the Similarities of the dS/CFT and AdS/CFT Correspondences, Nucl.Phys. B627 (2002) 311, see http://arxiv.org/abs/hep-th/0110062
48. The dS/CFT Correspondence and the Big Smash, JHEP 0208 (2002) 029, see http://arxiv.org/abs/hep-th/0112066
49. Stringy Instability of Topologically Non-Trivial AdS Black Holes and of Asymptotically deSitter S-brane Spacetimes, Nucl. Phys. B660 (2003) 373, see http://arxiv.org/abs/hep-th/0205103
50. What if w < -1?, page 265 of the proceedings of the XVIIIth Colloquium of the Institut d'Astrophysique de Paris, July 2002, "On the Nature of Dark Energy",
edited by P. Brax, J. Martin, J.-P. Uzan, see http://arxiv.org/abs/astro-ph/0210321
51. The Covariant Entropy Bound, Brane Cosmology, and The Null Energy Condition, JHEP 0212(2002)053, see http://arxiv.org/abs/hep-th/0212014
52. The Strong Energy Condition and the S-Brane Singularity Problem, JHEP 0306 (2003) 043, see http://arXiv.org/abs/hep-th/0305107
53. de Sitter and Schwarzschild-de Sitter According to Schwarzschild and de Sitter, JHEP 0309 (2003) 009, see http://arxiv.org/abs/hep-th/0308022
54. Orbifold Physics and de Sitter Spacetime, Nucl Phys B 692 (2004), 270-300, see http://arxiv.org/abs/hep-th/0311055
55. APS Instability and the Topology of the Brane-World, Physics Letters B 593 (2004) 10, http://arxiv.org/abs/hep-th/0401035
56. Quintessential Maldacena-Maoz Cosmologies, J High Energy Physics 0404(2004)036, http://arxiv.org/abs/hep-th/0403104
57. Answering a Basic Objection to Bang/Crunch Holography, JHEP 0410(2004)018, http://arxiv.org/abs/hep-th/0407189
58. Inflation, Large Branes, and the Shape of Space, Nucl.Phys. B709 (2005) 213-240, http://arxiv.org/abs/hep-th/0410115
59. The Phantom Divide in String Gas Cosmology, Nucl.Phys. B718 (2005) 55-82, http://arxiv.org/abs/hep-th/0502209
60. The Most Probable Size of the Universe, Nucl.Phys. B730 (2005) 50-81, http://arxiv.org/abs/hep-th/0509035.
61. Pre-Inflationary Spacetime in String Cosmology, Nuclear Physics B748 (2006) 309-332, http://arxiv.org/abs/hep-th/0511227
62. The Geometry of The Entropic Principle and the Shape of the Universe, JHEP10(2006)029, http://arxiv.org/abs/hep-th/0604150
63. Unitarity at Infinity and Topological Holography, Nuclear Physics B754 (2006) 91-106, http://arxiv.org/abs/hep-th/0606068
64. Inaccessible Singularities in Toral Cosmology, Class. Quantum Grav. 24 (2007) 1605-1613 http://arxiv.org/abs/gr-qc/0611101
65. Arrow of Time In String Theory, Nuclear Physics B782 (2007) 1-25, http://arxiv.org/abs/hep-th/0611088
66. Initial Conditions for Bubble Universes, Physical Review D77 (2008) 123530-123545, http://www.arxiv.org/abs/0705.4141
67. The Arrow of Time in the Landscape, R. Vaas (ed.): Beyond the Big Bang. Springer: Heidelberg 2010, http://arxiv.org/abs/0711.1656
68. Black Hole Final State Conspiracies, Nucl. Phys. B807 (2009) 33-55, http://arXiv.org/abs/0806.3818
69. Bounding the Temperatures of Black Holes Dual to Strongly Coupled Field Theories on Flat Spacetime, JHEP09(2009)048, http://arxiv.org/abs/0905.1180
70. Decoupling Inflation From the String Scale, Classical and Quantum Gravity 27 (2010) 165001, http://arxiv.org/abs/0911.4583
71. Holography of the Quark Matter Triple Point, Nuclear Physics B832 (2010) 323-341, http://arxiv.org/abs/0910.4456
72. Fragile Black Holes, Nuclear Physics B 842 (2011) 86–106, http://arxiv.org/abs/1008.0231
73. A Universal Lower Bound on the Specific Temperatures of AdS-Reissner-Nordstrom Black Holes with Flat Event Horizons, Nucl.Phys.B848 (2011) 474, http://arxiv.org/abs/1012.4056
74. Kerr Black Holes Are Not Fragile, Nucl.Phys. B857 (2012) 362, http://arxiv.org/abs/1108.6234
75. Fragile Black Holes and an Angular Momentum Cutoff in Peripheral Heavy Ion Collisions, Nucl.Phys. B861 (2012) 236-258 arXiv:1201.6443
76. Universality of the Holographic Angular Momentum Cutoff, http://arxiv.org/abs/1206.0120, to appear in Nuclear Physics B
I was an invited speaker at a cosmology conference in beautiful Wuerzburg, Germany. The talk was based on Paper 65, above.
I was invited to Cambridge
to speak at this: The Very Early Universe 25 Years On , celebrating the 25th anniversary of the famous Nuffield conference at which the foundations of the theory of Inflation were laid.....and it was a great conference! See the PROGRAMME page for the slides of the talks, including mine.
CURRENT RESEARCH INTERESTS:
• PLUGGING A BIG HOLE IN THE HOLOGRAPHIC ANGULAR MOMENTUM CUTOFF PREDICTION
The results in Paper 75 are interesting….. but you shouldn’t really believe them! Why? 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!] The rate at which this happens depends on direction, so it’s not like a rigid body in general. 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 75. The problem is that in that paper I had to use a specific black hole, the one discovered long ago by Klemm et al. But this black hole does *not* induce a realistic velocity profile at infinity! 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. [a] Specify a realistic profile. [b] Find a black hole inducing that profile at infinity. [c] Re-run the calculations in Paper 75. In reality, however, this is utterly beyond reach! [a] We don’t actually *know* the exact shape of the profile, just its general form. [b] This is extremely hard! [c] Therefore, this too cannot be done in practice!
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. This special case is still hard, because it requires us to specify the [exact] shape of the solutions of a certain *extremely* nasty third-order non-linear differential equation. 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! [There was a certain perverse pleasure in subjecting myself to the same kind of torture to which I put my first-year students… and a good deal of slightly childish satisfaction in succeeding…] Anyway, the conclusion was that any profile which is even vaguely realistic will lead to exactly the same kind of instability that we found for the Klemm et al. black hole, so the angular momentum cutoff is a universal property of the holographic approach. Very satisfying to use such extremely abstract mathematics to solve such a concrete problem.
• HOW FAST CAN A QUARK PLASMA SPIN?
When lead ions collide, at extremely high energies, at the LHC in Geneva, the collisions are not always head-on.
If they aren’t, then the quark-gluon plasma produced by the collision [see below], and observed by the ALICE collaboration, may acquire a huge amount of angular momentum. By the
AdS/CFT correspondence, questions about the QGP can be translated into questions about AdS black holes, so one can hope to investigate the rapidly spinning QGP by looking at rapidly spinning AdS black holes. We know [from my work, see below] that AdS black holes with topologically spherical event horizons are stable no matter how rapidly they spin. But the AdS black holes that one studies as dual systems to the QGP don’t have topologically spherical event horizons --- their event horizons are planar. And in fact it turns out that they *aren’t* stable. So if the AdS/CFT correspondence is not misleading us, this means that the QGP produced in off-centre collisions is unstable! This is not as dramatic as it seems, however, because it takes a tiny but non-zero amount of time for the instability to develop…. 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. [This is an example showing why the whole duality idea is so interesting – things that are hard to compute on one side are relatively easy to compute on the other.] It turns out that the instability only develops very rapidly when the angular momentum density is very high. In principle this should mean that it may not be possible to observe arbitrarily large angular momentum densities in these collisions: so I call this the “Holographic Angular Momentum Cutoff”. Whether such a thing is really observable at the LHC is very hard to say, both for theoretical reasons [we don’t really understand how accurate the duality is] and for experimental ones [because I don’t know enough experimental physics to understand what really goes on in these collisions! ]. All this is explained in Paper 75.
• HOW FAST CAN A BLACK HOLE SPIN?
Black holes *do* often spin extremely rapidly, and, as you’d expect, this distorts their shape very drastically. But one knows [see “Fragile Black Holes” below] that distorting the shape of a black hole, at least in Anti-de Sitter spacetime, is a recipe for disaster --- they can only be distorted up to a point, beyond which they become unstable. So there is every reason to fear that a rotating black hole might likewise become unstable. In paper 74 I showed that this, very surprisingly, doesn’t happen. This is very fortunate because black holes have been observed which rotate at a rate which is at least 99% of the maximal rate permitted by cosmic censorship.
• HOW COLD CAN A BLACK HOLE GET?
I’m very interested in the minimum possible temperatures of black holes --- specifically, black holes in Anti-de Sitter spacetime, with flat event horizons [see below]. Paper 73 shows that there is actually a *universal* lower bound on the temperatures of these objects, universal in the sense that it doesn’t even matter what the dimensionality is. That’s about as definitive as you can get: *no* black hole of this sort, in any number of dimensions, can ever have a [mass-normalised] temperature below about 0.156875.
• FRAGILE BLACK HOLES?
Intuitively it seems clear that the event horizon of a black hole has to be “round”, that is, have the topology of a sphere. In our Universe, that is indeed true, as a beautiful theorem due to Stephen Hawking shows. However, Hawking’s theorem assumes that energies cannot be negative, so it doesn’t work in Anti-de Sitter spacetime [see below]. And in fact, strange as it seems, black holes in AdS-like spacetimes can have non-spherical topologies and geometries! This is just as well, actually, because the kinds of field theories to which the famous AdS/CFT duality [see below] are applied normally live on flat spacetimes, and that means that one wants to consider AdS black holes with *flat* event horizons.
However, that’s only an approximation. 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. Well, why should anything special happen? The answer is that, when you look at black holes in AdS, they turn out to be *fragile* --- that is, if you deform them too much, they become unstable. Even spherical AdS black holes are somewhat fragile, as was shown in a very nice paper by Keiju Murata, Tatsuma Nishioka, and Norihiro Tanahashi. So the question is: how fragile are AdS black holes with *flat* event horizons? It turns out to be surprisingly complicated to answer this question --- you can’t just re-use the methods of Murata et al, though you can gain some insight that way. However, by using some fancy methods from global differential geometry you can answer the question, and the answer is rather shocking: flat AdS black holes are *completely* fragile: even the slightest distortion of the event horizon makes them unstable. 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.
It’s now universally accepted that protons and neutrons and many other “fundamental” particles are made up of quarks. Under normal circumstances you can never see a quark --- they are said to be confined. But at unimaginable temperatures, quarks [and the related particles called gluons] can break free and become “deconfined”. This can also happen at unimaginably high pressures. The situation is summarized by a “phase diagram” like the one shown:
Here T is the temperature and you can think of the horizontal axis as pressure. Don’t worry about the letters scattered around, just look at the upper part of the diagram, labeled QGP: the “Quark-Gluon Plasma”. This is the state of matter at the highest temperatures. Now notice that you can have this plasma at relatively low temperatures around the point B --- this is called the “triple point” of quark matter. The temperature there is the lowest possible temperature for a quark plasma --- that’s as cold as it can get. 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. [Black holes have a temperature, as Stephen Hawking taught us.] Anyway, by using geometry to study the rates at which surface areas and volumes grow in such spacetimes, you can use the duality to compute the temperature at the triple point. After a long and complicated argument I find that this temperature is a chilly 70 MeV, or around 800 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. [Though even there they aren’t reaching pressures as high as the pressures at B in the diagram; they are probing the region around A, called the critical point.]
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. 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? Andrei Linde, one of the founding fathers of Inflation, has an ingenious solution to that: assume that the spatial geometry is a compactified version of hyperbolic space [see below, under “The Most Probable Size of the Universe”]. 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. Like all good ideas, Linde’s suggestion has a number of problems which need to be overcome if it is to be made to work. 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! These are black holes in “anti-de Sitter spacetime”; all you have to do is to throw charged particles into them and they get colder and colder. 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. It turns out that, in string theory, there is an equivalence between a gravitational theory in AdS and a *non-gravitational* theory living in a space of one dimension lower, which is attached as a sort of boundary, as in the picture below:
This means that one can use facts about *gravity in five dimensions* to learn things about other theories in four dimensions, where we live. 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. It turns out that when you squash protons and neutrons together at fantastically high pressures, they “melt” into a state in which the quarks are no longer bound together. 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]. These objects are so dense that even a part of the core the size of a 10-cent coin would be much more massive than the entire suburb where you live; the cores of these things have the highest pressures in the Universe…. 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. So that’s why we care about very cold black holes in five dimensions --- they may tell us about what goes in in the cores of neutron stars in *our* universe! 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. On the neutron star side of the equivalence, this means that something strange must happen to quark matter at low temperatures but with very high values of something called the “chemical potential.” This fits in with earlier guesses about what happens under these extreme conditions. See paper 69 above.
When things fall into black holes, the information they contain is apparently lost. That may not seem to be a problem, but in fact it leads to one of the deep mysteries in physics, one that has defeated some of the best minds: the "Black Hole Information Loss Paradox." I don't know how to solve it either, but I have an idea that might make a small contribution towards solving it. See paper 68, above. The basic idea is due to Horowitz and Maldacena, who proposed that the "lost" information is actually saved by means of a special version of "quantum teleportation".
It's a brilliant idea, and in my opinion the most likely solution of the puzzle. But it is generally thought that there are serious problems with it. Paper 68 points out that the problems may not be as bad as people think, if you are willing to accept the possibility of pretty weird behaviour inside certain kinds of black holes [in Anti-de Sitter spacetime]. My theory of the arrow of time predicts exactly that kind of behaviour......and the paper appeared in Nuclear Physics B. By the way, Nuclear Physics B is an outstanding journal and it is extremely difficult to get papers accepted by them.
While I was in Wuerzburg in September 2007, I was invited to contribute a chapter to a book, to be published by Springer Verlag in 2009, titled "Beyond the Big Bang". Writing that chapter gave me an opportunity to think through a lot of things which I had previously been unclear about, so I am very grateful for the invitation. See paper 67 above. The upshot is the following picture. Our Universe is believed by many to be a baby universe, which split off from some larger, older universe. There could be a lot of babies around, all with different properties. This is like the different micro-environments in a landscape here on earth, as in this beautiful landscape by Pieter Bruegel [the elder]:
String theory does allow for a very large variety of possible universes, brought into existence by the birth of baby universes. This ensemble of many universes is called the String Landscape. Now while the Landscape allows for a large number of possibilities, that doesn't mean that "anything goes". The extremely rich chemistry of carbon allows for a fantastic number of carbon-based objects [like us] to be constructed, but nobody would use this as evidence that organic chemistry is a waste of time because it allows *anything* to be built out of carbon compounds. We probably need a "large" Landscape to account for the famous Cosmological Constant, but for some purposes the Landscape is actually very very small. One of those purposes is the problem of accounting for the Arrow of Time, the fact that time seems to pass, resulting in a future which differs from the past. It was shown a long time ago by Roger Penrose that our Universe is ``special" --- that is, its initial conditions were somehow selected fantastically accurately --- so accurately that there is no hope whatever of finding a universe like ours in the Landscape by mere chance. I argue that the origin of this specialness is that the ``mother universe" of our Universe was even more special, and her mother was
more special still.....and so on....right back to the Mother of All Universes, which I call Eve [Seen here courtesy of Albrecht Duerer and Lukas Cranach.]. So the problem is to understand the extreme specialness of Eve. Here "special" means "incredibly smooth". Using recent developments in string theory, I argue that Eve was "born" when time itself emerged from some purely stringy state which itself was timeless. A subtle argument [discussed below] shows that Eve had to be born in a state of extreme spatial smoothness. This is supposed to explain the existence of an Arrow of Time in Eve, and thus in at least some of her descendants. The overall picture of the Landscape is then like an ancestral tree, with babies branching off Eve, and inheriting an Arrow. Some of the descendants, by the way, will die [if their vacuum energy becomes negative] but Eve herself may well be immortal, as promised by you-know-who, pictured above.
Babies are bad....... ....see paper 66 above.
Well, not that kind of baby actually, but rather baby *universes*. It's possible for a tiny region of our Universe to split off and go its own way. Thus even if you start with one universe, you might end up with lots of them. Some people like this idea, because it may give you a way of explaining the value of the cosmological constant: each time a baby splits off, the cosmological constant decreases, so eventually you might arrive at the absurdly small value we observe today. The question, however, is whether a baby universe can ever look like *our* universe --- if it can't, then the possible existence of babies is of no concern to us. Now [see below] *our* Universe began in an extremely special way --- with a very nice smooth geometry. That is extremely hard to organise, especially for a baby universe. For baby universes start out small, which may mean that whatever goes into making them may get squashed when the baby is born. Squashed things, however, are rarely nice and smooth. The simplest way to avoid this problem is to start out with a big universe that is *already* very smooth, and try to arrange things so that the smoothness is "inherited" by the baby. Paper 66 is concerned with this very tricky question of inheritance. It turns out that string theory *may* allow for the existence of a very exotic form of matter which does permit inheritance [by forbidding the baby to be too small]. So now we have passed the buck back to the mother universe: the baby can be smooth if the mother was smooth. But how can the mother be smooth in the first place? That was explained in paper 65.......see below. [By the way, saying that the baby is "small" when it is born is slightly misleading....actually the baby is indeed very small when seen from the *outside*, and it is true that to get inside it from the outside you would have to be squashed. But the baby itself is actually *large* when viewed from the inside --- in fact, it is *infinitely* large inside! Strange but true.] [Another note: some people object to the name "baby universe" for the things I describe as such. They want to confine this term to apply to a different kind of universe which [maybe] can split off and then remain completely inaccessible to the mother universe. I must say that these people have very peculiar notions regarding the raising of children. One lives in hope, at least, that children can be influenced by their parents long after they are born. In Asia, in fact, children remain firmly under the influence of their parents for an indefinite period. Similarly, the point I want to make is that "baby" universes are permanently exposed to influences from their mother.]
Another one of those questions that seemingly have no scientific answers, yet some cosmologists do have ideas about it! [See also the very useful AOT FAQ .] And so do I, and I will write about that when I can find the......you know. Well, it seems that that time has arrived. Everyone knows that it is easier to make a mess than to clear it up. But if that is the case, then why do we ever see any situation that ISN'T a mess? It's easier to break an egg that to put it back together....but then, why do we ever see unbroken eggs? This may seem to be a strange question, but, if you think about it, you will see that it is at the core of one of our most basic experiences...the experience of time passing. We know that time passes precisely because things break and almost never unbreak. And it's easy to see why: there are more ways to have a broken egg than to have an unbroken one! Messed-up situations simply outnumber non-messy situations. In physics terms, this is why we have the Second Law of Thermodynamics: entropy [nearly] always increases in a closed system, simply because "special" arrangements are likely to evolve into "generic" ones; the reverse, while possible, is highly unlikely:
So it's actually easy to understand why time *passes*. The catch is that, by *exactly the same argument*, it's extremely tough to understand WHY the world was less messed-up in the past. By the way, in case you are worrying that "messed up" is in the eye of the beholder, if you go to this webpage Molecules and play with the applet there, you will get an idea of what I really mean by this. Alternatively, look at this [but turn off the "music"!]
Anyway, the point is that we recognise the past by the fact that it was less messed up. The question is why was the past so nice and non-messy. This is an extremely tough problem. Eventually you can trace it right back to the very beginning of time: the geometry of space at that time was fantastically nice and smooth. It didn't have to be: it could have been a weirdly distorted, horrible three-dimensional lump. But of all the infinitely many possible initial geometries, our Universe chose something smoother and more regular than anything that has ever existed since then. Why? This is the Question of Questions: we will understand why time passes only if we understand HOW THE UNIVERSE BEGAN!
One possibility is that if, when the Universe is born, space is already infinitely big [this *is* possible!], then everything that can happen must happen somewhere, so just by chance there was a smooth patch somewhere, and that small bit of space grew into the observable Universe. Personally I think that this kind of argumentation is sheer nonsense. You could equally well argue that all the so-called "laws" of nature are an illusion: in an infinitely large Universe, everything that can happen will happen, so all of the lawful behaviour we see around us, and that has been going on for billions of years, is JUST A COINCIDENCE, something that is bound to happen in an infinitely large world. This sort of argument is so lame that I adduce it as evidence that the Universe *cannot* be infinite. Jokes aside, though, can one do better? Can one put together an argument that says in effect that the Universe HAD to be born in an ultra-smooth state? I tried to do that in Paper 65 above. Basically the argument is very simple, though the details aren't. It turns out that the initial value problem for General Relativity has an unusual structure,
[see http://lombok.demon.co.uk/mathToolkit/home.svc for the nifty gadget that converts tex to gif] and in order for it to be internally consistent, the scalar curvature of the initial three-dimensional space has to be related to the total energy density at that time. This is well-known, but what is not well-known to most physicists is that imposing conditions on the scalar curvature can, if the topology of the space is right, actually constrain the geometry in an amazingly drastic way. I argue that these amazingly drastic restrictions on the initial geometry are ultimately responsible for the Arrow of Time. The theory implies, by the way, that time does not reverse inside a black hole, as some have argued it might. The key point is that the Arrow of Time is ultimately a global, topological effect.
who are born with the Universe --- that is, they enter the Universe simultaneously at the initial singularity [apologies to the mother of the twins]. One of them is stationary, the other moves. Then we examine them much later, and find to our dismay that their ages are different. This is a famous observation that everyone learns when they are taught Special Relativity. What is spectacularly different here is that the twins not only have different ages: the one who stays home is infinitely older than the traveller! The travelling twin has discovered the ultimate anti-aging product. In order for this to work, however, she has to circumnavigate the Universe infinitely many times in what [to her] is a finite amount of time. That may sound impossible, but in fact it is almost inevitable: remember that, as we move back in time, the torus is shrinking, and in fact it shrinks so fast that it can be circumnavigated infinitely many times in a finite amount of proper time! [If you try to trace back without doing all those circumnavigations, you will find that the Universe is infinitely old. Actually, however, that is very difficult to do: the torus gets so small that the slightest motion zips you around it infinitely many times.] So the singularity is there, but it can only be reached in a finite amount of time if you are willing to travel right around the Universe a literally infinite number of times. The question is whether an actual physical object can really do that. I argue that it can't, for reasons that will become clearer when you read the next paragraph. So the singularity does exist, but it is physically irrelevant. I also try to argue that this conclusion remains valid even if the early Universe is dominated by [tachyonic] closed strings. See paper 64 for the details.
mathematical machinery to study
what properties the Universe is likely to have [its size and spacetime geometry] under the assumption that it can be
born as a torus. After all, the world
is flat, as you can plainly see. [This is a joke.....] A flat
three-dimensional torus is what you get if you take ordinary three-dimensional
space and identify opposite faces of a cube [in
the most straightforward way]. The space is flat in the sense that the local
geometry is the same as ordinary geometry; however, the topology is
different. You might think that changing the
topology without changing the geometry should have no physical effects, but you
would be very wrong. In particular, changing from a sphere to a torus tends to
make the spacetime both singular and unstable in
string theory. This may sound bad, but actually it's good, because it means
that string theory constrains the spacetime much more
strongly in the toral case, and so we can hope to get
some predictions out. See paper 61 above. Readers who think that I have gone
out of my mind should notice that, if so, I am in good company: there are
famous professors at Harvard
who also believe that it is worth investigating the idea that the Universe came
into being out of "nothing". Prof Tye and
his co-workers at Cornell University have
done a particularly nice
job of explaining this idea.
Well, no, not that kind of Inflation..... it is generally accepted that, very early in the history of the Universe, the expansion was incredibly fast for a short time....we say that the Universe INFLATED. This simple idea allows us to explain some otherwise very puzzling facts....for example, the fact that the Universe looks the same in opposite directions. The idea is that although the Universe is huge, the part of it we can see was originally tiny, so it is not so surprising that it looks pretty much the same in all directions. However, there are technical complications here: if you believe that the Universe simply came into existence out of NOTHING, then it is a lot easier to believe that it was born small than large. In fact, my calculations show that it should have been born at a size which is about 100 times smaller than its estimated size at the time Inflation began. But assuming that it was born nice and smooth --- and this assumption too has to be thought about --- how do we stop it from becoming lumpy during the time when it expands by that factor of 100? Here the great cosmologist Andrei Linde comes to the rescue: he points out that if the Universe is a torus and if the spacetime geometry is right, then signals can propagate all the way around the torus many times during that period, nicely maintaining the uniformity until Inflation is ready to begin. Of course the catch is that the Universe does indeed have to be born at just the right size and with just the right spacetime geometry, for otherwise signals will not be able to circulate around the torus. In Paper 60 I showed that, taking into account certain quantum-mechanical subtleties discussed by Prof Tye and his group [see above] and by means of a geometric trick, I could actually calculate the most probable initial size and the most probable initial geometry......and they come out just exactly right! [Of course in Quantum Mechanics, probabilities are all you ever get, but in this case the "just right" answers really are a lot more probable than any alternative.] The whole calculation is probably too simplified to be completely realistic, but things come out so beautifully that it seems that this may be a real contender for a proper theory of the very earliest Universe --- even before Inflation!
collapse, including space itself, as the saying goes. That is, your (great)^n grandchildren would see a *contracting* universe,
getting smaller all the time, and shrinking to "zero" size in a
finite time, as shown in the diagram above. That may seem rather sad, but
there are worse possibilities [as for example a Big Smash/Rip, see below].
However, a theorem due to Witten and Yau [both of them Fields Medal
winners, by the way] forbids this kind of behaviour
under apparently physically reasonable conditions. So how did M&M manage
it? Explaining this is a rather subtle exercise in global
differential geometry. Explaining how it works in the more realistic
case in which the Universe accelerates is even more tricky.
This is the subject of paper 56 above.
Unfortunately, I see that Homer has the
wrong value for the cosmic density parameter [Capital Omega] --- it should
actually be 1, or perhaps slightly less than 1, not
greater than 1. Doh!
However, it gets even weirder: because the matter content of this cosmology is rather strange [it has to be, to get a non-trivial topology] it turns out that the gluing has to be done in a non-trivial way. In fact what you really get is a sort of four-dimensional analogue of the famous Klein Bottle. Two Klein bottles are shown above: one is a bottle, the other a hat......if you want to gain some intuition about how Klein bottles work, and at the same time drive yourself completely insane, try playing "tic-tack-toe" [= "noughts and crosses"] on a Klein bottle. Anyway, the point is that a Klein bottle is defined in much the same way as a torus, but you do a reflection of one end before joining them. Now I want to do this in four dimensions, starting with flat three-dimensional spaces. There are in fact 10 topologically distinct flat compact three-manifolds. One of them is the torus, and it's obvious that you can do the Klein bottle construction for it, but the other nine are more tricky [they are themselves obtained by performing various kinds of twists one dimension lower down]. It turns out that the construction won't work for most of them........ which is good, because it means that all of this may help us to find out which of the possibilities has actually been realised in our Universe! In fact, you end up with just three candidates: the torus and the ones named by the famous geometer [etc] John Conway the "dicosm" and the "didicosm". So in this roundabout way we have narrowed the topology of space down to just these three candidates. The didicosm is a particularly fascinating candidate for the shape of 3-dimensional space, and one can only hope that Nature has had the good taste to select it. All this is explained in paper #58, above.
Underlying all this is the extremely deep [and hard!] theorem, due to Gromov and Lawson, which says that, no matter how you twist and turn it, you can never turn a flat space into one with positive scalar curvature: also, the only way for the scalar curvature to be zero is for the space to be *exactly* flat. In other words, flat spaces, despite their apparent simplicity, actually have an extremely deep geometry, and it is very nice indeed that they seem to be the underlying structure both for *space* and for *spacetime*.
This is called a *Penrose Diagram*. Here time is up and down. The wavy lines represent the singularities inside a black hole [at top] and a white hole [at bottom]. The stars at right are copies of the *TWO*-dimensional real projective space. [The ones on the left are in another universe, which, sadly, we can never visit, because the singularity will tear you to pieces if you try to go there!] If you try to travel through the RP^2 on the right, with your back to the black hole, you will immediately re-emerge [still on the right] but now *facing* the black hole....and you will be upside-down! [You can see this if you remember the definition of RP^2 and if you have a good visual imagination.] Of course, to you, it will seem that the world has suddenly turned upside-down! This is portrayed in the following diagram, in which the images represent the cosmic microwave background radiation [as seen by NASA's WMAP satellite, see below] and in which Felix [see http://www.everwonder.com/david/felixthecat/about.html ] is a cosmological explorer in a topologically non-trivial de Sitter spacetime. [If your screen is narrow, you may not see all three images on the same line---imagine it! ] If you look closely you will see that the image on the right is upside-down, as above. In reality this picture is not quite right, because poor Felix would never be able to return home---the acceleration of the Universe implies that the Universe is getting larger faster than he can travel! See paper 53 above for details, including references to the fascinating history of "de Sitter spacetime". This paper was accepted, with laudable punctuality, by JHEP, one of the two top theoretical/mathematical physics journals.
This is what you get if you take a
dodecahedron, like the ones shown here, and then imagine that if you try
to escape from it, you instantly find yourself back inside, but rotated through
an angle of 36 degrees! [This is just a more complicated version of the
experiences of Felix the Cat, above.See http://www.nature.com/nsu/031006/031006-8.html
. Unfortunately, subsequent analyses of the data have shown that this
wonderful model probably isn't true. But *why* did Nature miss out on this golden opportunity to use one of the most
beautiful of all spaces? Recently [see paper 55 above] I tried to explain this
using the DESTROYING THE UNIVERSE idea described above. You can easily modify
de Sitter spacetime so that it has the Poincare space
as spatial sections, and it is not too hard to extend this into the fifth
dimension in the way I described above. The question is whether this triggers
off a disturbance in the five-dimensional spacetime
that might destroy our world. To determine this, you have to work out whether
certain symmetries of anti-de Sitter spacetime are
preserved or not. The group that defines the Poincare space --- it is called
the *binary icosahedral group* --- is quite a lot
more complicated than the ones I discussed in paper 54, so I had to develop
some new method to handle it. The first thing to do is to simplify the problem.
It's clear from the picture that the dodecahedron has a huge group of
symmetries --- in fact it has 60 [rotational] symmetries. *But* if you go to http://www.toonz.com/personal/todesco/java/polyhedra/dodecahedron_tetrahedron.html
you will find a beautiful picture of a tetrahedron inside a dodecahedron,
and if you play with the applet you can easily convince yourself that all of
the symmetries of the tetrahedron are also symmetries of the dodecahedron.
Using this, and using something called "quaternions", see http://mathworld.wolfram.com/Quaternion.html
, I showed that certain symmetries of the tetrahedron break all of the relevant
symmetries in anti-de Sitter spacetime, and therefore
the same must be true of the dodecahedral symmetry group. And combining
all this stuff, that's why we can't live in the Poincare homology sphere!
That is, if you try to put that kind of Universe
into a suitably modified version of anti-de Sitter spacetime,
then string theory tells you that the situation is unstable and your carefully
constructed Universe self-destructs --- and you wouldn't be sitting there
reading this....intriguingly, however, the details of the argument suggest that
we *might* live in a simpler kind of version of the sphere, called a *lens
space*. In such a space, Felix might see something like this on his return from
Stay tuned for the next episode.....meanwhile, a compressed version of the paper I wrote about this has been published in Physics Letters B [see paper 55 above].
The Escher woodcut [above; thanks to http://sunsite.icm.edu.pl/cjackson/escher/index.html] shows the hyperbolic plane, a "Euclidean" version of Anti-deSitter space [which is a maximally symmetric spacetime of constant negative curvature]; it is clear that it is very natural to regard this space as the interior of a manifold-with-boundary, the boundary [which is infinitely far away from an internal point of view] having the topology of a circle. [Note: in physics, "Euclidean" means "with a positive [or perhaps negative] definite metric", as opposed to the Lorentzian signature of ordinary spacetime. Thus "Euclidean" spacetimes are in fact usually curved!] This works in higher dimensions too: an (n+1) dimensional hyperbolic space has a natural "boundary" which has the topology [and conformal geometry] of an n-dimensional sphere. Now String Theory, which is the focus of virtually all work on the problem of reconciling General Relativity with quantum mechanics, works particularly well on Anti-deSitter space. The AdS/CFT correspondence, put forward by Juan Maldacena, gives a duality between a gravitational theory in the "bulk" [where the angels and demons live] and a kind of gauge theory on the spherical boundary at infinity. The illustration shows that even though the bulk has negative curvature, its boundary at infinity is *positively* curved. This is interesting in physics because, as I said, string theory seems to work best on manifolds of *negative* curvature, but astronomical observations suggest that our world is *positively* curved. [This is deduced from the fact that the expansion of the Universe is *accelerating*, one of the great discoveries in the history of astronomy.] The illustration suggests how this apparent contradiction might be resolved. Namely, it is clear that it is natural to divide the picture into an infinite set of concentric circles [spheres in higher dimensions]. So maybe our four-dimensional, positively curved Universe sits inside [as a "membrane" or "brane-world"] a five-dimensional, negatively curved space like the one above.
Getting the Universe to be "Alice-like" at one stage of its history but not later is a surprisingly delicate mathematical exercise, which is particularly subtle and interesting on the real projective space RP^3 [described above; note that RP^3 *is* orientable in the usual sense.]. I will soon be returning to this subject, because I believe that the mysterious de-localised charge ["Cheshire Charge" --- see the picture of the eponymous cat above] that one gets in Alice physics is relevant to the question of the existence of charged black holes in RP^3 de Sitter space....
The basic cosmological model corresponding to a positive cosmological constant [now apparently observed, as mentioned above, in the form of the cosmic acceleration] is deSitter space. But it is hard to obtain deSitter space from string theory. In particular, one needs to understand the "dS/CFT" correspondence better. One key to a deep understanding of AdS/CFT was Maldacena and Witten's use of "Euclidean" techniques. I want to use Euclidean techniques to investigate dS/CFT---but the problem is that the usual Euclidean version of deSitter space is a sphere, not a space like the one illustrated at the top of this page! The trick here is to realise that "Riemannian" metrics don't really have to be *positive* definite---negative definite works equally well. This apparently trivial generalisation allows you to use the picture at the top of the page for a Euclidean version of deSitter space too---but you have to use an unorthodox foliation of that picture. [Instead of the obvious foliation by spheres, you foliate by hyperbolic spaces of one dimension less---think of what you get by visualising "phases of the moon" moving from left to right across the woodcut.] The nice thing about this foliation is that the *apparently* disconnected conformal boundary of deSitter space is revealed, in the Euclidean picture, as being connected. [The point is that the slices, being hyperbolic, are "infinitely large", so they extend right out to the boundary.] This is in agreement with the Witten-Yau theorem. Thus the one-to-one bulk/boundary correspondence is maintained in the Euclidean formulation. See paper 47 above.
Evidence for a positive cosmological constant keeps piling up. To be more precise, the evidence keeps piling up for some kind of "dark energy" with a *negative* pressure which is close in absolute value to its [positive] density. Of course, to say that the ratio of pressure to density is observationally close to -1 is to say that it might be slightly greater than -1.....or maybe slightly *less* than -1. In fact, there has been a lot of work on the former, which can be modelled by a "quintessence" field, but very little on the latter. Now in fact the "less than -1" cosmologies, called "Phantom Cosmologies" by R Caldwell http://www.dartmouth.edu/~caldwell/ [who was one of the quintessence pioneers] are very interesting indeed, for several reasons.
Caldwell's *particular* phantom cosmologies have the bizarre property that they come to an end eventually, like the familiar "Big Crunch" cosmologies----but in the phantom world, the end comes not because the Universe collapses to zero size, but rather because the distance between any two given galaxies becomes infinitely large in a finite proper time! [I like to call this the "Big Smash". Caldwell's "Big Rip" would really destroy the universe, but I think that it is more likely that the universe would shatter into disconnected pieces---something more like this:
which is why I prefer "smash" to "rip". Of course, I don't know what happens to the pieces---but, since the pieces are themselves infinitely large, I would expect them to shatter too, and that the process would continue forever.] More bizarre still is the fact that in these cosmologies, the density *increases* as the Universe expands, and in fact it diverges as the Universe expands "infinitely fast". That seems a little too extravagant for me, so I have constructed a family of [extremely simple] Phantom cosmologies with no beginning or end---they are in fact asymptotically deSitter in a certain sense. Like deSitter space, they have a disconnected conformal boundary, but *unlike* deSitter space, their Euclidean versions *still* have a disconnected boundary, so they are ideal for exploring the question as to whether the one-to-one bulk/boundary correspondence can really be maintained. In fact, I claim that this disconnectedness of the Euclidean boundary is a signal that, contrary to appearances, these spacetimes are "effectively disconnected". Using recent ideas of Andrew Strominger and Vijay Balasubramanian et al, I argue that, in these spacetimes, time flows away from *both* boundaries. But there is an alternative: maybe it is OK to have a disconnected boundary if there is a quantum-mechanical "entanglement" between the conformal field theories on the two connected components. This would be analogous to Maldacena's rather amazing analysis of the quantum mechanics of asymptotically anti-deSitter black holes. If this version is correct, then the evolution of our Universe is controlled by strictly quantum-mechanical effects in theories which are defined on our *infinite* future and *infinite* past! This ties in with the recent efforts of Turok and Steinhardt et al to abolish the Big Bang and replace it with a Big Bounce. My paper on this is number 48 above. A less technical and more literate version, complete with insults directed at the Null Energy Condition, may be found at http://arxiv.org/abs/astro-ph/0210321. This is a version of the talk I gave at the Institut d'Astrophysique de Paris, which is a nice place, in July 2002.
The event horizon of a black hole is the boundary of the region from which there is no escape. In "ordinary" spacetime, that surface is a 2-dimensional sphere. In five dimensions, which is particularly interesting in string theory, it would be a 3-dimensional sphere. But in a background with a negative cosmological constant---again, this is interesting in string theory, particularly in connection with the AdS/CFT correspondence---the topology doesn't have to be spherical. However, in the context of string theory, I have shown that there is a very severe restriction on the possible topology of the event horizon---only a tiny minority of topologies are possible for a *stable* black hole. This uses a kind of string-theoretic instability discovered by Seiberg and Witten, involving the unstable production of "membranes". Now actually this kind of argument is usually regarded as dubious, because, according to Einstein's equations, the unstable production of membranes [or anything else] would be expected to modify the geometry of spacetime, and this modification might invalidate the condition for the Seiberg-Witten instability to be present----in other words, the instability would be expected to limit itself, as instabilities normally do. The amazing and beautiful thing here is that this *does not* happen---no matter what the membranes do to the geometry of spacetime, the Seiberg-Witten condition continues to hold! Only if the membranes change the *topology* of spacetime can this conclusion be avoided---but it is well known that such topology changes also lead to very unpleasant phenomena such as closed timelike worldlines.. A similar argument imposes restrictions on the structure of asymptotically deSitter versions [if any] of the spacetimes corresponding to the so-called "spacelike branes" introduced by Gutperle and Strominger; again, the conclusion holds *even if back-reaction is taken into account*. The amazing thing here is not just that such a statement is true, but also that it is possible to prove it! [It is certainly not possible to predict in detail what will happen to the spacetime geometry when back-reaction becomes important.] The proof involves an application of the concept of "enlargeable" manifolds in global differential geometry. It's very satisfying that such pure [and extremely deep] pure mathematics can be used for something so physical.
If you want to prove something about the structure of spacetime ---for example, if you want to prove the Penrose-Hawking theorems, which claim that singularities in spacetime are inevitable---then you need to assume something about the kinds of matter that can occur in the real world. For many years we were all trained to believe that if pressure can be negative at all, it cannot be very negative. The usual assumption was that, in units such that the speed of light is unity, the pressure cannot be less than (-1/3) times the density. [In these units, the ordinary pressures that you encounter are vastly smaller than this [in magnitude] so this is a very reasonable thing to claim---apart from the fact that "negative pressure" is pretty weird anyway!] But the discovery of cosmic acceleration proves that this is wrong; the dark energy has a pressure which is negative but almost equal in magnitude to its density! Question: why can't it be even lower? Answer: maybe it can! However, I have recently argued as follows. HOLOGRAPHY is the extremely fashionable idea [ see picture above! ] that [roughly speaking] claims that everything happening inside a *volume* can be understood by studying its surface. That is, the stick figure can be completely understood by studying his shadow on the wall. Clear, eh? Anyway, I claim that holography *predicts* w >= -1, where w is the pressure-to-density ratio. See paper 51 above. This paper has a good list of references for papers discussing violations of the General-Relativistic energy conditions, a field of research that is just catching on, especially in the astrophysics community, which tends to be more open-minded about such things than the mathematical physics community.
Pictures of me:
Strangely penetrating gaze of guess-who as a small child [19??]
A mountain of sand in Namibia 
Me, at summit of that mountain of sand 
Quadbike in Namibian desert--a lot of fun until I crashed it 
Pictures of Wanmei:
Photo of wanmei in Venice 
Wanmei on the Grossglocknerhochalpenstrasse 
Wanmei's First Snowball 
Wanmei In Innsbruck 
Wanmei in Kitzbuhel 
Wanmei in a Tea Plantation, Malaysia 
Wanmei enjoying a fireplace in the Tropics [Malaysia!] 
Another photo of Wanmei 
Wanmei at 3000 metres in the Dolomiti 
Wanmei in Corvara, in the Dolomiti 
Wanmei at Bled, in Slovenia 
The castle overlooking the lake at Bled 
Wanmei at San Gimignano, a beautiful small town in Tuscany 
Wanmei at one with the tourist zeitgeist in Pisa 
Wanmei at Lake Bohinj, Slovenia 
Wanmei in Siena 
Wanmei in Florence 
Wanmei with me [wearing a Hannibal Lecter hat] on the Ponte Vecchio, Florence 
Wanmei at the Tre Cime di Lavaredo, the Dolomiti 
Wanmei in Piran, Slovenia 
Another view of Piran 
Wanmei at the Weissensee, in Austria 
CROCODILES IN AUSTRIA?! 
Wanmei in Rome, at St Peter's 
Wanmei at the Colosseo 
Oops! Forgot about the swatch 
Wanmei in Indonesia 
Wanmei in Jacuzzi 
Vietnamese Wanmei 
Wanmei in Trieste 
In nearby Muggia 
In Miramare 
In Rovinj, Croatia 
Also in Rovinj 
The Pordoi Pass, near Arabba, Dolomiti 
Lake Como 
Varenna on Lake Como 
Did I mention that we liked Varenna? 
Bellagio on Lake Como, with the Hotel Metropole 
Dinner at the Metropole 
The Giant's Tooth, Italian Alps 
Courmayeur, Val d'Aosta 
Lake Garda 
Mali Losinj, Croatia 
Venice Again! 
Venice is So Cacciatore, as Granny Used to Say 
She's Been There, Done That [sort of] 
What News from the Rialto? 
Me Rowing in Austria 
Wanmei Rowing in Austria 
The Passenger 
The Weissensee at Stockenboi 
Wanmei and Friend on the Nockalmstrasse 
The Valley of the Soca, Slovenia 
Wanmei finally ceases to fear dogs! Armidale, Australia 
Inspecting a Koala 
Wanmei learns to ride a horse 
Valentine's day, Venice, 2005
Hard at work at the International Centre for Theoretical Physics, Italy 2005
The view from our apartment in the Via Zamboni, Trieste, Italy 2005
Another view 2005
Wanmei learns to skate, Weissensee, winter 2005
Getting a bit more confident 2005
View from our hotel, Weissensee, March 2005
Cold butt good...the ice was 1 metre thick on the Weissensee 2005
In Saudi Arabia...not really: the Isle of Rab, Croatia, 2005
On the car ferry, leaving Rab 
At the Plitvice lakes [world heritage site] Croatia 2005
And you thought the one in Pisa was leaning....London 2005
No, I'm not squashing her arm! Paris 2005
Lake Bohinj, Slovenia 2005
Feeding the trout in Lake Bohinj 2005
Looking guilty at Lake Bohinj 2005
Vintgar Gorge, Slovenia 2005
The Mondsee, near Salzburg, Austria 2005
This has something to do with Julie Andrews. Or so I'm told. Salzburg 2005
View from my balcony
Scenic WC in Bukit Tinggi, Sumatra, May 2006
Wanmei about to attack the Suao-Hualien Highway, Eastern Taiwan, July 2006
At Taroko Gorge, Taiwan
View From Jioufen, Taiwan
Bangkok, December 2006
Wat Phra Keao
Maya bay, Krabi, Thailand Feb 2007
Asteroid falling into sea
Sawtell, NSW, Australia, June 2007
On the beach...in winter.
Armidale, Australia, winter [June 2007]
I swear I wasn't that nervous 
The Flower Girl
Beauty and the Beast....and if anyone else calls her a beast....
It's all uphill from here on
Reload That Matrix!
Well, it's still a great picture of the bride anyway.....
Just keep your eye on the bride, ok?
Staying Cool by the Pool
Jasmine, who [with James] prevented all disasters, and to whom we are eternally grateful
A loving embrace
Wanmei from a different angle!
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