The most beautiful thing we can experience is the mysterious. It is the source of all true art and science. He to whom this emotion is stranger, who can no longer pause to wonder and stand rapt in awe, is as good as dead: his eyes are closed. Albert Einstein (1879-1955) University of Alberta Higher-Dimensional Gravitational Objects with External Fields by Shohreh Abdolrahimi A thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Physics c Shohreh Abdolrahimi Fall 2010 Edmonton, Alberta Permission is hereby granted to the University of Alberta Libraries to reproduce single copies of this thesis and to lend or sell such copies for private, scholarly or scientific research purposes only. Where the thesis is converted to, or otherwise made available in digital form, the University of Alberta will advise potential users of the thesis of these terms. The author reserves all other publication and other rights in association with the copyright in the thesis and, except as herein before provided, neither the thesis nor any substantial portion thereof may be printed or otherwise reproduced in any material form whatsoever without the author’s prior written permission. Examining Committee Don N. Page, Physics Valeri P. Frolov, Physics Sharon Morsink, Physics Dmitri Pogosyan, Physics Eric Woolgar, Mathematics Kayll Lake, Physics, Queens University To My Parents Mansour and Rosa, My Sisters Shiva and Shahrzad Abstract This thesis summarizes a study of higher-dimensional distorted objects such as a distorted 5-dimensional Schwarzschild-Tangherlini black hole. It considers a par- ticular type of distortion corresponding to an external, static distribution of matter and fields around this object. The corresponding spacetime can be presented in the generalized Weyl form which has an R U(1) U(1) group of isometries. This × × is a natural generalization of the 4-dimensional Weyl form which was presented in the paper by Emparan and Reall [1]. In the frame of this generalized Weyl form one can derive an exact analytic solution to the Einstein equations which describes the non-linear interaction of the black hole with external matter and gravitational fields. This research focuses on the effects of such interaction on the event horizon and the interior of the black hole. A similar study was presented in the papers [2] for 4-dimensional neutral black holes, where special duality relations between a neutral black hole horizon and singularity were derived. In relation to this work it is interesting to study which properties of distorted black holes remain present in the 5-dimensional case. This thesis also gives an investigation of the d-dimensional Fisher solution which represents a static, spherically symmetric, asymptotically flat spacetime with a massless scalar field. This solution has a naked singular- ity. It is shown that the d-dimensional Schwarzschild-Tangherlini solution and the Fisher solution are dual to each other. [1] R. Emparan and H. S. Reall, Phys. Rev. D, 65, 084025 (2002). [2] V. P. Frolov and A. A. Shoom, Phys. Rev. D, 76, 064037 (2007). Acknowledgements I am very much thankful to my parents and sisters who have been always a support for me despite the great geographical distance. I passed through the difficult times with their love. I am grateful to my supervisor, Professor Don N. Page, for his advice and support during my PhD program. I have learnt enthusiasm from him. I am very much grateful to Professor Valeri P. Frolov for the things I have learnt from him, for his support, and kindness, for his advice and encouragement. I really thank him for his very interesting lectures. It has been a great privilege to be his student. I would like to thank Professor Sharon Morsink for her kindness. I thank Professor Dmitri Pogosyan for his lectures on Cosmology. I would like to thank Professor Eric Woolgar for his kindness, and some nice discussions in the quad. I am happy to have found wonderful friends in Edmonton, from whom I have also learnt many things both about physics and the world: Giang Beach, Patrick Connell, Long Dinh Dang, Dan Gorbonos, Rituparno Goswami, Stepan Grinyok, David Kubizˇn´ak, Rodrigo Rocha Cuzinatto, Neda Naseri, Elaheh Poureslami Ar- dakani, Abdorreza Samarbakhsh, Andrey Shoom, Xing Wu, Shima Yaghoobpour Tari, and Hirotaka Yoshino. I thank the members of the Theoretical Physics Institute, and other colleagues and friends: Faqir Khanna, Eric Poisson, Frans Pretorius, Suneeta Vardarajan, and Andrei Zel’nikov. I would like to thank Elizabeth Berends, Gordana Brouilette, Patty Chu, Lee Grimard, Sandra Hamilton, Linda Jacklin, Nandi Khanna, Mary Jean Smallman, Carolyn Steinborn, Ruby Swanson, Roseann Whale, Maya Wheelock and Dr. Isaac Isaac for their numerous help and support. My especial thanks goes to Sarah Derr for answering all my questions during my Ph.D. program. I also would like to thank some of my first teachers: Amir Aghamohammadi, Amir H. Fatollahi, Kamran Kaviani, Mohammad Khorrami, Farinaz Roshani, and Ahmad Shariati. Table of Contents 1 Introduction 5 2 Background Material 10 2.1 Mathematicalpreliminaries . 10 2.1.1 Manifold............................. 10 2.1.2 Curve .............................. 10 2.1.3 Function............................. 11 2.1.4 Tangentvector ......................... 11 2.1.5 One-form ............................ 12 2.1.6 Tensors ............................. 12 2.2 Curvature ................................ 13 2.2.1 Paralleldisplacement . 14 2.2.2 Covariant differentiation and Christoffel symbols . .... 16 2.2.3 Curvaturetensor . .. .. 18 2.2.4 PropertiesoftheRiemanntensor . 20 2.2.5 PropertiesoftheWeyltensor . 22 2.2.6 Energyconditions. 23 2.2.7 3-sphere ............................. 24 2.3 A review of the generalized Weyl solutions . .. 26 2.4 A distorted 4-dimensional Schwarzschild black hole . ...... 32 3 Distorted 5-dimensional vacuum black hole 39 3.1 Introduction............................... 39 3.2 5-dimensionalWeylsolution . 43 3.3 Distorted 5-dimensional vacuum black hole . ... 46 3.3.1 5-Dimensional Schwarzschild-Tangherlini black hole..... 46 3.3.2 Metric of a 5-dimensional distorted black hole . .. 48 3.4 Solution ................................. 51 3.5 Symmetries and boundary values of the distortion fields . ...... 55 3.6 Space-timenearthehorizon . 59 3.6.1 Intrinsic curvature of the horizon surface . .. 59 TABLE OF CONTENTS 3.6.2 Shapeofthehorizonsurface . 61 3.6.3 Metricnearthehorizon . 63 3.7 Space-timeinvariants . 65 3.8 Space-time near the singularity . 69 3.8.1 Metricnearthesingularity . 69 3.8.2 Stretchedsingularity . 70 3.8.3 Geometry of the stretched singularity surface: duality trans- formation ............................ 72 3.9 Proper time of free fall from the horizon to the singularity ..... 74 3.10 Summary of results and discussion . 75 4 Distorted 5-Dimensional Charged Black hole 81 4.0.1 The 5-dimensional Reissner-Nordstr¨om solution . .... 81 4.0.2 Chargingvacuumsolutions. 84 4.0.3 Distorted 5-dimensional charged black hole . .. 86 4.0.4 Dimensionlessformofthemetric . 89 4.0.5 Duality relations between the inner and outer horizons ... 90 4.1 Space-timeinvariants . 91 5 Analysis of the Fisher Solution 97 5.1 Introduction............................... 97 5.2 TheFishersolution ........................... 99 5.2.1 Metric .............................. 99 5.2.2 Duality .............................101 5.2.3 TheFisheruniverse. 105 5.3 Curvaturesingularities . 108 5.3.1 Spacetimeinvariants . 108 5.3.2 Strengthofthesingularities . 109 5.4 Causal Properties of the Fisher solution . 115 5.4.1 Closedtrappedsurfaces . 115 5.4.2 Misner-Sharpenergy . 119 5.4.3 Causalstructure . 122 5.5 Isometricembedding . 127 5.6 The Fisher spacetime and the Fisher universe . 131 5.7 Summaryanddiscussion . 133 6 Conclusion 140 A Gaussian curvatures 142 B U, W , and V near the horizon and singularity 143 B.1 Geodesics near the singularity . 146 b c b TABLE OF CONTENTS C The Einstein and the Klein-Gordon Equations 148 D The Riemann Tensor and the Kretschmann Invariant 149 List of Figures 2.1 The relation of the Cartesian coordinates (x1, x2, x3, x4) and the Hopf coordinates. A is any point on a unit sphere. The semi-axis ∂ η = 0 corresponds to the orbit of the Killing vector ∂φ , and the ∂ semi-axis η = π/2 corresponds to the orbit of the Killing vector ∂χ . 25 2.2 Distribution of Uj sources on the z-axis (a) for a uniform black string in 5-dimensions; U2 has semi-infinite rod sources which extend to z = and z = , (b) for 5-dimensional Schwarzschild black ∞ −∞ hole; U2 corresponds to a semi-infinite rod source which extends to z = , and U corresponds to a semi-infinite rod source which ∞ 3 extends to z = . .......................... 30 −∞ 2.3 Distribution of Uj sources on the z-axis (a) for black ring in 5- dimensions; U2 corresponds to a semi-infinite rod source which ex- tends to z = , and U3 corresponds to a semi-infinite rod source which extends∞ to z = , and a finite rod source, (b) for black hole −∞ plus a KK bubble; U3 has semi-infinite rod sources which extend to z = and z = . .......................... 30 ∞ −∞ 2.4 Distribution of Uj sources on the z-axis (a) for two black hole con- figuration, (b) for three black hole configuration. .. 31 2.5 Distribution of Uj sources on the z-axis for a finite number of multi blackholes(see,[36]) ......................... 31 3.1 The Hopf coordinates (λ,χ,φ). The fixed points of the Killing vec- α α tors ξ(φ) and ξ(χ) belong to the “axes” defined by λ = 0 and λ = π/2, respectively. The coordinate origin O is a fixed point of the isom- etry group U (1) U (1). Planes 1, 2, and 3, embedded into 4- χ × φ dimensional space, are orthogonal to each other. .. 47 3.2 Conformal diagram for the (ψ,θ) plane of orbits corresponding to the black hole interior.