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Black Holes in Quantum Gravity Hidefumi Nomura THESIS Black Holes in Quantum Gravity 量子重力理論におけるブラックホール Hidefumi Nomura 野村英史 A doctoral dissertation submitted to Department of Physics, Waseda University March, 2007 Contents I Black Hole in Brane World Scenario 9 1 Brane World Scenario and Black Hole Production 11 1.1 Brane World Scenario . ................................. 11 1.2 Black Hole Production in Brane World Scenario ...................... 13 2 Five-Dimensional Rotating Black Hole 15 2.1 Quantum Radiation of Five-Dimensional Rotating Black Hole . ........... 15 2.1.1 Non-minimal Coupled Scalar Field in 5D Rotating Black Hole .......... 16 2.1.2 Superradiance . ................................. 18 Λ=0( =0).................................... 19 Λ > 0( = −1) .................................. 20 Λ < 0( =1) ................................... 21 2.1.3 Quantization of scalar field and particle production rate . ........... 22 2.2 Evaporation of Five-Dimensional Rotating Black Hole . ................ 27 II Canonical Approach to Quantum Gravity : Loop Quantum Gravity 35 3 Formalism of Loop Quantum Gravity 37 3.1 Hamiltonian Formulation . ................................. 38 3.1.1 (3+1) decomposition ................................ 38 3.1.2 Action of Gravitational Field . ........................... 39 3.1.3 The New Variables -from metric to connection- . ................ 41 3.1.4 Holonomy and Wilson Loop (Loop Variables) . ................ 43 3.2 The Programme of Canonical Quantization . ...................... 44 3.2.1 Cylindrical Function ................................ 45 3.2.2 Orthonormal Basis of Hilbert space . ...................... 46 3.2.3 Gauge Invariant States ............................... 47 3.2.4 Spin Network . ................................. 47 3.2.5 Diffeomorphism Invariant States .......................... 50 3.3 Quantum Geometry in Loop Quantum Gravity . ...................... 50 3 4 CONTENTS 4 Black Holes in Loop Quantum Gravity 55 4.1 Black Hole Thermodynamics and Entropy ......................... 55 4.1.1 The Laws of Black Hole Mechanics ........................ 55 4.2 Origin of Black Hole Entropy . ............................... 57 4.2.1 Isolated Horizon . ............................... 58 4.2.2 Quantum Theory of Isolated Horizon . .................... 60 Hamiltonian framework .............................. 60 Quantum Horizon Geometry . ......................... 61 Entropy Calculation . ............................... 65 4.3 Semiclassical Approach to Quantum Black Hole . .................... 67 4.3.1 Bohr’s Correspondese Principle and Quasinormal Modes of Black Hole . 68 4.3.2 Universal Area Spectrum in Single-Horizon Black Holes ............. 69 WKB Analysis for single-horizon black holes ................... 69 4.3.3 Continuous Area Spectrum in Regular Black Hole . .............. 75 Highly damped QNMs of Regular Black Hole ................... 76 4.4 Reconsideration of Black Hole Entropy . ......................... 82 4.4.1 Summary of the ABCK framework ........................ 85 4.4.2 number counting . ............................... 86 5 Summary and Conclusion 89 A Angular Functions and eigenvalues 93 Introduction The quantization of gravity is probably the most important open problem in theoretical physics. Such a theory is needed not only for conceptual reasons such as unification of quantum theory with general relativity, but also for the understanding of fundamental issues such as the origin of the Universe or space and time, and the final fate of evaporating black holes. In 1915, Albert Einstein has recognized gravity as the geometry of spacetime and proposed general relativity. General relativity has predicted properly the advance of Mercury’s perihelion, gravitational deflection of light around the sun, and so on. On the other hand, quantum field theory has achieved the remarkable experimental success as the standard model in the description of fundamental interactions except gravitational interaction. The standard model indeed provides a very good description of particle physics. General relativity and quantum mechanics have changed our understanding of Nature dramatically. Both theories have destroyed and revolutionized the picture of the world before. General relativity has modified the notions of space and time. General relativity has required the general covariance of four dimensional spacetime and disproved the notion of Newton’s absolute time and space. General relativity is the discovery that spacetime and the gravitational field are the same entity. Spacetime is itself a dynamical and physical object. In turn, Quantum mechanics has changed the notions of deterministic law, measurements, and wave and particle. The term quantum refers to discrete units assigned to certain physical quantities. In other words, the quantum world is discrete. It has revealed wave-particle duality, uncertainty principle, and probabilistic law in the microscopic world. In spite of their success, the novel modified notions of those theories don’t fit together easily. We want to combine what we have learned from the two theories, however. This is an exciting challenge in fundamental theoretical physics. Fortunately, gravity is negligible in microscopic scale. Actually, the ratio of gravity to coulomb force between two electrons 1 femto meter apart is only 10−43. Then we usually neglect the gravitational force. When should we consider quantum effects of gravity? The characteristic scale in general relativity is the Schwarzschild radius 2Gm R = . (1) Sch c2 On the other hand, the characteristic scale of the elementary particle is Compton wavelength λ = . (2) mc We set the above equations equal to each other. We can get what is called Planck length lPl, Planck time 5 6 CONTENTS tPl, and Planck mass mPl, G −33 lPl = 3 10 cm, (3) c Gh −44 tPl = 5 10 s, (4) c c m = 10−5g 1019GeV. (5) Pl G The Planck length is indeed very small. If one imagines atom to be of the size of the Moon’s orbit, Planck length would only be as small as about a tenth of the size of a nucleus. In this Planck scale we can not handle the gravitational force classically and have to consider quantum effects. We have several candidates for quantum gravity at present. Those, however, are far from the com- plete one. The main approach to quantum gravity is string theory. It is based on the idea that the problem of constructing a viable quantum theory of gravity can be solved naturally within a unification of all in- teractions. String theory suggests the existence of 10-dimensional spacetime. Motivated from this string theory, a “brane world scenario” has attracted much attention [1, 2] recently. In the brane world sce- nario our four-dimensional world could itself be a “brane” (membrane) embedded in higher spacetime. In this scenario the fundamental gravitational energy scale might be low down to O(TeV) if we admit the large scale of the extra dimensions. The size of the extra dimensions could be as large as 0.1mm compatibly with gravitational experiment. Then the hierarchy problem about the discrepancy between the Planck and the electro-weak scales could be resolved. We call this “TeV scale scenario”. From a phenomenological point of view, the most exciting possibility in such a scenario is that mini-black holes might be produced in particle colliders or be found in cosmic ray events. If such a TeV scale gravity is fundamental in nature, a black hole will be produced in two body scattering above a TeV energy scale, with a geometrical cross section of the order of Schwarzschild radius, because the Schwarzschild radius of higher-dimensional black hole becomes larger than that of four-dimensional one [4, 5]. The produced black hole will evaporate soon and emit a lot of standard particles throughout quantum radiation. These phenomena can test quantum radiation, determine the number and the scale of extra dimensions, and glimpse quantum effects of gravity. Therefore, it is important that we study the general property of a higher-dimensional black hole in advance. The first part of the thesis we focus on the evolution of a higher-dimensional black hole by Hawking radiation, especially a five-dimensional black hole. The five-dimensional rotating black hole has two rotation parameter while four-dimensional one has only one rotation parameter. This property causes an interesting observation. On the other hand, another main approach to quantum gravity is the direct quantization of general rel- ativity. This includes covariant methods such as path-integral quantization and canonical methods like the Wheeler-DeWitt approach or the more recent approach so-called loop quantum gravity. We are interested in loop quantum gravity in particular. Loop quantum gravity is an attempt to construct a mathematically rigorous, non-perturbative, background independent quantum theory of four-dimensional, Lorentzian general relativity plus all known matter in the continuum. Loop quantum gravity is a straightforward quantization of general relativity with its conventional matter couplings, so the program of loop quan- tum gravity seems to be conservative and of small ambition. No claim of being the final “Theory of everything” is made. Loop quantum gravity, however, has a radical innovative picture; it predicts the minimal structures of spacetime. In the latter part of the thesis we focus on loop quantum gravity and CONTENTS 7 its application to a black hole. A black hole is one of the most interesting objects which general rela- tivity predicts. The outstanding feature of a black hole is that it has a singularity and
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