Geometric Relativity

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GRADUATE STUDIES IN MATHEMATICS 201

Dan A. Lee 10.1090/gsm/201

Geometric Relativity

GRADUATE STUDIES IN MATHEMATICS 201

Geometric Relativity

Dan A. Lee EDITORIAL COMMITTEE Daniel S. Freed (Chair) Bjorn Poonen Gigliola Staﬃlani Jeﬀ A. Viaclovsky

2010 Mathematics Subject Classiﬁcation. Primary 53-01, 53C20, 53C21, 53C24, 53C27, 53C44, 53C50, 53C80, 83C05, 83C57.

For additional information and updates on this book, visit www.ams.org/bookpages/gsm-201

Library of Congress Cataloging-in-Publication Data Names: Lee, Dan A., 1978- author. Title: Geometric relativity / Dan A. Lee. Description: Providence, Rhode Island : American Mathematical Society, [2019] | Series: Gradu- ate studies in mathematics ; volume 201 | Includes bibliographical references and index. Identifiers: LCCN 2019019111 | ISBN 9781470450816 (alk. paper) Subjects: LCSH: General relativity (Physics)–Mathematics. | Geometry, Riemannian. | Differ- ential equations, Partial. | AMS: Differential geometry – Instructional exposition (textbooks, tutorial papers, etc.). msc | Differential geometry – Global differential geometry – Global Riemannian geometry, including pinching. msc | Differential geometry – Global differential geometry – Methods of Riemannian geometry, including PDE methods; curvature restrictions. msc | Differential geometry – Global differential geometry – Rigidity results. msc — Differential geometry – Global differential geometry – Spin and Spin. msc | Differential geometry – Global differential geometry – Geometric evolution equations (mean curvature flow, Ricci flow, etc.). msc | Differential geometry – Global differential geometry – Lorentz manifolds, manifolds with indefinite metrics. msc | Differential geometry – Global differential geometry – Applications to physics. msc | Relativity and gravitational theory – General relativity – Einstein’s equations (general structure, canonical formalism, Cauchy problems). msc | Relativity and gravitational theory – General relativity – Black holes. msc Classification: LCC QC173.6 .L44 2019 | DDC 530.1101/516373–dc23 LC record available at https://lccn.loc.gov/2019019111

Copying and reprinting. Individual readers of this publication, and nonproﬁt libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Requests for permission to reuse portions of AMS publication content are handled by the Copyright Clearance Center. For more information, please visit www.ams.org/publications/pubpermissions. Send requests for translation rights and licensed reprints to [email protected]. c 2019 by the author. All rights reserved. Printed in the United States of America. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at https://www.ams.org/ 10987654321 242322212019 For my parents, Rupert and Gloria Lee

Contents

Preface ix

Part 1. Riemannian geometry

Chapter 1. Scalar curvature 3 §1.1. Notation and review of Riemannian geometry 3 §1.2. A survey of scalar curvature results 17

Chapter 2. Minimal hypersurfaces 23 §2.1. Basic deﬁnitions and the Gauss-Codazzi equations 23 §2.2. First and second variation of volume 26 §2.3. Minimizing hypersurfaces and positive scalar curvature 38 §2.4. More scalar curvature rigidity theorems 54

Chapter 3. The Riemannian positive mass theorem 63 §3.1. Background 63 §3.2. Special cases of the positive mass theorem 76 §3.3. Reduction to Theorem 1.30 86 §3.4. A few words on Ricci ﬂow 104

Chapter 4. The Riemannian Penrose inequality 107 §4.1. Riemannian apparent horizons 107 §4.2. Inverse mean curvature ﬂow 121 §4.3. Bray’s conformal ﬂow 142

Chapter 5. Spin geometry 159

vii viii Contents

§5.1. Background 159 §5.2. The Dirac operator 166 §5.3. Witten’s proof of the positive mass theorem 169 §5.4. Related results 175 Chapter 6. Quasi-local mass 181 §6.1. Bartnik mass and static metrics 181 §6.2. Bartnik minimizers 187 §6.3. Brown-York mass 193 §6.4. Bartnik data with η = 0 199

Part 2. Initial data sets Chapter 7. Introduction to general relativity 207 §7.1. Spacetime geometry 207 §7.2. The Einstein ﬁeld equations 214 §7.3. The Einstein constraint equations 221 §7.4. Black holes and Penrose incompleteness 228 §7.5. Marginally outer trapped surfaces 240 §7.6. The Penrose inequality 249 Chapter 8. The spacetime positive mass theorem 255 §8.1. Proof for n<8 256 §8.2. Spacetime positive mass rigidity 275 §8.3. Proof for spin manifolds 275 Chapter 9. Density theorems for the constraint equations 285 §9.1. The constraint operator 285 §9.2. The density theorem for vacuum constraints 292 §9.3. The density theorem for DEC (Theorem 8.3) 295 Appendix A. Some facts about second-order linear elliptic operators 301 §A.1. Basics 301 §A.2. Weighted spaces on asymptotically ﬂat manifolds 318 §A.3. Inverse function theorem and Lagrange multipliers 337 Bibliography 343 Index 359 Preface

The mathematical study of general relativity is a large and active field. This book is an attempt to introduce students to just one part of this field. Specifically, as the title suggests, this book deals primarily with problems in general relativity that are essentially geometric in character, meaning that they can be attacked using the methods of Riemannian geometry and partial differential equations. However, since there are still so many topics that match this description, we have chosen to further narrow the focus of this book to the following concept. This book is primarily about the positive mass theorem and the various ideas that surround it and have grown from it. It is about understanding the interplay between mass, scalar curvature, minimal surfaces, and related concepts. Many geometric problems in general relativity specialize to problems in pure Riemannian geometry. The most famous of these is the positive mass theorem, first proved by Richard Schoen and Shing-Tung Yau in 1979 [SY79c, SY81a], and later by Edward Witten using an unrelated method [Wit81]. Around two decades later, Gerhard Huisken and Tom Ilma- nen proved a generalization of the positive mass theorem called the Pen- rose inequality [HI01], which was later proved using a different approach by Hubert Bray [Bra01]. The goal of this book is to explain the background context and proofs of all of these theorems, while introducing various related concepts along the way. Unfortunately, there are many topics and results that would fit together nicely with the material in this book, and an argument could certainly be made that they belong in this book, but for one reason or another, we had to leave them out. At the top of the wish list for topics we would have liked to include are: a thorough discussion of the Jang equation as in [SY81b, Eic13, Eic09, AM09], a

ix x Preface complete proof of the rigidity of the spacetime positive mass theorem as in [BC96, HL17] (see Section 8.2), compactly supported scalar curvature deformations as in [Cor00, CS06, Cor17] (see Theorems 3.51 and 6.14), and a tour of constant mean curvature foliations and their relationship to center of mass [HY96, QT07, Hua09, EM13]. The main prerequisite for this book is a working understanding of Riemannian geometry (from books such as [Cha06, dC92, Jos11, Lee97, Pet16, Spi79]) and basic knowledge of elliptic linear partial differential equations, especially Sobolev spaces (various parts of [Eva10,GT01,Jos13]). Certain facts from partial differential equations are recalled in the Appendix, with special attention given to the topics which are the least “standard”— most notably the theory of weighted spaces on asymptotically flat manifolds. A modest amount of knowledge of algebraic topology is assumed (at the level of a typical one-year graduate course such as [Hat02,Bre97]) and will typically only be used on a superficial level. No knowledge of physics at all is required. In fact, the book has been structured in such a way that Part 1 contains almost no physics. Although the Riemannian positive mass theorem was originally motivated by physical considerations, it is the author’s conviction that it eventually would have been discovered for purely mathematical reasons. Part 2 includes a short crash course in general relativity, but again, only the most shallow understanding of physics is involved. Despite the level of prerequisites, this book is still, unfortunately, not self-contained. We will typically skip arguments that rely on a large body of specialized knowledge (e.g., geometric measure theory). More generally, there are many places in the book where we only give sketches of proofs. This is sometimes because the results draw upon a wide variety of facts in geometric analysis, and it is not realistic to include all relevant background material. In other cases, it is because our goal is less to give a complete proof than to give the reader a guide for how to understand those proofs. For example, we avoid the most technical details in the two proofs of the Penrose inequality in Chapter 4, partly because the author has little to offer in terms of improved exposition of those details. The interested reader can and should consult the original papers [HI01, Bra01, BL09]. Since this book is intended to be an introduction to a field of active research, we are not shy about presenting statements of some theorems without any proof at all. We hope that this will help the reader to understand the current state of what is known and offer directions for further study and research. In order to simplify the discussion, most definitions and theorems will be stated for manifolds, metrics, functions, vector fields, etc., which are smooth. Except where explicitly stated otherwise, the reader should assume that everything is smooth. (Despite this, because of the use of elliptic theory, Preface xi we will of course still need to use Sobolev spaces for our proofs.) The reason for this is to prevent having to discuss what the optimal regularity is for the hypotheses of each theorem. The reader will have to refer to the research literature if interested in more precise statements. When we refer to concepts or ideas that are especially common or well known, instead of citing a textbook, we will sometimes cite Wikipedia. The reasoning is that in today’s world, although Wikipedia is rarely the best source, it is often the fastest source. Here, the reader can get a quick introduction (or refresher) on the concept and then seek a more traditional mathematical text as desired. These citations will be marked with the name of the relevant article. For example, the citation [Wik, Riemannian geometry] means that the reader should visit

http://en.wikipedia.org/wiki/Riemannian geometry.

There are many exercises sprinkled throughout the text. Some of them are routine computations of facts and formulas that are used heavily throughout the text. Others serve as simple “reality checks” to make sure the reader understands statements of definitions or theorems on a basic level. Finally, there are some exercises (and “check this” statements) that ask the reader to fill in the details of some proof—these are meant to mimic the sort of routine computations that tend to come up in research. The motivation for writing this book came from the fact that, to the author’s knowledge, there is no graduate-level text that gives a full account of the positive mass theorem and related theorems. This presents an un- necessarily high barrier to entry into the field, despite the fact that the core material in this book is now quite well understood by the research com- munity. A fair amount of the material in Part 1 was presented as a series of lectures during the Fall of 2015 as part of the General Relativity and Geometric Analysis seminar at Columbia University. I would like to thank Hubert Bray, who is the person most responsible for shepherding me into this field of research. He taught me much of what I know about the subject matter of this book and strongly shaped my intuition and perspective. He also encouraged me to write this book and came up with the title. I thank Richard Schoen, my doctoral advisor, for teaching me about geometric analysis and supporting my research in geometric relativity. I have also learned a great deal about this subject from him through many private conversations, unpublished lecture notes, and talks I have attended over the years. Similarly, I thank my other collaborators in the field, who have taught me so much throughout my career: André Neves, Jeffrey Jauregui, Christina Sormani, Michael Eichmair, Philippe LeFloch, and especially Lan-Hsuan Huang, who kindly discussed certain technical issues related to this book. xii Preface

I also thank Mu-Tao Wang for inviting me to give lectures at Columbia on the positive mass theorem at the very beginning of this project, and Greg Galloway for explaining to me various things that made their way into the introduction to general relativity in Part 2. Indeed, the exposition there owes a great deal to his excellent lecture notes [Gal14]. I thank Pengzi Miao for some helpful conversations while writing this book, as well as the anonymous reviewers who oﬀered constructive feedback on an earlier draft. As an undergraduate, I wrote my senior thesis on Witten’s proof of the positive mass theorem under the direction of Peter Kronheimer, and in some sense this book might be thought of as the culmination of that project, which began nearly two decades ago.

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ADM energy-momentum, 225, 257 cosmic censorship, 238 ADM mass, 68, 72–74, 91, 226 apparent horizon d.o.c., see also domain of outer in initial data sets, 228, 240, 245–247 communication Riemannian, 107, 109, 110, 112 DEC, see also dominant energy asymptotically flat, 66 condition initial data sets, 225 deformation vector field, 27 axisymmetric, 81, 220 density theorem for DEC, 295 background metric, 12 for vacuum initial data, 292 nonnegative scalar curvature case, Bartnik mass, 182 101 Bianchi identities, 11, 12 scalar-flat case, 89 black hole, 228 Dirac operator, 166 Bochner formula, see also Weitzenböck divergence, 9, 25 formula divergence theorem, 10 Bondi mass, see also Trautman-Bondi domain of outer communication, 228 mass dominant energy condition, 222, 223 boost, 209 Bray flow, 142 Einstein constraint equations, see also Brown-York mass, 197 constraint equations Einstein equations, see also Einstein Cauchy hypersurface, 213 field equations causal, 211 Einstein field equations, 214 causal future, 211 Einstein tensor, 12 causal structure, 211 Einstein-Hilbert action, 216 Clifford algebra, 161 elliptic estimates, 303, 304 coframe, 4 enclosed region, 109 conformal, 21 enclosing, 109 conformal Laplacian, 22 enclosing boundary, 109 conformally flat, 77, 83 exceptional set, 316 constraint equations, 221 constraint operator, 285 first variation of mean curvature, 32 modified, 296 first variation of volume, 28, 29, 32

359 360 Index

frame, 4 Minkowski space, 208 Fredholm, 306 MOTS, see also marginally outer Fredholm index, see also index trapped surface

Gauss curvature, 12 NEC, see also null energy condition Gauss equation, 25, 26 null, 210 Gauss-Bonnet Theorem, 17 null energy condition, 232 Gauss-Codazzi equations, 25 null expansion, 230, 233 Geroch monotonicity, 123, 132 null generators, 230 globally hyperbolic, 213 null hypersurface, 213 graphical hypersurfaces, 78, 119 null second fundamental form, 233

Hölder inequality, 320 outermost minimal hypersurface, 109 harmonic functions, 10, 315 outward minimizing, 109 harmonic polynomial, 315 Hawking area theorem, 240 Penrose incompleteness, 234 Hawking mass, 121 Penrose inequality, 113, 121, 249 Hodge Laplacian, 186 perimeter, 109 Hopf maximum principle, see also Peterson-Codazzi-Mainardi equation, 25 maximum principle Poincaré group, 209 Poisson kernel, 317 index, 306 principal eigenfunction, 307 index form, 15 principal eigenvalue, 307 initial data set, 223 inverse mean curvature flow, 121, 123, quasi-local mass, 121 125 Raychaudhuri equation, 231 isotopic, 27 Rayleigh quotients, 308 Kähler, 84 Rellich-Kondrachov compactness, 321 Kelvin transform, 318 Riccati equation, 231, 232 Kerr spacetime, 220 Ricci curvature, 12 Killing field, 8 Ricci flow, 48, 97, 104, 117 Krein-Rutman Theorem, 310 Riemann curvature tensor, 11 Kruskal-Szekeres, 219 Riemannian case, see also time-symmetric Laplace-Beltrami operator, 10 Laplacian, 10 scalar curvature, 12, 14, 17 Legendre polynomials, 317 Schrödinger-Lichnerowicz formula, 166, Levi-Civita connection, 8 277 Lichnerowicz formula, see also Schwarzschild Schrödinger-Lichnerowicz formula space, 63, 66 Lie derivative, 7 spacetime, 217 linearization, 26 second fundamental form, 23 Lorentz transformations, 208 null, 230 Lorentzian, 207 second variation of volume, 31–33, 35 sectional curvature, 12 marginally outer trapped surface, 234, shape operator, 23 240 null, 230 maximum principle, 302 shear scalar, 231 mean curvature, 24 Sobolev embedding, 320 min-max, 61 spacelike, 210 minimal, 29 spacelike hypersurface, 213 minimizing hull, 109 spacetime, 211 Index 361

special relativity, 208 spectral theorem, 313 spherical harmonics, 315 spherically symmetric, 63, 77, 251 spinor, 164 spinors, 161 stability inequality, 33, 34 stability operator for minimal hypersurfaces, 33 for MOTS, 243 stable minimal submanifold, 33 MOTS, 243 static, 214 stationary, 220 stress-energy tensor, 214 strong maximum principle, see also maximum principle time-symmetric, 225 timelike, 210 timelike hypersurface, 213 trapped surface, 234 Trautman-Bondi mass, 250 two-sided, 24 uniformization theorem, 21 vacuum, 216, 223 static, 184, 218

Wang-Yau mass, 198 Weitzenb¨ock formula, 48, 96, 185 Weyl tensor, 83 Willmore inequality, 122

Yamabe positive, 18 Yamabe problem, 21 zonal harmonic, 317

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For a complete list of titles in this series, visit the AMS Bookstore at www.ams.org/bookstore/gsmseries/. Many problems in general relativity are essentially geometric in nature, in the sense that they can be understood in terms of Riemannian geometry and partial differential equations. This book is centered around the study of mass in general relativity using the techniques of KISQIXVMGEREP]WMW7TIGM½GEPP]MXTVSZMHIWEGSQTVILIRWMZIXVIEXQIRXSJXLITSWMXMZIQEWW theorem and closely related results, such as the Penrose inequality, drawing on a variety of tools used in this area of research, including minimal hypersurfaces, conformal geometry, MRZIVWI QIER GYVZEXYVI ¾S[GSRJSVQEP ¾S[WTMRSVW ERH XLI (MVEG STIVEXSVQEVKMREPP] SYXIVXVETTIHWYVJEGIWERHHIRWMX]XLISVIQW8LMWMWXLI½VWXXMQIXLIWIXSTMGWLEZIFIIR gathered into a single place and presented with an advanced graduate student audience in mind; several dozen exercises are also included. The main prerequisite for this book is a working understanding of Riemannian geometry and basic knowledge of elliptic linear partial differential equations, with only minimal prior knowledge of physics required. The second part of the book includes a short crash course SRKIRIVEPVIPEXMZMX][LMGLTVSZMHIWFEGOKVSYRHJSVXLIWXYH]SJEW]QTXSXMGEPP]¾EXMRMXMEP data sets satisfying the dominant energy condition.

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