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Conformal Symmetry for Black Holes Subtracted Geometry & Hidden Conformal Symmetry

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Conformal Symmetry for Black Holes Subtracted Geometry & Hidden Conformal Symmetry FACULTY OF SCIENCE UNIVERSITY OF COPENHAGEN Master Thesis Isak Buhl-Mortensen Conformal Symmetry for Black Holes Subtracted Geometry & Hidden Conformal Symmetry Main supervisor: Niels Obers Co-supervisor: Jelle Hartong Submitted on October 6, 2014 Abstract In this thesis we give a brief review of modern black hole physics. We point out that a microscopic derivation of entropy for black holes, is facilitated by the mere presence of some conformal field theory, whose central charges and conformal weights give us the entropy via the Cardy formula. Ultimately we are interested in the continuation of the recently proposed subtracted geometry approach, which manifests hidden conformal symmetry for non-extremal asymptotically flat black holes. We review this approach in detail, reproducing several of the results found in the literature, amongst which is the fact that for the four-charged black holes in the N = 2 minimal supergravity model (STU model), the warp factor ∆0 may be altered to ∆ = ∆¯ + Θ G, leaving the thermodynamic potentials unchanged while altering the asymptotics. This change in warp factor maintains separability of the massless Klein-Gordon equation, and gives the radial equation a hypergeometric form; making the conformal symmetry manifest. In the literature, the minimal warp factor ∆¯ (going like r for large r) has been studied extensively, thus we focus our attention on the class of subtracted geometries with warp factors ∆ with Θ 6= 0. We identify a set of warp factors ∆NHEK;A given by Θ = Am2 + (7 − A)a2 + a2 cos2θ, for which the near-horizon extremal Kerr limit coincides with the very same limit on ∆0. This is significant as the same limit on ∆¯ failed to coincide with the limit on ∆0. In pursuit of matter supporting the subtracted geometry with ∆NHEK;A, we find matter supporting a large class of warp factors in the simpler case Θ = Θ0 (a constant for static geometries). In the rotating case we find matter for the 2 2 2 warp factor ∆− ≡ Ared = 4m ((Πc − Πs)r + 2mΠs) via a scaling limit on the original matter. It remains to find matter for ∆NHEK;A, as a scaling limit is not applicable. We also present the perspective of subtracted geometry from the dual CFT point of view, where in an interpolating black hole family, the flow from subtracted to original geometry is understood as the effect of irrelevant deformations. We conclude with an outlook, presenting several interesting future directions. Acknowledgments First and foremost I thank Niels Obers and Jelle Hartong for taking me on as a master student. They have guided me through this process, and made my first encounter with this incredibly grand subject manageable. I thank them in particular for their patience during numerable meetings. I also thank Niels Obers in particular for nominating me to the Lørup scholar stipend. The stipend has allowed me to focus entirely on my thesis during its final stage. I am very grateful for this. I would also like to thank the students at the Niels Bohr Institute for a pleasant study environment. In particular I would like to thank the basement dwellers, Callum Kift and Svavar Gunnarsson for making the last year particularly memorable. I also thank Patryk Kusek and Jochen Heinrich for sharing their enthusiasm for physics and being great company. Lastly I thank my family for their support, and Meeli for making my life in Copenhagen so much brighter. One geometry cannot be more true than another; it can only be more convenient. ∼ Henri Poincar´e Contents 1 Introduction 1 1.1 The Birth of Black Hole Physics . .1 1.2 Black Holes in General Relativity . .2 1.3 Black Hole Thermodynamics . .5 1.4 A Theory of Quantum Gravity . .9 1.5 Outline of the Thesis . 10 2 Black Holes in String Theory 13 2.1 The Low-Energy Effective Action of String Theory . 13 2.2 Supersymmetry and the BPS Bound . 15 2.3 Solitons in Supergravity . 19 2.3.1 Reissner-Nordstr¨om as a SUGRA Soliton . 19 2.3.2 More General p-brane Solitons . 21 2.4 Black Holes as Strings and Dp-branes . 23 3 Entropy Through the Looking Glass 25 3.1 Entropy Prelude . 25 3.2 Microscopic Counting for BPS Black Holes . 26 3.3 Entropy Without Counting . 27 4 Conformal Symmetry and CFTs 29 4.1 Symmetry Prelude . 29 4.2 Conformal Transformations . 29 4.3 Conformal Algebra in Two Dimensions . 31 4.4 The Energy-Momentum Tensor . 33 4.5 Approaching Operators: The OPE . 34 4.6 Quantizing a 2D CFT . 35 4.7 The Cardy Formula . 37 5 Subtracted Geometry - Hidden Symmetry 41 5.1 Introducing Subtracted Geometry . 41 5.2 The Four-Dimensional Black Holes . 42 5.3 Separability of the Massless Wave Equation . 45 5.4 Singular Points of the Radial Equation . 47 5.5 The Minimal Warp Factor ∆¯ ∼ r ......................... 49 5.6 General Warp Factors . 50 5.7 NHEK Limit on Warp Factor . 51 5.8 Asymptotic Behavior . 52 5.9 Hidden Conformal Symmetry . 55 6 Supporting Matter - Static Case 59 6.1 Einstein's Field Equations . 59 6.2 Matter Supporting the Original Geometry . 62 6.3 Matter Supporting the Minimally Subtracted Geometry . 63 6.4 Toward Matter for General Warp Factors . 64 2 6.5 The Warp Factor ∆− = Ared and its Matter . 65 6.6 Matter for General Warp Factors . 67 7 Irrelevant Deformations 69 7.1 The Static Four-Parameter Family . 69 7.2 Interpolating: Subtracted and Original Geometry . 73 7.3 Irrelevant Deformations: Summary of the Procedure . 74 7.4 Linearized Analysis for ∆¯ ............................. 75 7.5 Perturbation Theory around ∆¯ - Identifying Sources . 77 7.6 Kaluza-Klein Reduction . 78 7.7 Explicit Expressions for the 3D Fields . 79 7.8 Subtracted Solution for 3D Fields . 80 7.9 Linearizing 3D Fields . 81 7.10 Linearizing around ∆− ............................... 82 7.11 Perturbation Theory around ∆− ......................... 83 8 Supporting Matter - Rotating Case 85 8.1 Scaling Limits in the Static Case . 85 8.2 Scaling Limit for ∆¯ in the Rotating Case . 87 8.3 Scaling Limit for ∆− in the Rotating Case . 89 8.4 Geometrical Interpretation for Graviphoton . 91 8.5 No Scaling Limit for NHEK-Like Warp Factor . 92 9 Conclusion and Outlook 95 A Form Notation 99 B Dimensional Reduction 103 B.1 Motivating Dimensional Reduction . 103 B.2 Dimensional Reduction of Einstein Gravity . 104 B.3 Reduction of n-form Field Strength . 110 C The STU Model 113 C.1 A Chain of Dimensional Reduction . 113 C.2 Two Magnetic two Electric . 114 C.3 Truncated STU Model . 115 D Checking EOMs for ∆− Matter 117 E Spherical Reduction of 5D EOMs 119 1 Introduction The introduction is divided into five sections. The first gives a short historical introduction to black holes in theoretical physics, with focus on general relativity. The second section scratches the surface of the Schwarzschild solution, in a manner that allows for a fluid introduction to concepts central to black hole physics. The third gives a brief review of black hole thermodynamics, with emphasis on the questions that it allows us to ask. The fourth section sets the stage for the remainder of this thesis, briefly introducing novel ideas that may answer the questions of black hole mechanics, and resolve the microscopic structure of black holes. Finally, the last section gives an overview of the remainder of this thesis. 1.1 The Birth of Black Hole Physics The advent of black hole physics is hard to pinpoint, but perhaps it was the idea put forward by John Michell in a letter to Henry Cavendish in 1783. Michell reasoned that particles of light emitted from distant stars would slow down due to the attractive nature of gravity. He argued that in extreme cases, when the star was sufficiently dense, the escape velocity would be greater than the speed of light effectively making the star invisible to distant observers. Perhaps not surprisingly, Michell's idea did not catch on at the time, it was too wild! A lucid treatment remained out of reach until Einstein shared his new insight regarding time and space, which lead him to a novel understanding of gravity in the context of the theory of general relativity (GR) published in 1916. GR is a pillar of modern physics, and a short review of its principles is in order. GR follows from a set of postulates and principles. Primarily, Einstein postulated the constancy of the speed of light, which lead to the theory of special relativity. Unfortunately, special relativity did not incorporate gravity, ultimately it treated spacetime as a flat space- time, Minkowski-space, where coordinates and distances have their usual intuitive global aspects. He extended the theory to explain gravity by an incredibly simple, and ingenious insight, which we refer to as the equivalence principle. The equivalence principle asserts that an accelerating frame and a stationary frame in a gravitational field, are locally indistin- guishable: An observer accelerating in a space ship experiences the same "force" as another observer stationary in a gravitational field1. GR was as such born out of an adaptation of special relativity, by discarding the idea of global inertial frames, and replacing it with the notion of local inertial frames. A patchwork of such local inertial frames, translates into the abstract mathematical construction of a differentiable manifold, the curvature of which manifests itself locally as what we experience as a gravitational "force". As a side note, the mathematics used to deal with the notion of spacetime in GR, is the mathematics of differential geometry. An account of differential geometry and other important mathematical results are largely omitted in this thesis. The literature available is vast as well as brilliant, and the interested reader should not find accessibility a problem.
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