The polygon representation of three dimensional gravitation and its global properties The polygon representation of three dimensional gravitation and its global properties De polygoonrepresentatie van de zwaartekracht-theorie in drie dimensies en haar globale eigenschappen met een samenvatting in het Nederlands Proefschrift ter verkrijging van de graad van doctor aan de Universiteit Utrecht, op gezag van de Rector Magnificus, Prof. dr. W. H. Gispen, ingevolge het besluit van het College voor Promoties in het openbaar te verdedigen op maandag 18 april 2005 des middags te 14.30 uur door Zoltan´ Kad´ ar´ geboren op 26 oktober 1976 te Boedapest, Hongarije¨ Promotor: Prof. dr. G. ’t Hooft Copromotor: dr. R. Loll Instituut voor Theoretische Fysica en Spinoza Instituut Universiteit Utrecht ISBN 90-393-2698-3 Contents Preface vii 1Introductionto21gravity 1 1.1Firstorderformalism...................... 4 1.2Zerocosmologicalconstant.................. 8 1.3 Phase space reduction: second order formalism . 12 1.4Phasespacereduction:firstorderformalism......... 17 1.5Pointparticles.......................... 19 2 Polygon model 23 2.1Geometricstructures...................... 25 2.2Particles............................. 28 2.3Timeslicing,phasespace................... 30 2.4Dynamics............................ 35 2.5Vertexconditions........................ 38 2.6Thedualgraph......................... 40 2.7 One polygon tessellation OPT ................ 43 2.8Uniformizingsurfacewithoutparticles............ 46 2.8.1PropertiesofanOPT.................. 48 2.8.2Geodesicpolygon.................... 49 2.8.3TheZVCcoordinates.................. 50 2.8.4Theexchangetransition................ 51 2.8.5Lorentztransformation................. 51 2.8.6Thecomplexconstraint................ 53 2.8.7OPTwithaconcaveangle............... 56 2.8.8Multi-polygontessellation............... 57 2.9Uniformizingsurfacewithparticles.............. 59 2.10Discussion............................ 62 2.11Cosmologicalsingularities................... 65 v Contents 3 Polygon model from first order gravity 69 3.1Reductiontofinitedegreesoffreedom............ 70 3.2Gaugefixingandsymplecticreduction............ 75 3.3Statusofquantization..................... 79 Appendix 85 A.1Modelsofhyperbolicspace.................. 85 A.2 Classification of globally hyperbolic spacetimes . 86 A.3 Boost parameters from Teichmullerspace..........¨ 89 A.4Thecomplexconstraint.................... 90 A.5Eliminatingpolygonsbygauge-fixing............. 93 Bibliography 96 Samenvatting 101 Curriculum Vitae 103 vi Preface There are four fundamental interactions in nature. The electromagnetic force describes not only electric and magnetic fields, but also optical phe- nomena and other forms of radiation. The strong force, active in atomic nuclei, keeps the quarks together in protons and neutrons. Finally, the weak force is responsible for phenomena such as the beta decay of neutron into a proton while emitting an electron and a neutrino in some radioactive nuclei. These first three forces dominate small scales from millimeters to 10−16 millimeters. Together they describe the dynamical features of the so-called Standard Model. This theory requires a full use of quantum mechanics, which is very different from classical physics, like Newtonian mechanics, Maxwell’s electromagnetism or general relativity. For example, it is an es- sential feature of quantum mechanics, that simultaneous measurement on all observables of a system cannot be made with arbitrary precision, even in principle. A classical theory is deterministic: knowing all initial condi- tionsofasystemenablesustopredictitsstateinalatertimeexactly.A quantum theory is non-deterministic, it predicts probabilities for the state of a system at a later time. For the fourth force, that of gravitation, we only have a classical the- ory called General Relativity. General relativity is an extremely success- ful model of gravitation in four dimensions agreeing with all experimental data from cosmological distances to millimeter scales. So there is a classical theory describing gravitation on macroscopic scales and a quantum theory describing matter on microscopic scales. No deviations from their predic- tions have ever been observed, however, this situation cannot be logically consistent. There is one scale, called the Planck length, where effects com- ing from all theories are expected to be important. At this scale all forces should act together and therefore be in harmony with one another. It has been a long-standing open issue to unify the standard model and general relativity or to find the quantum theory of gravitation, which would predict new physics at that scale. The value of the Planck length is 10−32 millime- vii Preface ter, which is 16 orders of magnitude smaller than the best resolution we can achieve in modern particle physics, where large underground particle accelerators are being used to probe small scales using particles with the highest available energy. The Planck regime is therefore inaccessible using today’s technology. It may seem that we do not need a quantum gravity theory and there is no experimental evidence for its existence. However, we do have theoretical evidence that such a theory should exist. In quantum mechanics the position x and the momentum p of a particle cannot be measured with arbitrary precision. The more accurately its posi- tion is determined, the less precise our knowledge is about its momentum and vice versa. The uncertainty around a sharp, exact value of the location oftheparticleis∆x and the uncertainty of its momentum is ∆p. Accord- ing to quantum mechanics the product ∆x · ∆p cannot be smaller then ħ, a fundamental constant in quantum mechanics. On macroscopic scales it is a tiny number: for an object of weight 1 kg moving with 1 meter/s, the uncertainty is around 10−34 meter. Classical general relativity nevertheless can be used to measure both x and p precisely which would be a contradic- tion. This indicates that there is an inconsistency of the combined theory of quantum mechanics of matter and classical general relativity as a funda- mental theory of nature. There has to be a quantum gravity theory, which reduces to general relativity on macroscopic scales. Among others it should explain the formation and evaporation of black holes and the physics of the early universe. Physicists have been trying to construct quantum gravity since the 60’s. The situation is a bit different from the birth of fundamental theories of physics in the last couple of centuries. At that time, modifications about the current view about nature were forced upon us by numerous experimental observations. Now, for quantum gravity, the only guiding principle in the construction of the theory is that it must reduce to general relativity in the macroscopic regime. We can guess a great deal about features of quantum gravity, but the full theory itself has not yet been not found. To summarize the fundamental difficulty: the conventional quantized particle theories are formulated in a given spacetime, whereas in general relativity spacetime is an outcome of the dynamics of the theory. In general relativity, spacetime is determined or created by the type and the distribution of matter. How to reconcile these two fundamentally different approaches is a deep and fundamental problem. In modern science, if a problem turns out to be too difficult to solve, one can try to solve a simpler model of similar nature, with the intention to generalize it to the original later. In this thesis, we have chosen to re- duce the dimensionality of spacetime from four to three. That is to say, viii Preface Glue B B A O A O the number of space dimensions went from three to two. We studied grav- itating point-like particles moving in two space dimensions. For example, we have a sheet of paper, or a surface, like the surface of the Earth or a donut. The point-like particles have no size, but they have masses so they interact gravitationally. There are no such objects in the four dimensional world, since, if we try to “squeeze” matter in a too small region, it becomes a black hole. A black hole, however, is an extended object with a horizon separating its “inside” from the rest of the world. In other words, black holes have a finite size. This, however, cannot happen in three dimensions, where particles can be truly point-like. Furthermore, what is peculiar about particles living in three dimen- sions, is that they do not really feel each other in a way planets do in nature, they follow straight lines. Both space and time surrounding a point particle in three dimensions are locally flat. However, a particle creates a “cone” around itself. Consequently, the locally straight trajectory of a parti- cle may seem to be bent when regarded from a distance. This phenomenon is illustrated in the figure on the top of this page. The right side of the figure can be reproduced from the left with scissors and glue as indicated. We are at point O in the “two dimensional sea” and look to the left towards asmallshipA. There is a heavy tanker at point B. We see that the ship is goingstraightandweseeitfrombehind,butlookingtotherightwesee the front of the ship far away coming towards us! This was an illustration of the effect caused by a point-like massive particle B when Einstein’s general relativity is applied in three dimensions. An important motivation for studying three dimensional quantum grav- ity is that this theory is classically soluble. Quantization of classical systems has been studied and a general experience