Conserved charges, surface degrees of freedom, and black hole entropy Ali Seraj Under supervision of M.M. Sheikh-Jabbari A Thesis presented for the degree of Doctor of Philosophy arXiv:1603.02442v1 [hep-th] 8 Mar 2016 Institute for Research in Fundamental Sciences (IPM) Department of Physics Tehran, Iran February 2016 Abstract In this thesis, we study the Hamiltonian and covariant phase space description of gravitational theories. The phase space represents the allowed field configurations and is accompanied by a closed nondegenerate 2 form- the symplectic form. We will show that local/gauge symmetries of the Lagrangian formulation will fall into two different categories in the phase space formulation. Those corresponding to constraints in the phase space, and those associated with nontrivial conserved charges. We argue that while the former is related to redundant gauge degrees of freedom, the latter leads to physically distinct states of the system, known as surface degrees of freedom and can induce a lower dimensional dynamics on the system. These ideas are then implemented to build the phase space of specific gravitational systems: 1) asymptotically AdS3 spacetimes, and 2) near horizon geometries of extremal black holes (NHEG) in arbitrary dimension. In the AdS3 phase space, we show that Brown-Henneaux asymptotic symmetries can be extended inside the bulk of spacetime and hence become symplectic symmetries of the phase space. We will show that in the NHEG phase space, surface gravitons form a Virasoro algebra in four dimensions, and a novel generalization of Virasoro in higher dimensions. The central charge of the algebra is proportional to the entropy of the corresponding extremal black hole. We study the holographic description of NHEG phase space and show that the charges can be computed through a Liouville type stress tensor defined over a lower dimensional torus. We will discuss whether surface gravitons can serve as the microscopic origin of black hole entropy. Keywords: Black hole microstates, Conserved charges, Surface degrees of freedom, Symplectic mechanics, Holography Contents 1 Introduction 2 1.1 Symmetries in physics . .2 1.2 Different notions of conserved charges in Gravitational physics . .4 1.3 The mystery of black hole entropy and approaches . .5 1.4 Outline; What is new in this thesis? . .8 2 Hamiltonian formulation of gauge theories and gravity 10 2.1 Introduction . 10 2.2 Constrained Hamiltonian dynamics . 11 2.3 Hamiltonian dynamics of field theories . 17 2.4 Open spacetimes and conserved charges . 22 2.5 Hamiltonian formulation of General Relativity . 24 3 Covariant phase space formulation of gauge theories and gravity 29 3.1 Introduction . 29 3.2 Symplectic Mechanics . 30 3.3 Covariant phase space formulation of gauge theories . 34 3.4 Local symmetries and their generators . 37 3.5 Explicit form of generators and charges in gravity . 41 3.6 Discussion . 48 4 AdS3 phase space and its surface degrees of freedom 52 4.1 Introduction and outline . 52 4.2 Asymptotically AdS spacetimes . 55 iv Contents v 4.3 AdS3 phase space . 57 4.4 Symplectic symmetries and charges . 58 4.5 The two Killing symmetries and their charges . 65 4.6 Phase space as Virasoro coadjoint orbits . 69 4.7 Extremal phase space and decoupling limit . 73 4.8 Microscopic counting of entropy . 77 4.9 Discussion . 79 5 Near horizon geometries of extremal black holes 81 5.1 Introduction . 81 5.2 Extremal black holes and near horizon limit . 83 5.3 Quick Review on NHEG . 84 5.4 Global charges and first law of NHEG mechanics . 91 5.5 The Kerr/CFT correspondence . 94 5.6 No dynamics in NHEK . 96 5.7 NHEG perturbation uniqueness . 99 6 NHEG phase space and its surface degrees of freedom 102 6.1 Introduction . 102 6.2 Rationale behind the NHEG phase space . 104 6.3 Generator of infinitesimal transformations . 105 6.4 Finite transformations and phase space configurations . 108 6.5 Symplectic Structure . 112 6.6 Algebra of Charges: the \NHEG Algebra" . 115 6.7 Charges on the phase space . 117 6.8 Discussion . 120 7 Summary and Outlook 125 A Technical Proofs. 129 A.1 Proof of proposition 5 . 129 A.2 Charge expression for field dependent gauge transformations . 131 A.3 On the generator of Killing horizon in NHEG . 133 References 136 Contents vii The work in this thesis is based on the results in the following publications 1. G. Comp`ere,P. Mao, A. Seraj and M.M. Sheikh-Jabbari, \Symplectic and Killing Symmetries of AdS3 Gravity: Holographic vs Boundary Gravitons", [arXiv:1511.06079 [hep-th]]. 2. G. Comp`ere,K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, \Wiggling Throat of Extremal Black Holes," JHEP 1510, 093 (2015) [arXiv:1506.07181 [hep-th]]. 3. G. Comp`ere,K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, \Extremal Rotating Black Holes in the Near-Horizon Limit: Phase Space and Symmetry Algebra," Phys. Lett. B 749, 443 (2015) [arXiv:1503.07861 [hep-th]]. 4. K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, \Near Horizon Extremal Geom- etry Perturbations: Dynamical Field Perturbations vs. Parametric Variations," JHEP 1410, 111 (2014) [arXiv:1407.1992 [hep-th]]. 5. K. Hajian, A. Seraj and M. M. Sheikh-Jabbari, \NHEG Mechanics: Laws of Near Horizon Extremal Geometry (Thermo)Dynamics," JHEP 1403, 014 (2014) [arXiv:1310.3727 [hep-th]]. 6. K. Hajian, A. Seraj, \Symplectic structure of extremal black holes", Proceedings of the 2nd Karl Schwarzschild Meeting on Gravitational Physics, 2015, To appear Dedicated to My wife for her patience and encouragement and to my little daughter for joining our family in the last 3 months of this project Acknowledgements The work presented in this thesis was completed in the Physics department of Institute for research in fundamental sciences (IPM) during the last three years. I am extremely grateful to all who helped me finish this work and apologize anyone I have forgot to mention below. I want to specially thank Prof. M. M. Sheikh Jabbari. He was an excellent teacher, an excellent supervisor, and an excellent collaborator. I can never forget our extended discussions in his office during which I learned how to think and research in physical problems. He was also supportive in many scientific activities I had during these years. I also appreciate other professors in the department specially Y. Farzan and A. Naji for many things I learned in their lectures. I am grateful to everyone who made IPM a pleasant place for scientific activities. I specially thank N. Pilehroudi, M. Babanzadeh, S. Jam, J. Aliabadi, H. Zarei and Ms Bagheri for making the physics department a lovely place. I also thank all past and current PhD students and Postdocs at IPM including: K. Hajian, A. Mollabashi, R. Mohammadi, A. A. Abolhassani, M.H. Vahidinia, M. Noorbala, H. Afshar, H. Ebrahim, R. Fareghbal, M. Ali-Akbari, P. Bakhti, S. Sadeghian, A. Maleknejad. and B. Khavari. We had many useful discussions and spent beautiful moments together. Thanks also to the new students R. Javadinejad, E. Esmaili, H.R. Safari, J. Ebadi and M. Rajaee. During our scientific projects, I enjoyed many discussions with Kamal Hajian. Thanks to him, and his novel ideas and intuitions. I am also grateful to Hamid Afshar for great discussions we had in the course of writing this thesis. I spent 6 months at the \Physique Math´ematiquedes Interactions Fondamentales" in \Universit`eLibre de Bruxelles" between Feb. to Aug. 2015. I am specially grateful ix Contents 1 to Geoffrey Comp`erefor many things I learned from him and for the very nice collab- orations we had. I also thank all the friendly professors and students there, specially Prof. G. Barnich, and graduate students B. Oblak, V. Lekeu, P. Mao and A. Ranjbar. Special thanks go to M. F. Rogge for her great helps during my visit. In the end, I really appreciate the unconditional love from my parents and my wife and thank them for all their support and encouragement during the years. Chapter 1 Introduction 1.1 Symmetries in physics Symmetry has played a critical role in the development of modern physics. The notion of symmetry appears in different settings. Specially we would like to distinguish two notions: I) symmetry of a theory, II) symmetry of a special solution. A symmetry of a solution is a transformation which keeps the field configuration of the solution intact. In gravitational physics, these symmetries are usually called isometries and are represented by Killing vectors. The other notion of symmetry is attributed to a theory which is determined by its action (or equivalently its field equations) together with a set of initial/boundary conditions. A symmetry in this sense is a transformation in fields that does not change the action for all field configuration allowed by the initial/boundary conditions. Note that such a symmetry transformation maps a solution of the theory to another solution. A symmetry can be labeled by either a discrete or a continuous parameter (or set of parameters). While the former can also lead to charges with discrete values (like parity, time reversal, etc.) here we are interested in the latter. Continuous symmetries are best described by Lie groups, a rigorous construction in mathematics. Continuous symmetries of a theory can be either global or local. A local symmetry is by definition a transformation in fields parametrized by one or more arbitrary func- tions of spacetime, while global symmetries are specified by a set of parameters. For example any field theory in the context of Special Relativity has a global symmetry 2 1.1. Symmetries in physics 3 with Poincar´ealgebra, wherein the parameters of transformation are rotation angles, boost parameters, and displacement vectors.