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Durham E-Theses Durham E-Theses Spacelike Geodesics and Other Puzzles in the Mixmaster Universe KENWAY, ANGHARAD,SONIA How to cite: KENWAY, ANGHARAD,SONIA (2012) Spacelike Geodesics and Other Puzzles in the Mixmaster Universe, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/3457/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk Spacelike Geodesics and Other Puzzles in the Mixmaster Universe Angharad Sonia Kenway A Thesis presented for the degree of Doctor of Philosophy Centre for Particle Theory Department of Mathematical Sciences University of Durham England October 2011 Dedicated to my family Spacelike Geodesics and Other Puzzles in the Mixmaster Universe Angharad Sonia Kenway Submitted for the degree of Doctor of Philosophy October 2011 Abstract In this thesis we are going to investigate the behaviour of geodesics in a metric with a singularity known as the \Mixmaster Universe". This was motivated from previous work done, where the now well-known AdS/CFT correspondence was used to extract information about an AdS Schwarzschild black hole singularity beyond the horizon by studying correlators on the boundary that correspond to spacelike geodesics which bounce off the singularity. It was then shown that when the singularity was a cosmological one (in this case a Friedmann Robertson Walker cosmology with a Big Crunch), this was no longer possible as it is impossible for spacelike geodesics to bounce off this kind of singularity. This raises the question of whether, when an example of a more general singularity (the \Mixmaster Universe") is considered, it is possible for the spacelike geodesics to bounce away from this kind of singularity. This would enable us to potentially extract information about the singularity from the boundary correlators. Unfortunately, it will be shown that bouncing of such geodesics is extremely unlikely (if not impossible) and thus we would be unable to extract information about the singularity in the mixmaster universe using such a technique. We also discuss another aspect of the evolution of the mixmaster universe which shows that it does indeed have a very complicated evolution. Declaration The work in this thesis is based on research carried out at the Centre for Particle Theory, Department of Mathematical Sciences, Durham University, England. No part of this thesis has been submitted elsewhere for any other degree or qualification and it is all my own work unless referenced to the contrary in the text. Copyright c 2011 by Angharad Kenway. \The copyright of this thesis rests with the author. No quotations from it should be published without the author's prior written consent and information derived from it should be acknowledged". iv Acknowledgements It is a long and difficult road to a PhD and it is a journey that would not have been possible without the help of many people. Firstly, thanks goes to my supervisor Veronika Hubeny for all her help and guidance over the four years. I would also like to thank all the students and staff in the Maths Department for the many lively discussions we have had over coffee throughout the years on every possible subject. Thanks to all my family for its support especially in the last few stressful days of writing this thesis and thanks to all my friends around Durham (and beyond) for all the great times we have had in the last four years when I wasn't in the office working. An honourable mention goes to \Team Sporcle" for helping me learn so much general knowledge trivia outside the world of maths and physics. I'm sure that the knowledge of all the US state capitals, presidents and countries of the world will be invaluable one day. v Contents Abstract iii Declaration iv Acknowledgements v 1 Introduction 1 1.1 A Short Introduction to String Theory and the AdS/CFT Correspon- dence . 2 1.1.1 String Theory . 3 1.1.2 The AdS/CFT Correspondence . 4 1.2 Probing Singularities . 7 1.3 Summary . 10 2 Background 12 2.1 The Kasner Metric . 12 2.2 The Mixmaster Universe . 14 2.3 Summary . 21 3 Bouncing Geodesics in the Kasner Metric 22 3.1 A Specific Example . 25 3.2 Numerical Results . 29 3.2.1 Compactifying the Constants . 30 3.2.2 Volume of Bouncing Geodesics in Kasner Universe Varying u . 33 3.3 Spacelike Geodesics Which Are Nearly Null . 35 3.4 Summary . 36 vi Contents vii 4 Geodesics in the Mixmaster Universe 40 4.1 Modelling the Mixmaster Universe . 40 4.1.1 Matching Up the Geodesic Constants . 41 4.1.2 Transition Time Schemes . 44 4.2 Bouncing Geodesics in a \Regular" Time Transition Scheme . 46 4.3 Numerical Results . 48 4.4 Impossibility of Bouncing After the First Epoch in Regular Time Transitions . 49 4.5 A Caveat . 51 4.6 Summary . 52 5 Moving Away From \Regular" Time Transitions 53 5.1 An Example . 53 5.2 Some Inequalities . 55 5.2.1 Upper Bounds on Two Constants . 56 5.2.2 A Crude Lower Bound . 57 5.3 Epsilon Perturbations in Transition Times . 58 5.4 Plotting the Bouncing Region . 59 5.5 Initially Purely K1 Geodesics . 60 5.5.1 One Situation . 61 5.6 Average Length of Bouncing Region in Pure K1 Geodesics . 64 5.6.1 Different Initial Values of u ................... 67 5.7 Summary . 71 6 The Puzzle of a Periodic Parameter 72 6.1 Square Roots . 73 6.1.1 The Simplest Scenario . 74 6.2 The Golden Ratio . 74 6.3 How to Generate Some Periodic Values . 75 6.3.1 Largest and Smallest Periodic Values . 75 6.3.2 Generating Quadratics . 76 6.3.3 Some Observations . 78 Contents viii 6.4 Further Investigation of the Quadratics . 78 6.5 Generation Via Matrices . 79 6.6 Back to the Square Roots . 83 6.7 Near Periodic Values . 84 6.8 Summary . 85 7 Discussion 88 Bibliography 92 List of Figures 2.1 A plot of the circle of parameters fp1; p2; p3g which satisfy the condi- tions for the Kasner metric. 13 2.2 A contour plot showing the equipotential lines of the potential V (β+; β−) demonstrating the triangular symmetry of this potential. 19 3.1 A plot of the potential of a geodesic in a Kasner metric (u = 2) with K2 = 1, K1 = K3 =0. .......................... 26 3.2 A plot of the potentials of a geodesic in a Kasner metric (u = 2) with K2 = 1, K3 = 0 and small values of K1 ranging from 0.01 to 0.05. 27 3.3 A plot of the potential of a geodesic in a Kasner metric (u = 2) with K1 = 1, K2 = K3 =0. .......................... 28 3.4 A plot of the potentials of a geodesic in a Kasner metric (u = 2) with K1 = 1, K3 = 0 and small values of K2 ranging from 0.01 to 0.05. 29 3.5 A plot of the potentials of a geodesic in a Kasner metric (u = 2) with K1 = 1, K3 = 0 and small values of K2 ranging from 0.01 to 0.05 plotted near the bounce time. 30 3.6 A sketch of the region of Ki-space which will give bouncing geodesics in a Kasner metric. 31 3.7 Plots of the region of Ki space for geodesics corresponding to bouncing geodesics in a Kasner metric with parameter u where (a)u = 2, (b)u = 3, (c)u = 10, (d)u = 15. 32 −ki 3.8 A plot of various compactification functions of the form Ki = 2n ki −1 for the geodesics' constants with different values of n. 33 ix List of Figures x 3.9 A plot of the proportion of ki-space corresponding to bouncing geodesics for varying u. ............................... 34 3.10 A plot of ki-space with each point coloured by the relative path length of the corresponding geodesic. Those with the shortest path lengths are coloured black and those with longest are coloured red. 38 3.11 A sketch summarising the behaviour of geodesics in a Kasner metric with respect to bouncing and \nullness". 39 4.1 An illustration of the model of the mixmaster universe made by matching together a set of Kasner metrics at transition times ti. 41 4.2 A picture of a bouncing geodesic in the mixmaster Universe . 42 4.3 A plot of the proportion of ki space which corresponds to bouncing p geodesics for increasing numbers of epochs with initial u = 7 . 49 5.1 A plot of the potential of a geodesic with constants f0:71; 0:08; 0:1g p in a mixmaster universe with 3 epochs, with initial u = 7. 54 5.2 An example of the region in Mi-space in which it may be possible to find bouncing geodesics. 60 5.3 A sketch of a potential in a mixmaster universe with nearly regular transition times demonstrating how it may be possible for a geodesic in such a setup to bounce.
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