Lectures on Mathematical Relativity
Justin Corvino and Pengzi Miao
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; a treatment of the Penrose singularity theorem; an introduction to scalar curvature deformation and the conformal method; a detailed introduction to asymptotically flat spaces and the Riemannian positive mass theorem; gluing construction of initial data sets which are Schwarzschild near infinity; constant mean curvature surfaces and the center of mass for asymptotically flat initial data sets; 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 included in the text and exercises. A brief treatment of elliptic partial differential equations is designed to help the reader navigate through the applications of geometric analysis to the Einstein constraint equations discussed in the analysis-heavy second half of the book.
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.