Description
Book SynopsisMany 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 geometric analysis. Specifically, it provides a comprehensive treatment of the positive mass 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, inverse mean curvature flow, conformal flow, spinors and the Dirac operator, marginally outer trapped surfaces, and density theorems. This is the first time these topics have been 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 part
Trade ReviewGeometric Relatively is refreshing in its narrative approach to this topic. The author is open and honest about the material included and the material excluded in the text, explaining when certain material is omitted or glossed over. Indeed, oftentimes finer technical details will be omitted from a proof for the sake of narrative clarity. Overall, this book is a nice textbook for a graduate student to study from or a great reference for a research mathematician. Anyone who is interested in exploring relativity from a geometry perspective or simply interested purely in geometric analysis can gain something from this text." —John Ross, Southwestern University
Table of Contents
- Riemannian geometry: Scalar curvature
- Minimal hypersurfaces
- The Riemannian positive mass theorem
- The Riemannian Penrose inequality
- Spin geometry
- Quasi-local mass
- Initial data sets: Introduction to general relativity
- The spacetime positive mass theorem
- Density theorems for the constraint equations
- Some facts about second-order linear elliptic operators
- Bibliography
- Index
