This book provides an introduction to the mathematics and physics of general relativity, its basic physical concepts, its observational implications, and the new insights obtained into the nature of space-time and the structure of the universe. It introduces some of the most striking aspects of Einstein's theory of gravitation: black holes, gravitational waves, stellar models, and cosmology. It contains a self-contained introduction to tensor calculus and Riemannian geometry, using in parallel the language of modern differential geometry and the coordinate notation, more familiar to physicists. The author has strived to achieve mathematical rigour, with all notions given careful mathematical meaning, while trying to maintain the formalism to the minimum fit-for-purpose. Familiarity with special relativity is assumed.
The overall aim is to convey some of the main physical and geometrical properties of Einstein's theory of gravitation, providing a solid entry point to further studies of the mathematics and physics of Einstein equations.
Author(s): Piotr T. Chruściel
Series: Compact Textbooks in Mathematics
Edition: 1
Publisher: Birkhäuser
Year: 2019
Language: English
Tags: General Relativity
Preface
Contents
1 Introduction to Tensor Calculus and Riemannian Geometry
1.1 Introduction to Tensor Calculus
1.1.1 Manifolds
1.1.2 Scalar Functions
1.1.3 Vector Fields
1.1.4 Covectors
1.1.5 Bilinear Maps, Two-Covariant Tensors
1.1.6 Tensor Products
1.1.6.1 Contractions
1.1.7 Raising and Lowering of Indices
1.2 Covariant Derivatives
1.2.1 Functions
1.2.2 Vectors
1.2.3 Transformation Law
1.2.4 Torsion
1.2.5 Covectors
1.2.6 Higher Order Tensors
1.2.7 The Levi-Civita Connection
1.2.8 Geodesics and Christoffel Symbols
1.3 Local Inertial Coordinates
1.4 Curvature
1.4.1 Symmetries
2 Curved Spacetime
2.1 The Heuristics
2.2 Summary of Basic Ideas
2.2.1 Geodesic Deviation (Jacobi Equation) and Tidal Forces
2.3 Einstein Equations and Matter
2.3.1 Dust in Special and General Relativity
2.3.2 The Continuity Equation
2.3.3 Einstein Equations with Sources
3 The Schwarzschild Metric
3.1 The Metric, Birkhoff's Theorem
3.1.1 r=0
3.2 Stationary Observers and the Parameter m
3.3 Time Functions and Causality
3.4 The Eddington–Finkelstein Extensions
3.5 The Kruskal–Szekeres Extension
3.6 The Flamm Paraboloid
3.7 Conformal Carter-Penrose Diagrams
3.8 Keplerian Orbits
3.9 Geodesics
3.9.1 The Interpretation of E
3.9.2 Gravitational Redshift
3.9.3 Shapiro Delay
3.9.4 Circular Causal Geodesics
3.9.5 Weak-Field Light Bending
3.9.6 The Shadow of a Black Hole
3.9.7 Perihelion/Periastron Precession
3.10 Gyroscope Precession
3.10.1 General Considerations
3.10.2 The Parallel Transport Equation
3.10.3 Geodetic Precession on Circular Geodesics
3.10.4 Fermi–Walker Transport and Thomas Precession
3.10.5 The Lense–Thirring Effect
4 Weak Fields and Gravitational Waves
4.1 Weak Gravitational Fields
4.1.1 Small Perturbations of Minkowski Spacetime
4.1.2 Coordinate Conditions and Wave Coordinates
4.1.3 Linearized Einstein Equations in Wave Coordinates
4.1.4 First-Order Post-Newtonian Expansion
4.1.5 Newton's Equations of Motion, and Why 8πG Is 8πG
4.2 Linearized Plane Waves
4.3 Remarks on Submanifolds, Integration, and Stokes' Theorem
4.3.1 Hypersurfaces
4.4 Energy-Emission by Solutions of Wave Equations
4.4.1 Conservation of Energy on Minkowskian-Time Slices
4.4.2 Energy-Radiation on Hypersurfaces Asymptotic to Light-Cones
4.5 The Quadrupole Formula
4.6 Backreaction, the Chirp Mass
4.7 Multipole Expansions
5 Stars
5.1 Perfect Fluids
5.1.1 Some Newtonian Thermodynamics
5.2 Spherically Symmetric Static Stars
5.2.1 The Tolman–Oppenheimer–Volkov Equation
5.2.2 The Lane–Emden Equation
5.2.3 The Buchdahl–Heinzle Inequality
5.3 End State of Stellar Evolution: The Chandrasekhar Mass
6 Cosmology
6.1 The Lie Derivative
6.1.1 A Pedestrian Approach
6.1.2 The Geometric Approach
6.1.2.1 Transporting Tensor Fields
6.1.2.2 Flows of Vector Fields
6.1.2.3 The Lie Derivative Revisited
6.2 Killing Vectors and Isometries
6.2.1 Killing Vectors
6.2.2 Maximally Symmetric Manifolds
6.3 The Cosmological Principle, Congruences
6.4 Friedman-Lemaître-Robertson-Walker Cosmologies
6.4.1 Hubble Law
6.4.2 The Cosmological Redshift
6.4.3 The Einstein Equations for a FLRW Metric
6.4.4 Vacuum Solutions
6.4.5 Dust (``Matter-Dominated'') Universe
6.4.6 Radiation-Fluid Universe
6.4.7 Radiation-and-Dust Universe
6.4.7.1 Asymptotic Behavior
6.4.7.2 Time-Independent Solutions
A Some Reminders: Minkowski Spacetime
A.1 Minkowski Metric, Lorentz Transformations
A.2 Electromagnetic Fields
B Exercises
Bibliography
