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NON-LINEAR ASPECTS of BLACK HOLE PHYSICS Thesis Submitted NON-LINEAR ASPECTS OF BLACK HOLE PHYSICS Thesis submitted for the degree of Doctor of Philosophy (Sc.) In Physics (Theoretical) by Arindam Lala Department of Physics University of Calcutta 2015 To The memory of my uncle Anup Kumar Lala Acknowledgments This thesis is the outcome of the immense efforts that have been paid for the last four and half years during my stay at Satyendra Nath Bose National Centre for Basic Sciences, Salt Lake, Kolkata, India. I am happy to express my gratitude to those people who have constantly supported me from the very first day of my joining at the S. N. Bose Centre. I would like to covey my sincere thanks to all of them. First of all, I would like to thank Prof. Rabin Banerjee for giving me this nice opportunity to work under his supervision. The entire research work that I did for the last four and half years is based on the platform provided by Prof. Banerjee. I am indebted to the Council of Scientific and Industrial Research (C.S.I.R.), Government of India, for providing me financial support under grant no. 09/575 (0086)/2010-EMR-I. I am also thankful to Dr. Dibakar Roychowdhury, Dr. Sunandan Gangopadhyay, and Mr. Shirsendu Dey who have actively collaborated with me in various occasions and helped me to accomplish various projects in different courses of time. I am specially thankful to Mr. Subhajit Sarkar with whom I have discussed lots of basic physics as well as condensed matter physics in various occasions. Also, the discussions with Dr. Samir Kumar Paul and Dr. Debaprasad Maity helped me a lot in gaining insight into certain physical ideas. I am glad to thank my senior group members Dr. Bibhas Ranjan Majhi, Dr. Sujoy Kumar Modak, and Dr. Debraj Roy for some fruitful discussions. I am particularly thankful to my group mates Dr. Biswajit Paul, Mr. Arpan Kr- ishna Mitra, and Ms. Arpita Mitra for their active participation in various academic discussions. I also thank the organizers of the conference The String Theory Universe, held during 22-26 September, 2014 at the Johannes Gutenberg University Mainz for giving me the opportunity to present my work and for providing me an environment to discuss with people working in the direction of my research. Finally, I would like to convey my sincere thanks to my family members, espe- cially to my parents, in-laws, and my wife, for their constant support. 2 List of publications 1. \Ehrenfests scheme and thermodynamic geometry in Born-Infeld AdS black holes," Arindam Lala, D. Roychowdhury, Phys. Rev. D 86 (2012) 084027 [arXiv[hep-th:1111.5991]]. 2. \Critical phenomena in higher curvature charged AdS black holes," Arindam Lala, AHEP Vol. 2013, 918490 (2013) [arXiv[hep-th:1205.6121]]. 3. \Holographic s-wave condensate with non-linear electrodynamics: A non-trivial boundary value problem," R. Banerjee, S. Gangopadhyay, D. Roychowdhury, Arindam Lala, Phys. Rev. D 87 (2013) 104001 [arXiv[hep-th:1208.5902]]. 4. \Holographic s-wave condensation and Meissner-like effect in Gauss-Bonnet gravity with various non-linear corrections," S. Dey, Arindam Lala, Annals of Physics 354 (2015) 165-182 [arXiv[hep-th:1306.5167]]. 5. \Magnetic response of holographic Lifshitz superconductors: Vortex and Droplet solutions," Arindam Lala, Phys. Lett. B 735 (2014) 396-401 [arXiv[hep- th:1404.2774]]. This thesis is based on all the above mentioned papers whose reprints are at- tached at the end of the thesis. NON-LINEAR ASPECTS OF BLACK HOLE PHYSICS Contents 1 Introduction . 7 1.1 Overview . 7 1.2 Outline of the thesis . 24 2 Thermodynamic Phase Transition In Born-Infeld-AdS Black Holes 28 2.1 Overview . 28 2.2 Phase transitions and their classifications in ordinary thermodynamics 30 2.3 Thermodynamic phases of the Born-Infeld AdS black hole . 31 2.3.1 Geometric structure of the black hole . 31 2.3.2 Thermodynamic variables of the black hole . 32 2.4 Study of phase transition using the Ehrenfest's scheme . 36 2.5 Study of phase transition using state space geometry . 39 2.6 Conclusive remarks . 42 3 Critical Phenomena In Higher Curvature Charged AdS Black Holes . 44 3.1 Overview . 44 3.2 Geometric and thermodynamic properties of Lovelock-Born-Infeld- AdS black holes . 45 3.2.1 Gravity action and metric structure . 45 3.2.2 Thermodynamic quantities . 48 3.3 Phase structure and stability of third order LBI-AdS black hole . 52 3.4 Critical exponents and scaling hypothesis . 59 3.4.1 Critical exponents . 59 3.4.2 Scaling laws and scaling hypothesis . 66 3.4.3 Additional exponents . 68 3.5 Conclusive remarks . 69 4 Holographic s-wave Superconductors with Born-Infeld Correction 71 4.1 Overview . 71 4.2 Ingredients to construct holographic superconductors . 73 4.3 Critical temperature for condensation . 75 4.4 Order parameter for condensation . 78 4.5 Conclusive remarks . 81 5 CONTENTS 6 5 Gauge and Gravity Corrections to Holographic Superconductors: A Comparative Survey . 84 5.1 Overview and motivations . 84 5.2 Basic set up . 86 5.3 s-wave condensation without magnetic field . 89 5.4 Magnetic response: Meissner-like effect and critical magnetic field . 96 5.5 Conclusive remarks . 101 6 Holographic Lifshitz Superconductors and Their Magnetic Re- sponse . 104 6.1 Overview . 104 6.2 Lifshitz holographic superconductors: a brief review . 105 6.3 Vortex and droplet solutions in holographic Lifshitz superconductors . 108 6.3.1 Holographic vortex solution . 108 6.3.2 Holographic droplet solution . 112 6.4 Conclusive remarks . 116 7 Summary and Outlook . 118 Bibliography . 121 Chapter 1 Introduction 1.1 Overview We live in a universe surrounded by four types of forces: weak, strong, electro- magnetic, and gravitational. While there is considerable amount of theoretical and experimental data that enables us to understand the nature of first three of these forces, there remains substantial ambiguity in understanding the true nature of the gravitational force[1]-[3]. Gravity has puzzled scientific community for centuries, and still it is a rather challenging avenue to make unprecedented expeditions. In the beginning of the twentieth century, Albert Einstein proposed a new the- ory, the Special Theory of Relativity, which changed the age-old notion of space and time. Alike Newtonian mechanics, special relativity also depicts the structure of space-time, but the inconsistency of Newtonian mechanics with the Maxwell's elec- tromagnetism was removed in the latter. Nevertheless, there is an apparent subtlety in the formulation of special relativity, it is a theory formulated entirely in inertial (i.e. non-accelerating) frames of references and it involves flat space-times having no curvature and hence there is no gravity. The notion of gravity can easily be procured by shifting to the General Theory of Relativity (henceforth GTR). In this framework, gravity emerges quite naturally from the curvature of space-time which indeed implies that gravity is inherent to space-time. The core idea of GTR is encoded in the principle of equivalence which states that, the motion of freely falling particles are the same in a gravitational field and a uniformly accelerated frame in small enough regions of space-time. In this region, it is impossible to detect the existence of gravitational filed by means of local experiments[1]. Mathematically, the curvature of space-time is described by the metric tensor. The metric tensor encapsulates all the geometric and causal structure of space-time. In the classical GTR the dynamics of metric in the presence of matter fields is described by the celebrated Einstein's equations. These equations in fact relate the curvature of space-time with energy of matter fields[1]. The most fascinating thing about GTR is that it is not only a theory of gravity. There are more than that. This theory has been verified with success through ex- periments. On the other hand, with the thrill to understand Nature, many dazzling 7 1.1. Overview 8 theories such as String Theory[4], the Standard Model of particle physics[2], etc. have been proposed. While each one of them deserves appreciation in their own right, there are close connections among GTR and these theories. As a matter of fact, classical GTR is an indispensable tool to explain many of the features of these theories. Despite the tremendous successes of GTR, a genuine quantum theory of gravity is still missing. This makes gravity more enigmatic than other forces of Na- ture, although with the advancement of our knowledge even more exciting features are expected to be disclosed in the future. There are several useful applications of GTR such as, black holes, the early universe, gravitational waves, etc. While these fall into the regime of high energy physics and astrophysics, it is worth mentioning that GTR is being profoundly applied in modern technology, such as the Global Positioning System (GPS), and many more. However, this thesis is solely devoted to the study of several crucial aspects of black hole physics which are one of the striking outcomes of the solutions of the Einstein's equations[1]. Black holes are usually formed from the gravitational collapse of dying stars. Due to the immense gravitational pull, no information can escape from within a black hole while information can enter into it. A black hole is characterized by at least one gravitational trapping surface, known as the event horizon, surrounding the re- gion of intense gravitational field. Moreover, there exists a singularity of space-time within the event horizon, guaranteed by the Hawking-Penrose singularity theorem, which arises from the geodesic incompleteness of space-time[1]. Over the past sev- eral decades, different types of black hole solutions have been formulated.
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