Lectures on Mathematical Relativity

A mathematical introduction emphasizing the Einstein constraint equations from the initial-value problem for Einstein's field equation.

Justin Corvino (Author), Pengzi Miao (Author)

9781107439252, Cambridge University Press

Paperback / softback, published 10 April 2025

426 pages
23.5 x 15.5 x 2.4 cm, 0.632 kg

'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

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.

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.

Subject Areas: Maths for scientists [PDE]