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 in Riemannian geometry (for students of Mathematics) or in general relativity (for those in Physics) can benefit from this book.
Author(s): Jose Natario
Series: Latin American Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 186
City: Cham
Tags: General Relativity
Preface
Contents
1 Preliminaries
1.1 Special Relativity
1.2 Differential Geometry: Mathematicians vs Physicists
1.3 General Relativity
1.4 Exercises
2 Exact Solutions
2.1 Minkowski Spacetime
2.2 Penrose Diagrams
2.3 The Schwarzschild Solution
2.4 Friedmann–Lemaître–Robertson–Walker Models
2.4.1 Milne Universe
2.4.2 de Sitter Universe
2.4.3 Anti-de Sitter Universe
2.4.4 Universes with Matter and =0
2.4.5 Universes with Matter and >0
2.4.6 Universes with Matter and <0
2.5 Matching
2.6 Oppenheimer–Snyder Collapse
2.7 Exercises
3 Causality
3.1 Past and Future
3.2 Causality Conditions
3.3 Exercises
4 Singularity Theorems
4.1 Geodesic Congruences
4.2 Energy Conditions
4.3 Conjugate Points
4.4 Existence of Maximizing Geodesics
4.5 Hawking's Singularity Theorem
4.6 Penrose's Singularity Theorem
4.7 Exercises
5 Cauchy Problem
5.1 Divergence Theorem
5.2 Klein–Gordon Equation
5.3 Maxwell's Equations: Constraints and Gauge
5.4 Einstein's Equations
5.5 Constraint Equations
5.6 Einstein Equations with Matter
5.7 Exercises
6 Mass in General Relativity
6.1 Komar Mass
6.2 Field Theory
6.2.1 Klein–Gordon Field
6.2.2 Electromagnetic Field
6.2.3 Relativistic Elasticity
6.3 Einstein–Hilbert Action
6.4 Gravitational Waves
6.5 ADM Mass
6.6 Positive Mass Theorem
6.7 Penrose Inequality
6.8 Exercises
7 Black Holes
7.1 The Kerr Solution
7.2 Killing Horizons and the Zeroth Law
7.3 Smarr's Formula and the First Law
7.4 Second Law
7.5 Black Hole Thermodynamics and Hawking Radiation
7.6 Exercises
A Mathematical Concepts for Physicists
A.1 Topology
A.2 Metric Spaces
A.3 Hopf–Rinow Theorem
A.4 Differential Forms
A.5 Lie Derivative
A.6 Cartan Structure Equations
References
Index
