An Introduction to Mathematical Relativity

1. Front Matter

   Pages i-viii

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2. Preliminaries

   • José Natário

   Pages 1-13

3. Exact Solutions

   • José Natário

   Pages 15-57

4. Causality

   • José Natário

   Pages 59-66

5. Singularity Theorems

   • José Natário

   Pages 67-95

6. Cauchy Problem

   • José Natário

   Pages 97-118

7. Mass in General Relativity

   • José Natário

   Pages 119-152

8. Black Holes

   • José Natário

   Pages 153-169

9. Back Matter

   Pages 171-186

   PDF

This concise textbook introduces the reader to advanced mathematical aspects of general relativity, covering topics like Penrose diagrams, causality theory, singularity theorems, the Cauchy problem for the Einstein equations, the positive mass theorem, and the laws of black hole thermodynamics. It emerged from lecture notes originally conceived for a one-semester course in Mathematical Relativity which has been taught at the Instituto Superior Técnico (University of Lisbon, Portugal) since 2010 to Masters and Doctorate students in Mathematics and Physics. 


Mostly self-contained, and mathematically rigorous, this book can be appealing to graduate students in Mathematics or Physics seeking specialization in general relativity, geometry or partial differential equations. Prerequisites include proficiency in differential geometry and the basic principles of relativity. Readers who are familiar with special relativity and have taken a course either inRiemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.

• Penrose diagrams
• causality theory
• singularity theorems
• Cauchy problem
• Einstein equations
• positive mass theorem
• Penrose inequality
• black hole thermodynamics

“This is a well-organized book that provides advanced students with the appropriate background a comfortable first introduction to important mathematical results of general relativity. … The instructive reading will certainly motivate many students to subsequently expand their scope even further, both thematically and methodologically.” (Wolfgang Hasse, Mathematical Reviews, April, 2022)

• Department of Mathematics, Instituto Superior Técnico, University of Lisbon, Lisbon, Portugal

  José Natário

José Natário holds a DPhil in Mathematical Sciences (2000) from the University of Oxford, England. He has been an Associate Professor of Mathematics at Instituto Superior Técnico (University of Lisbon, Portugal) since 2010, where he teaches a course in Mathematical Relativity to Master and Doctorate students in Mathematics and Physics. He authored "General Relativity Without Calculus" (2011, ISBN 978-3-642-21451-6) and co-authored "An Introduction to Riemannian Geometry" (2014, ISBN 978-3-319-08665-1), both published by Springer.

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