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Black Hole Singularities in the Framework of Gauge/String Duality By Black Hole Singularities in the Framework of Gauge/String Duality by Guido Nicola Innocenzo Festuccia Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY May 2007 @ Guido Nicola Innocenzo Festuccia, MMVII. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author ... Department of Physics 1May 18, 2007 Certified by. Hong Liu C Assistant Professor Thesis Supervisor Accepted by.............. ..............A p .m .. 2homas eytak Professor and Associate Department Head for Education ARCHIVES LIBRARIES Black Hole Singularities in the Framework of Gauge/String Duality by Guido Nicola Innocenzo Festuccia Submitted to the Department of Physics on May 18, 2007, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract In this dissertation black hole singularities are studied using the AdS/CFT correspon- dence. These singularities show up in the CFT in the behavior of finite-temperature correlation functions. A direct relation is established between space-like geodesics in the bulk and momentum space Wightman functions of CFT operators of large dimen- sions. This allows to probe the regions inside the horizon and near the singularity using the CFT. Information about the black hole singularity is encoded in the expo- nential falloff of finite-temperature correlators at large imaginary frequency. We also find a UV/UV connection that governs physics inside the horizon. For the case the bulk theory lives in 5 dimensions the dual theory is an SU(N) Yang-Mills theory on a sphere, a bounded many-body system. The signatures of the singularity we found are only present as N -+ oo. To elucidate the emergence of the singularity in the gauge theory we further study the large N limit. We argue that in the high temperature phase the theory is intrinsically non-perturbative in the large N limit. At any nonzero value of the 't Hooft coupling A, an exponentially large (in N2 ) number of free theory states of wide energy range (or order N) mix under the interaction. As a result the planar perturbation theory breaks down. We argue that an arrow of time emerges in the gauge theory and the dual string configuration should be interpreted as a stringy black hole. Thesis Supervisor: Hong Liu Title: Assistant Professor Acknowledgments It is with deep gratitude that I thank my advisor Hong Liu for his unwavering support. The many discussions held with him in the last four years constitute an important part of my education as a physicist. I will look at his unprejudiced, open attitude toward research and his relentless aim to understand completely all the facets of a problem as a source of inspiration in my future work. Many thanks also go to my friend Mauro Brigante with whom I had the priv- ilege of collaborating on many projects. My dearest friends Antonello Scardicchio and Alexander Boxer have been a constant source of support as has been Qudsia Jabeen Ejaz. The discussions I had with all of them provided both entertainment and occasions to learn new physics or to get a clearer understanding of many issues. I also want to thank all my fellow students and in particular Sergio Benvenuti, Brian Fore, Claudio Marcantonini, Rishi Sharma and Leonardo Senatore. Finally I want to congratulate the staff and faculty of CTP for providing an incredible learning envi- ronment and in particular professors Amihay Hanany, and Daniel Freedman for being good teachers and answering many questions of mine. Contents 1 Introduction 15 1.1 Parametric relations in AdS/CFT .. ....... ...... ....... .. 21 1.2 Black holes in AdS/CFT ......................... 23 1.3 Time arrow and space-like singularities ................. 30 2 AdS Black Holes and AdS/CFT at Finite Temperature 35 2.1 Black hole geometry ........................... 35 2.2 Finite temperature correlation functions in boundary theories . .. 40 2.3 AdS/CFT correspondence in the black hole background ........ 43 2.3.1 Bulk propagators ......................... 46 2.3.2 An alternative expression ................... .. 49 2.3.3 Analytic properties ........................ 51 3 Two Point Correlators in the Black Hole Background 55 3.1 Approximate expressions for Lorentzian correlation functions for small 1 55 3.1.1 An approximation for d = 4 ................... 55 3.1.2 An alternative approximation .................. 62 3.2 A "semi-classical" approximation and relation with bulk geodesics .. 63 3.2.1 A "semi-classical" approximation . .......... .. .. 64 3.2.2 Relation with bulk geodesics. ................ ... 67 3.2.3 Analytic continuation . .................. ... .. 68 3.3 Quasi-normal modes ........................... 70 3.3.1 Locations of poles in the large v limit: infinite mass black hole 73 3.3.2 Long-lived quasi-particles for strongly coupled SYM theories on S3 ....... ........ 4 Excursions Beyond the Horizon 81 4.1 Decoding the bulk geometry ....................... 81 4.1.1 UV/UV connection for physics beyond the horizon .... .. 85 4.2 Asymptotics of G+ at finite k ...................... 86 4.2.1 Light-cone limit ...... ....... ........ ..... 90 4.3 Manifestations of singularities in boundary theories .. ........ 91 4.4 Discussions: Resolution of black hole singularities at finite N ? .... 93 5 The Arrow of Time and Thermalization in Large N Gauge Theory 97 5.1 Prelude: theories and observables of interest ..... ........ 98 5.1.1 Matrix mechanical systems ........... ........ 98 5.1.2 Energy spectrum ................ ......... 99 5.1.3 Observables ................... ......... 101 5.2 Non-thermalization in perturbation theory ... ... ........ 106 5.2.1 Free theory ................... ......... 106 5.2.2 Perturbation theory . ..... ........ ........ 109 5.3 Break down of Planar perturbation theory ...... 110 5.4 Physical explanation for the breakdown of planar expansion 118 5.5 A statistical approach ..................... 122 5.6 D iscussions ................. .......... 129 6 Conclusion 133 A Appendix A 137 A.1 Bulk propagators in the Hartle-Hawking vacuum . ..... .... 137 A .2 B T Z . ...... ...... ...... ...... ..... ...... 139 A.2.1 Exact solution ........................... 139 A.2.2 Structure of poles ......................... 141 A.2.3 Asymptotic behavior of G+ .. .... ... .... .... .. 142 A.2.4 Large v limit ................... ........ 142 A.2.5 Geodesic approximation .. .... ..... ..... ..... 143 A.3 Solutions to 3.2 ............................. 144 A.4 An alternative approximation . .. ... .. ..... ..... .... 146 A.4.1 Retarded propagators ....................... 150 A.5 A more sophisticated WKB analysis .... .... .... .... ... 153 A.5.1 general remarks .... ........ .. .. .. ... ... 153 A.5.2 k=0 ........................ .. ....... 156 A.5.3 The k=0 case for large u .......... .......... 159 A.5.4 k / 0 ... ................ .. .... ..... 161 A.6 Explicit expressions of the integrals 3.42 . .... .... .. .... .. 161 A.6.1 The d= 4 case ................... .. ... 162 A.6.2 d=4andk=0............... ..... ........ 164 A.6.3 d= 3, d= 6 .................... ... .... 165 A.6.4 conventions for elliptic integrals ... ..... ........ 166 A.7 Asymptotic behaviour of Z(E, q) .................... 167 A.7.1 q=0 ..... ............................ 168 A.7.2 q 5 0 case .................. ..... ....... 169 A.7.3 The light-cone limit ........................ 175 A.8 Tortoise coordinate ...... ... ..... ..... .......... 177 B Appendix B 181 B.1 Self-energy in the real time formalism .... .. ... .. .... .. 181 B.1.1 Analytic properties of various real-time functions .. .. 181 B.1.2 Self-energy in real-time formalism .......... .. ... 183 B.2 Energy spectrum and eigenvectors of sparse random matrices ... .. 186 B.3 Single anharmonic oscillator .. ................... 187 B.4 Estimate of various quantities .. .... ........ ... .. .. 190 B.5 Some useful relations .................. .... ... .. .. 192 B.5.1 Density of states . .......................... 192 9 B.5.2 Properties of XE(e) and p,(E) . ................. 193 B.5.3 A relation between matrix elements and correlation functions in free theory . .. ...... ..... ....... ... ... 194 B.6 Derivation of matrix elements ................... ... 196 List of Figures 1-1 Penrose diagram for the AdS5 Schwarzschild black hole. Each point in the diagram represents a three sphere S3 . The radius of this sphere shrinks to zero at the past or future curvature singularities which are represented by wavy lines. The diagram is separated in four regions by the red lines representing the horizon. Near the two vertical bound- aries the spacetime is asymptotic to AdS5. The Schwarzschild time coordinate maps the region outside the horizon on the right to the real line and is constant on the blue lines. The region on the left of the diagram can be associated with a Schwarzschild time having an imagi- nary part equal to +±T- and flowing from up to down. This geometry is considered in detail in chapter 2.1 ......... ......... 23 1-2 Spacelike geodesics connecting t + 0/ and -t at the boundary. As t -- t. (black arrow) the geodesics becomes null and approaches the singularity. The extremal geodesic is dashed in figure. .......... 26 2-1 Penrose diagram for the AdS black hole. A null geodesic going from the boundary to the singularity is indicated in the figure. ........ 36 2-2 A choice of fundamental domain in the Im z - Im t plane is
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