Exact Solutions and Black Hole Stability in Higher Dimensional Supergravity Theories

Exact Solutions and Black Hole Stability in Higher Dimensional Supergravity Theories by Sean Michael Anton Stotyn A thesis presented to the University of Waterloo in fulfillment of the thesis requirement for the degree of Doctor of Philosophy in Physics arXiv:1209.2726v1 [hep-th] 12 Sep 2012 Waterloo, Ontario, Canada, 2012 c Sean Michael Anton Stotyn 2012 I hereby declare that I am the sole author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. iii Abstract This thesis examines exact solutions to gauged and ungauged supergravity theories in space-time dimensions D ≥ 5 as well as various instabilities of such solutions. I begin by using two solution generating techniques for five dimensional minimal ungauged supergravity, the first of which exploits the existence of a Killing spinor to gener- ate supersymmetric solutions, which are time-like fibrations over four dimensional hyper- Kählerbase spaces. I use this technique to construct a supersymmetric solution with the Atiyah-Hitchin metric as the base space. This solution has three independent parameters and possesses mass, angular momentum, electric charge and magnetic charge. Via an analysis of a congruence of null geodesics, I determine that the solution contains a region with naked closed time-like curves. The centre of the space-time is a conically singular pseudo- horizon that repels geodesics, otherwise known as a repulson. The region exterior to the closed time-like curves is outwardly geodesically complete and possesses an asymptotic region free of pathologies provided the angular momentum is chosen appropriately. The second solution generating technique exploits a hidden G2 symmetry in five dimensional minimal supergravity. I use this hidden symmetry approach to construct the most general black string solution in five dimensions, which is endowed with mass, angular momentum, linear momentum, electric charge and magnetic charge. This general black string satisfies the first law of thermodynamics, with the Bekenstein-Hawking entropy being reproduced via a microstate counting in terms of free M-branes in the decoupling limit. Furthermore it reduces to all previously known black string solutions in its various limits. A phase diagram for extremal black strings is produced to draw conclusions about extremal black rings, in particular why supersymmetric black rings exhibit a lower bound on the electric charge. The same phase diagram further suggests the existence of a new class of supersymmetric black rings, which are completely disconnected from the previously known class. A particular limit of this general black string is the magnetically charged black string, whose thermodynamic phase behaviour and perturbative stability were previously studied but not very well understood. I construct magnetically charged topological solitons, which I then show play an important role in the phase structure of these black strings. Topolog- ical solitons in Einstein-Maxwell gravity, however, were previously believed to generically correspond to unstable \bubbles of nothing" which expand to destroy the space-time. I show that the addition of a topological magnetic charge changes the stability properties of these Kaluza-Klein bubbles and that there exist perturbatively stable, static, magnetically charged bubbles which are the local vacuum and the end-point of Hawking evaporation of magnetic black strings. v In gauged supergravity theories, bubbles of nothing are stabilised by the positive energy theorem for asymptotically anti-de Sitter space-times. For orbifold anti-de Sitter space- times in odd dimensions, a local vacuum state of the theory is just such a bubble, known as the Eguchi-Hanson soliton. I study the phase behaviour of orbifold Schwarzschild-anti- de Sitter black holes, thermal orbifold anti-de Sitter space-times, and thermal Eguchi- Hanson solitons from a gravitational perspective; general agreement is found between this analysis and the previous analysis from the gauge theory perspective via the AdS/CFT correspondence. I show that the usual Hawking-Page phase structure is recovered and that the main effect of the soliton in the phase space is to widen the range of large black holes that are unstable to decay despite the positivity of their specific heat. Furthermore, using topological arguments I show that the soliton and orbifold AdS geometry correspond to a confinement phase in the boundary gauge theory while the black hole corresponds to a deconfinement phase. An important instability for rotating asymptotically anti-de Sitter black holes is the superradiant instability. Motivated by arguments that the physical end point of this instability should describe a clump of scalar field co-rotating with the black hole, I construct asymptotically anti-de Sitter black hole solutions with scalar hair. Perturbative results, i.e. low amplitude boson stars and small radius black holes with low amplitude scalar hair, are presented in odd dimensions relevant to gauged supergravity theories, namely D = 5; 7. These solutions are neither stationary nor axisymmetric, allowing them to evade the rigidity theorem; instead the space-time plus matter fields are invariant under only a single helical Killing vector. These hairy black holes are argued to be stable within their class of scalar field perturbations but are ultimately unstable to higher order perturbative modes. vi Acknowledgements It is with great delight that I acknowledge the numerous people throughout my academic career that helped make the production of this thesis possible. First and foremost, I would like to thank my Advisor, Robert Mann, who has selflessly given much time and devotion to training me to become an independent researcher and future colleague. Without his continued guidance, support, and friendship, I would likely not have had the patience and courage to persist in the academe. Similarly I would like to express sincere gratitude toward my undergraduate mentors, Kristin Schleich and Don Witt, for their continued support and friendship, especially during periods of hardship and uncertainty. Thanks are due to my first academic supervisor, Jasper Wall, whom I worked for as a summer student at the end of my third year of undergraduate studies and who taught me important lessons about the nature and frustration of academic research. Furthermore my gratitude extends to Don Marolf, who supervised my six month visit to the University of California at Santa Barbara during the first half of 2010. In addition to those who have directly advised me in my research, Robert Myers, Raphael Sorkin, and Eric Poisson deserve special thanks for their service as members of my Ph.D. Advisory Committee. The work presented in Chapter 3 of this thesis would not have been possible without the collaboration of Geoffrey Compère,Sophie de Buyl, and Amitabh Virmani; I worked very closely with Geoffrey and Sophie during my stay at the University of California at Santa Barbara, while Amitabh was collaborating with us from l'UniversitéLibre de Bruxelles in Belgium. I am especially thankful to Geoffrey and Sophie for their guidance and support as well as their friendship. Similarly, the work presented in Chapter 6 would not have been possible without the devoted collaborative efforts of Paul McGrath and Miok Park. Although I very much spear-headed the project, their hard work and insights at various stages were indispensable to the timely outcome of the results. There are numerous peers in a context other than collaborative work who deserve named recognition for their inspiration, guidance and friendship throughout my graduate studies: Chanda Prescod-Weinstein and Jonathan Hackett, for their support and advice in academic and non-academic related matters respectively, and Paul McGrath, Miok Park, Eric Brown, Razieh Pourhassan, Danielle Leonard, Wilson Brenna, Uzair Hussein, and Aida Ahmadzadegan for their friendship, which has not only fostered my passion for physics by allowing an outlet to talk about it with people who will actually understand but has also provided necessary distraction from physics when needed. An apology is also due to Eleanor Kohler, Stephanie Crawford, Gregory Lyons, and any other non-physics person who has been the victim of seemingly boring and abstract physics talk during social outings. vii I certainly would not have gotten this far without the love and support of family and friends. For their continual support in the face of not understanding a single thing I do, I would like to thank my mother, Angela, my father, Dwayne, my two sisters, Jennifer and Michelle, my grandparents, Anne, Virginia, and Richard, my aunts, Pamela, Brenda, Candy, and Barb, and my uncles, Brad, Brent, Brad, and Terry. I am furthermore grateful to Gregory Lyons for the love, comfort, and steady support that only a partner can provide, Kevin Nixon for being a best friend and all around support structure, Andrew Zurell for being essentially the only non-physics friend who has shown a genuine interest in and a sincere attempt to understand the work that I do, Ricardo Fonseca for being an instant, inseparable, and true friend, Audrey Lopez and Jack Loehr for being needed silly distractions from reality, Simon Reid for being an incredible roommate and having an ability to match my sense of humour perfectly, Dwayne Appleby for mutual academic inspiration in vastly disparate fields, and Michael Kushnir for simply understanding me through and through. Finally, I am grateful to my love affair with art for keeping me sane and balanced; it has added another dimension of colour and beauty to the world that I simply could not live without. viii Dedication This is dedicated to the minority groups who are underrepresented in the physics com- munity. ix Table of Contents List of Figures xv 1 Introduction 1 1.1 A Brief Review of Quantising Gravity . 1 1.2 Supersymmetry and Supergravity . 2 1.2.1 Five Dimensional Minimal Supergravity .

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