Lectures on Mathematical Relativity: (Series Number 66 Mathematical Sciences Research Institute Publications)

This book introduces and explores some of the deep connections between Einstein's theory of gravitation and differential geometry. As an outgrowth of graduate summer schools, the presentation is aimed at graduate students in mathematics and mathematical physics, starting from the foundations of special and general relativity, and moving to more advanced results in geometric analysis and the Einstein constraint equations. Topics include the formulation of the Einstein field equation and the Einstein constraint equations; gluing construction of initial data sets which are Schwarzschild near infinity; and an introduction to the Riemannian Penrose inequality. While the book assumes a background in differential geometry and real analysis, a number of basic results in geometry are provided. There are well over 100 exercises, many woven into the fabric of the chapters as well as others collected at the end of chapters, to give readers a chance to engage and extend the text.

Table of Contents:
Preface; Notation and conventions; 1. Special relativity and Minkowski spacetime; 2. The Einstein equation; 3. Basics of Lorentzian causality; 4. The Penrose singularity theorem; 5. The Einstein constraint equations; 6. Scalar curvature deformation and the Einstein constraint equations; Excursus: first and second variation of area; 7. Asymptotically flat solutions of the Einstein constraint equations; 8. On the center of mass and constant mean curvature surfaces of asymptotically flat initial data sets; 9. On the Riemannian Penrose inequality; References; Index.

About the Author :
Justin Corvino is Marshall R. Metzgar Professor in the Department of Mathematics at Lafayette College, Pennsylvania. He earned his Ph.D. at Stanford in 2000 under the direction of Richard M. Schoen after completing his undergraduate degree at the Massachusetts Institute of Technology (MIT). He was previously an NSF Postdoctoral Research Fellow and Tamarkin Assistant Professor at Brown University. He spent the fall of 2008 at the Mittag-Leffler Institute in Sweden as a Fulbright Fellow. He has authored or co-authored numerous journal articles in geometric analysis and mathematical physics. Pengzi Miao is a professor in the Department of Mathematics at the University of Miami. He was previously a lecturer at Monash University in Melbourne, Australia. He obtained his Ph.D. from Stanford in 2003 with advisor Richard M. Schoen, after which he worked as a postdoctoral fellow at the Mathematical Sciences Research Institute in Berkeley, California in 2004, and as a Visiting Assistant Professor at UC Santa Barbara from 2004 to 2006. He has authored or co-authored dozens of research articles in geometric analysis and general relativity.

Review :
'This book will be a valuable source for any student planning to specialize in differential geometry, and especially for those preparing to work in mathematical relativity. It begins with a thorough introduction to relativistic spacetimes and the Einstein equations, and continues with some of the basic geometric results in the subject including the Penrose singularity theorem, the positive mass theorem, and geometric properties of solutions of the Einstein constraint equations.' Richard M. Schoen, Stanford University 'Corvino and Miao have written an excellent introduction to Special and General Relativity, from a geometric point of view not usually presented by most books on these topics. I recommend beginning students read this book to gain a beautiful overview of the subject, followed by Dan Lee's Geometric Relativity to master additional topics and details.' Hubert Lewis Bray, Duke University 'An impressive entry for graduate students into some of the deeper, and very important, topics in mathematical GR, written with care by leading experts in the field.' Greg Galloway, University of Miami (Emeritus), Fellow of the American Mathematical Society 'This book provides a rigorous and robust foundation for those seriously pursuing the mathematical theory of general relativity, with a particular focus on the Riemannian geometry of initial data sets. It meticulously develops the required mathematical framework and explains key concepts with clarity and precision. Readers will be well-prepared for more advanced studies in this fascinating field.' Mu-Tao Wang, Columbia University