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UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations UC Santa Cruz UC Santa Cruz Electronic Theses and Dissertations Title The Structure of Evaporating Black Holes Permalink https://escholarship.org/uc/item/0xg2n9g2 Author Schindler, Joseph Caulfield Publication Date 2019 Peer reviewed|Thesis/dissertation eScholarship.org Powered by the California Digital Library University of California UNIVERSITY OF CALIFORNIA SANTA CRUZ THE STRUCTURE OF EVAPORATING BLACK HOLES A dissertation submitted in partial satisfaction of the requirements for the degree of DOCTOR OF PHILOSOPHY in PHYSICS by Joseph Caulfield Schindler September 2019 The Dissertation of Joseph Caulfield Schindler is approved: Professor Anthony Aguirre, Chair Professor Howard Haber Professor Michael Dine Quentin Williams Acting Vice Provost and Dean of Graduate Studies Contents List of Figuresv Abstract vii Acknowledgments viii 1 Introduction1 2 Algorithms for the explicit computation of Penrose diagrams4 2.1 Introduction..............................5 2.2 Penrose diagrams in two and more dimensions...........8 2.2.1 Rigorous definition in two dimensions............8 2.2.2 Generalization to higher dimensions............. 11 2.3 A note about units.......................... 14 2.4 Strongly spherically symmetric spacetimes and their maximal ex- tensions................................ 15 2.4.1 Spacetimes with spherical symmetry about a fixed origin. 17 2.4.2 Coordinate naming conventions............... 17 2.4.3 Spacetimes with strong spherical symmetry........ 18 2.4.4 Horizons and blocks..................... 22 2.4.5 Ingoing/outgoing regions, Kruskal quad-blocks, and horizon vertices............................ 28 2.4.6 Block diagrams and maximal extension........... 33 2.4.7 Boundary points with undefined radius........... 35 2.4.8 True Penrose diagrams.................... 37 2.5 Explicitly computing Penrose diagrams for SSS spacetimes.... 38 2.5.1 Global tortoise function................... 38 2.5.2 Double-null and Penrose coordinates............ 42 2.5.3 Joining blocks in Penrose coordinates............ 45 2.5.4 Enumeration of the algorithm................ 46 2.6 Extension to piecewise-SSS spacetimes with null-shell junctions. 50 iii 2.6.1 Geometry of a radial null slice in SSS spacetime...... 52 2.6.2 Algorithm for implementing corner and shell junctions.. 53 2.6.3 Matter content at shell junctions.............. 57 2.6.4 Energy conservation at shell and corner junctions..... 58 2.7 Implementation and examples.................... 61 2.8 Concluding remarks.......................... 62 2.9 Appendix: Comparison to existing methods............ 69 2.10 Appendix: Unit conventions..................... 72 2.11 Appendix: Spherical symmetry with a fixed origin......... 75 2.12 Appendix: Proof of regularity at the horizons........... 78 2.13 Appendix: More properties of SSS spacetimes........... 85 3 Understanding black hole evaporation using explicitly computed Penrose diagrams 92 3.1 Introduction.............................. 93 3.2 Shell model of black hole formation and evaporation........ 99 3.3 Matter Content of Shell Models with Schwarzschild or Hayward Interior................................. 101 3.4 Diagram formalism.......................... 107 3.5 Diagrams for simple models..................... 110 3.6 Diagrams for a nonsingular model with more realistic time evolution114 3.7 Diagrams including background curvature............. 123 3.8 What is a black hole?......................... 124 3.9 Towards a self-consistent evaporation model............ 130 3.10 Possible implications for the physics of black hole evaporation.. 134 3.11 Concluding Remarks......................... 142 3.12 Appendix: Matter content of Shell Models: General Case..... 143 3.12.1 Quasistatic Contribution................... 143 3.12.2 Shell Contribution...................... 149 3.12.3 Energy Conservation and DTR............... 151 References 154 iv List of Figures 2.1 Minkowski space...........................9 2.2 Anatomy of SSS spacetimes..................... 16 2.3 Individual SSS blocks......................... 23 2.4 Conformal boundary of interior blocks............... 27 2.5 Kruskal quad-block.......................... 30 2.6 Block diagrams............................ 33 2.7 Boundary points with undefined radius............... 36 2.8 Tortoise function contour....................... 38 2.9 Tortoise function and pre-squishing function............ 39 2.10 Continuity properties of the global tortoise function........ 41 2.11 Joining blocks in Penrose coordinates................ 45 2.12 Radial null slice in SSS spacetime.................. 51 2.13 Piecewise junction schematic..................... 54 2.14 Positive mass Schwarzchild junctions................ 59 2.15 Example SSS diagrams........................ 63 2.16 Diagram detail view......................... 64 2.17 Understanding generic diagram features............... 66 2.18 Example piecewise-SSS diagrams.................. 68 2.19 Comparison to existing two-horizon methods............ 71 3.1 Conventional Penrose diagram for an evaporating black hole... 97 3.2 Schematic illustration of shell model................ 101 3.3 Matter content of the Hayward spacetime............. 103 v 3.4 Computed Penrose diagrams for a single-burst shell model.... 111 3.5 Computed Penrose diagrams for a more realistic model...... 115 3.6 Horizon detail view.......................... 121 3.7 Diagrams including background curvature............. 122 3.8 Shell collision illustration....................... 132 vi Abstract The structure of evaporating black holes by Joseph Caulfield Schindler A detailed model is constructed for the spacetime of an evaporating black hole. This model is used to analyze some persistent questions about black hole space- times in general, and to discuss the mechanism of Hawking radiation. The self- consistency of the model is studied and used to illuminate some general principles in semiclassical gravity. Along the way, a unified analysis is provided for the class of “strongly spherically symmetric” spacetimes, and a method for constructing a broad class of explicit Penrose diagrams is obtained. vii Acknowledgments I am hugely grateful to my advisor Anthony Aguirre, for his support, open- mindedness, and ability to direct our focus to the most important questions. Thank you for listening to ideas ranging from good to strange, and for encouraging me to pursue research directions which some might have dismissed. To my research group: Dominik, Amita, Ross — thank you for the many interesting discussions. The same to Cam, Mike, and Christoph, and to the many other students and faculty I’ve had the pleasure of interacting with at UCSC. Thanks to the many great teachers I have had at UCSC, from whom I have learned so much. Especially Howie, whose commitment to teaching a rigorous course is unrivaled, and to George Brown, who managed to teach me a lot without ever instructing a course I took. And finally, thanks to my former advisor Fred Ellis, who taught me how to do physics in the first place. The text of this dissertation includes reprints of the following previously pub- lished material: • J. C. Schindler and A. Aguirre. Algorithms for the explicit computation of Penrose diagrams. Class. Quant. Grav., 35(10):105019, 2018, 1802.02263. • Joseph C. Schindler, Anthony Aguirre, and Amita Kuttner. Understanding black hole evaporation using explicitly computed Penrose diagrams. 2019, 1907.04879. These publications are used with permission of all co-authors. The co-author A Aguirre listed in these publications directed and supervised the research which forms the basis for the dissertation. Both articles were written by J Schindler, and the research supporting both publications was conducted by J Schindler. viii Chapter 1 Introduction Black holes are common astrophysical objects, characterized by their strong gravitational field and extreme density. Recently, their existence has been con- firmed by increasingly direct observations [3,4]. But despite their being common from an observational perspective, a number of surprisingly basic theoretical and practical questions about black holes remain open to debate. For instance: • What is the proper definition of “black hole” within general relativity? • How should we model the spacetime structure of an evolving black hole? • In such a model, where is the physically meaningful “horizon”? • What happens inside a black hole? Is there any strange physics at the horizon? Can the process of black hole evaporation be understood locally? Does evaporation lead to a loss of information? And more ... These and related questions often have textbook answers based on oversim- plified considerations — namely, based on analysis of the Schwarzschild space- time, which represents a static and eternal black hole with unperturbed spherical 1 symmetry and a central singularity. But when more complicated dynamics are accounted for, the standard explanations break down. For instance, if a black hole is in theory allowed to accrete additional mass, then the globally defined “event horizon” (a black hole’s most popular “horizon”) has a location determined by events in the far future, and cannot be physically meaningful. Alternately, if spherical symmetry is perturbed (say by an orbit- ing accretion disk), then the location of the locally defined “apparent horizon” becomes ambiguous and impossible to asses. And if gravitational physics is as- sumed to be regulated by a Planck scale curvature cutoff, the global structure of a black hole spacetime can be drastically changed. Even the inclusion of charge
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