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Applications of Numerical Relativity Beyond Astrophysics

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Applications of Numerical Relativity Beyond Astrophysics Applications of Numerical Relativity Beyond Astrophysics ! This dissertation is submitted for the degree of Doctor of Philosophy SARAN TUNYASUVUNAKOOL EMMANUEL COLLEGE Supervisor: Pau Figueras January 2017 This dissertation is dedicated to my wife, Kathryn, whose love and support I have valued dearly… …and to my parents, who have been nurturing my sense of curiosity for as long as I can remember. Abstract Title: Applications of Numerical Relativity Beyond Astrophysics Author: Saran Tunyasuvunakool Numerical relativity has proven to be a successful and robust tool for non-perturbative studies of gravitational phenomena in the highly dynamical and/or non-linear regime. Perhaps the most prominent achievement in the field is the breakthrough success in simulating the merger of binary black hole systems. Gravitational waveforms resulting from these simulations serve as precise theoretical predictions of general relativity, which can be tested against observational data, such as those recently made by the LIGO experiment. This dissertation explores applications of numerical relativity which lie beyond the realm of astrophysics. One motivation for this comes from the AdS/CFT correspondence, which allows us to study strongly coupled quantum field theories by considering classical gravity with a negative cosmological constant. More concretely, we construct stationary asymptotically anti-de Sitter spacetimes by numerically solving the Einstein equations in a strongly elliptic form, subject to various boundary conditions corresponding to the physical setting of interest. Three applications of this technique are presented here. 1) A toroidal “black ring” in global AdS5, which provides a more complete phase diagram for AdS5 black holes. 2) A black hole on an AdS soliton background, which is dual to a localised ball of deconfined plasma surrounded by confined matter. 3) A rotating horizon extending to the AdS boundary, which allows us to the study the behaviour of the CFT in the presence of a rotating black hole. Outside of AdS/CFT, time-dependent numerical relativity in higher dimensions can also inform inquiries into the mathematical properties of general relativity as a theory of gravity. In particular, long, thin black hole horizons are known to be subject to the Gregory–Laflamme instability, and this is expected to result in an eventual violation of the weak cosmic censorship conjecture. A landmark simulation of the black string confirmed this in the Kaluza–Klein setting, however the generalisation of this setup to asymptotically flat black rings poses new challenges for numerical relativity. Even after a successful simulation, the resulting apparent horizons possess nontrivial geometries which are problematic for existing horizon finding methods. This dissertation also covers aspects of technical development in the GRChombo adaptive mesh refinement code which were necessary for the successful evolution and analysis of a black ring instability. Acknowledgements This dissertation is—in large parts—the product of the experience, patience and guidance of my supervisor, Pau Figueras. Through him, I have learned to become a better physicist and, ultimately, a better learner. I am also indebted to Prof. Harvey Reall for accepting me as a PhD student in the first instance, and for introducing me to Pau as I expressed an interest in numerical work. I am extremely grateful to my fellow PhD student Markus Kunesch, whose tireless effort and enthusiasm has been indispensable in riding out the rougher waves of our time-evolution work. I would also like to thank Katy Clough, Will Cook, Hal Finkel, Eugene Lim, and Ulrich Sperhake, for our thoroughly enjoyable and stimulating collaborations. My research was almost as much software engineering as it was gravitational physics, and on this side I have benefited tremendously from the friendship and the numerous, lengthy, and insightful discussions with Juha Jäykkä, Kacper Kornet, and James Briggs. Without their hard work, the smooth operation of the COSMOS supercomputer would have been impossible over the past few years. I must also acknowledge the hospitality of Hal Finkel during my visit to the Argonne Leadership Computing Facility, which was one of the most memorable trips of my PhD tenure. Throughout my time at DAMTP, I have greatly enjoyed the company of, among others, Chris Blair, Tom Eaves, Dejan Gajic, Davide Gerosa, Rahul Jha, Joe Keir, Emanuel Malek, Giuseppe Papallo, Alasdair Routh, Andrew Singleton, Carl Turner, and Helvi Witek. I am also very thankful for the longstanding friendship of Low Zhen Lin and Stephen Shaw. My PhD was supported by the European Research Council (ERC), initially through grant No. ERC-2011-StG 279363-HiDGR and later through grant No. ERC-2014-StG 639022-NewNGR. The research presented in this dissertation was made possible through the computational resources provided by the COSMOS Shared Memory system at DAMTP, University of Cambridge, operated on behalf of the STFC DiRAC HPC Facility (funded by BIS National E-infrastructure capital Grant No. ST/J005673/1 and STFC Grants No. ST/H008586/1, No. ST/K00333X/1), the SuperMike-II cluster at the Louisiana State University, and the Stampede cluster at the Texas Advanced Computing Centre (TACC), through the NSF-XSEDE Grant No. PHY-090003. Declarations This dissertation is the result of my own work and includes nothing which is the outcome of work done in collaboration except as declared below and specified in the text. It is not substantially the same as any that I have submitted, or, is being concurrently submitted for a degree or diploma or other qualification at the University of Cambridge or any other University or similar institution. I further state that no substantial part of my dissertation has already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification at the University of Cambridge or any other University or similar institution. Chapter 3 of this dissertation is based on the following co-authored publication: P. Figueras and S. Tunyasuvunakool, “CFTs in rotating black hole backgrounds,” Classical and Quantum Gravity 30 (2013) 125015 Chapter 4 of this dissertation is based on the following co-authored publication: P. Figueras and S. Tunyasuvunakool, “Localized Plasma Balls,” Journal of High Energy Physics 06 (2014) 025 Chapter 5 of this dissertation is based on the following co-authored publication: P. Figueras and S. Tunyasuvunakool, “Black rings in global anti-de Sitter space,” Journal of High Energy Physics 03 (2015) 149 Chapters 6 and 8 of this dissertation contains results from the following co-authored publications: K. Clough, P. Figueras, H. Finkel, M. Kunesch, E. A. Lim, and S. Tunyasuvunakool, “GRChombo: Numerical Relativity with Adaptive Mesh Refinement,” Classical and Quantum Gravity 32 no. 24, (2015) 245011 P. Figueras, M. Kunesch, and S. Tunyasuvunakool, “End Point of Black Ring Instabilities and the Weak Cosmic Censorship Conjecture,” Physical Review Letters 116 no. 7, (2016) 071102 Contents 1 Preliminaries 1 1.1 Introduction . 1 1.2 Gauge freedom in the Einstein equation . 3 2 Numerical Construction of Stationary Black Holes in AdS 7 2.1 The AdS/CFT correspondence . 7 2.2 The Einstein–DeTurck method . 11 2.3 Numerical methods . 14 3 CFTs on a Rotating Black Hole Background 23 3.1 Introduction . 23 3.2 Rotating boundary black holes in Poincaré AdS . 26 3.3 Numerical results . 35 3.4 Conclusions . 50 4 Localised Plasma Balls 53 4.1 Introduction . 53 4.2 Numerical setup . 56 4.3 Results . 65 4.4 Summary . 75 5 Black Rings in Global AdS 77 5.1 Introduction . 77 Contents xiii 5.2 Numerical construction of AdS black rings . 79 5.3 Calculating physical quantities . 86 5.4 Geometry . 89 5.5 Thermodynamics of AdS black holes . 94 5.6 Holographic stress tensor . 102 5.7 Summary . 107 6 Evolving Higher-Dimensional Black Holes with GRChombo 111 6.1 Introduction . 111 6.2 The CCZ4 formulation . 114 6.3 Adaptive mesh refinement and GRChombo . 119 6.4 Artificial dissipation and viscosity . 123 6.5 The modified cartoon method . 125 6.6 Technical considerations . 129 7 Finding Apparent Horizons 137 7.1 Introduction . 137 7.2 Level set parametrisation . 139 7.3 General parametrisation . 142 7.4 Interpolation of AMR data . 147 8 End Point of 5D Black Ring Instability 151 8.1 Introduction . 151 8.2 Numerical setup . 153 8.3 Analysing the black ring’s horizon geometry . 155 8.4 Summary of results . 158 9 Summary 165 References 167 Chapter 1 Preliminaries 1.1 Introduction General relativity (GR) has been an extremely successful theory of gravitation. Although the geometric nature of the Einstein equation provides an elegant and simple description of gravity, its nonlinearity means that numerical methods are often required to investigate physical phenomena in the strong-field regime. One early result of numerical relativity (NR) was Choptuik’s discovery of critical phenomena in the gravitational collapse of a massless scalar field [1]. The most noteworthy application of numerical relativity, however, is probably the simulation of black hole binaries. Initial progress in simulating black holes occurred as part of the Binary Black Hole Grand Challenge [2], and was spurred by the anticipated need for theoretical waveform predictions in a successful detection of gravitational waves at the then-upcoming Laser Interferometer Gravitational Wave Observatory (LIGO). A full ‘inspiral-merger-ringdown’ simulation of a binary black hole system only became possible after over a decade of
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