Durham E-Theses Thermodynamics of Accelerating Black Holes APPELS, MICHAEL,JOHN How to cite: APPELS, MICHAEL,JOHN (2018) Thermodynamics of Accelerating Black Holes, Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/12737/ Use policy The full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that: • a full bibliographic reference is made to the original source • a link is made to the metadata record in Durham E-Theses • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders. Please consult the full Durham E-Theses policy for further details. Academic Support Oce, Durham University, University Oce, Old Elvet, Durham DH1 3HP e-mail: [email protected] Tel: +44 0191 334 6107 http://etheses.dur.ac.uk Thermodynamics of Accelerating Black Holes Three years with the C-metric Michael Appels A thesis presented for the degree of Doctor of Philosophy Centre for Particle Theory Department of Mathematical Sciences Durham University United Kingdom July 2018 Thermodynamics of Accelerating Black Holes Three years with the C-metric Michael Appels Submitted for the degree of Doctor of Philosophy July 2018 Abstract: We address a long-standing problem of describing the thermodynamics of an accelerating black hole. We derive a standard first law of black hole thermo- dynamics, with the usual identification of entropy proportional to the area of the event horizon — even though the event horizon contains a conical singularity. We show how to generalise this result, formulating thermodynamics for black holes with varying conical deficits. We derive a new potential for the varying tension defects: the thermodynamic length, both for accelerating and static black holes. We discuss possible physical processes in which the tension of a string ending on a black hole might vary, and also map out the thermodynamic phase space of accelerating black holes and explore their critical phenomena. We then revisit the critical limit in which asymptotically-AdS black holes develop maximal conical deficits, first for a stationary rotating black hole, and then for an accelerated black hole, by taking various upper bounds for the parameters in the spacetimes presented. We explore the thermodynamics of these geometries and evaluate the reverse isoperimetric inequal- ity, and argue that the ultra-spinning black hole only violates this condition when it is nonaccelerating. Finally, we return to some of our earlier findings and adjust them in light of new results; a new expression for the mass is obtained by computing the dual stress-energy tensor for the spacetime and finding that it corresponds to a relativistic fluid with a nontrivial viscous shear tensor. We compare the holographic computation with the method of conformal completion showing it yields the same result for the mass. Declaration The work in this thesis is based on research carried out in the Department of Mathematical Sciences at Durham University. The results presented have in part already appeared in the following publications: • M. Appels, R. Gregory and D. Kubiznak, Thermodynamics of Accelerating Black Holes, Phys. Rev. Lett. 117 (2016) 131303, [1604.08812] • M. Appels, R. Gregory and D. Kubiznak, Black Hole Thermodynamics with Conical Defects, JHEP 05 (2017) 116, [1702.00490] • A. Anabalon, M. Appels, R. Gregory, D. Kubiznak, R. B. Mann and A. Övgün, Holographic Thermodynamics of Accelerating Black Holes, 1805.02687 Any section reproduced from these articles has been done so with the full consent and permission from the relevant collaborators. No part of this thesis has been submitted elsewhere for any degree or qualification. Copyright © 2018 Michael Appels. “The copyright of this thesis rests with the author. No quotation from it should be published without the author’s prior written consent and information derived from it should be acknowledged.” Acknowledgements Completing this PhD has been a long journey, and there are certain people whom I must recognise as being vital parts of that time spent. Furthermore, I would like to also take this opportunity to recognise people who I count as having contributed to this accomplishment either during or prior to my time as a postgraduate. First and foremost, I would like to extend my deepest thanks to Ruth Gregory for her invaluable guidance and supervision throughout the past three years. Ruth has given me the freedom to flourish as a physicist while giving me sound and most welcome advice in unfamiliar situations. My time as a PhD has been thoroughly exciting thanks to her, affording me countless opportunities to work alongside and interact with the brightest minds in the field, as well as providing me with near-VIP treatment at many international airport lounges. I would like to thank my collaborators Andrés Anabalón, Robert Mann, Ali Övgün, Pavel Krtouš and especially David Kubizňák, who has been extremely calm and patient and willing to explain many subtleties I had once failed to grasp. I would like to thank Aristos Donos for being my secondary supervisor and always being willing to have a quick chat. I would also like to thank Nigel Glover and Sebastián Franco for being so encouraging during my years as an undergraduate and providing me with the necessary mentoring in the lead-up to my PhD. Finally, I would like to thank the administrative and support staff both from the department in Durham and at the Perimeter Institute in Canada. On a personal level, I am extremely grateful to have such a supportive family. They have encouraged my curiosity from my earliest years all the way through to this day. I am also very fortunate to have so many kind friends, both back home in Belgium and in England from my time at Durham; thank you to all who have played a part in this journey. Leo, Omar and Akash have been amazing friends to have for the last four years, and I would most certainly not have had nearly as good a time as I have if it were not for the three of them. From Capriccio Mondays to sharing a house, trips to Canada, Denmark and Italy and an innumerable amount of hours spent helping each viii Acknowledgements other understand the minutia of theoretical physics, their collective impact on my PhD cannot be understated. Finally, perhaps the most important person to have helped me through this process, from beginning to end, Lara has been invaluable to me. I thank her for her endless patience and unwavering support, for the many adventures we shared together, but probably above all else for all the little things she probably thought went unnoticed; they didn’t. Thank you to the Science and Technology Facilities Council (STFC) for funding my research. Thank you to the Perimeter Institute for Theoretical Physics for hosting me three times during my PhD. Finally, thank you to the GGI in Florence, the NBI in Copenhagen, the Oviedo University, the FIAS in Frankfurt, the DAMTP in Cambridge and King’s College in London for hosting me at various times during this degree. Dedicated to Mum & Dad, Grandpa Ian & Granny Liz, Grandpa Bob & Granny Anne. Contents Preface 1 Conventions . 2 1 Black hole thermodynamics/mechanics 5 1.1 The laws of black hole mechanics . 5 1.2 The first law . 9 1.3 The Euclidean approach to black hole thermodynamics . 13 1.4 Incorporating Λ .............................................. 15 1.4.1 Thermodynamics of the Kerr-AdS black hole . 15 1.4.2 Pressure and volume of black hole spacetimes . 19 1.4.3 Thermodynamic stability and phase transitions. 21 2 Acceleration and the C-metric 27 2.1 Origins of the C-metric . 27 2.2 The Plebański-Demiański metric . 30 2.2.1 From Plebański-Demiański to the C-metric . 32 2.2.2 From Plebański-Demiański to the Kerr-Newman metric . 33 2.2.3 The factorised C-metric . 33 2.3 Reviewing the C-metric and its features . 36 2.3.1 The nonrotating C-metric . 36 2.3.2 Coordinate ranges and parametric restrictions . 38 2.3.3 The conical defect . 40 2.3.4 The rotating C-metric . 41 xii Contents 3 Thermodynamics of black holes with conical defects 45 3.1 Thermodynamics with conical deficits . 46 3.1.1 A concrete example — capture of a cosmic string . 48 3.2 Thermodynamic length for the rotating black hole . 49 3.3 The thermodynamic length . 52 4 Thermodynamics of accelerating black holes 55 4.1 Thermodynamics of the C-metric . 56 4.1.1 Establishing a first law . 56 4.1.2 Thermodynamics of the charged C-metric . 61 4.2 Critical behaviour of accelerating black holes . 66 4.3 Critical black holes with teardrop-shaped horizons . 72 4.3.1 Critical black hole geometries . 73 4.3.2 Thermodynamics of the critical C-metric . 77 4.3.3 Thermodynamics for the rotating C-metric . 80 5 Holographic thermodynamics of accelerating black holes 87 5.1 Mass of an accelerated black hole . 87 5.1.1 The Ashtekar-Das mass . 90 5.1.2 Holographic derivation of the mass . 91 5.2 Thermodynamics . 93 6 Concluding remarks and outlook 99 6.1 Thermodynamics of the rotating accelerating black hole . 101 Bibliography 103 Preface The 1970s brought in a new era for research into black holes by drawing parallels between the classical theory of thermodynamics and these gravitational solutions. This newly found application for a century old machinery opened many avenues for this research. Indeed, black holes have been discovered to display a wide and diverse variety of phenomena, much of which were made evident from studies into thermodynamics.