The aim of this textbook is to present in a comprehensive way several advanced topics of general relativity, including gravitational waves, tests of general relativity, time delay, spinors in curved spacetime, Hawking radiation, and geodetic precession to mention a few. These are all important topics in today's research activities from both a theoretical and experimental point of view.
This textbook is designed for advanced undergraduate and graduate students to strengthen the knowledge acquired during the core courses on General Relativity. The author developed the book from a series of yearly lectures with the intention of offering a gentle introduction to the field. This book helps understanding the more specialized literature and can be used as a first reading to get quickly into the field when starting research. Chapter-end exercises complete the learning material to master key concepts.
Author(s): Philippe Jetzer
Series: UNITEXT for Physics
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2022
Language: English
Pages: 195
City: Cham, Switzerland
Tags: General relativity, Gravitational Lensing, Stellar Structure, Gravitational Waves, Black Holes
Preface
Acknowledgments
Contents
1 Elements of General Relativity
1.1 Einstein's Field Equations
1.1.1 The Cosmological Constant
1.2 Static Isotropic Metric
1.2.1 Form of the Metric
1.2.2 Christoffel Symbols and Ricci Tensor for the Spherically Symmetric Case
1.2.3 Robertson Expansion
1.2.4 Schwarzschild Metric
1.3 General Equations of Motion
References
2 Some Applications of General Relativity
2.1 Gravitational Lensing
2.1.1 Historical Introduction
2.1.2 Point-Like Lens
2.1.3 Thin Lens Approximation
2.1.4 Lens Equation
2.1.5 Schwarzschild Lens
2.2 General Relativistic Stellar Structure Equations
2.2.1 Introduction
2.2.2 General Relativistic Stellar Structure Equations
2.2.3 Interpretation of the Gravitational Mass M
2.2.4 The Interior of Neutron Stars
2.3 Time Delay of Radar Echoes
2.4 Precession
2.4.1 Geodetic Precession
2.5 Linearized Field Equations
2.5.1 The Energy-Momentum Tensor of the Grav. Field
2.5.2 Linearized Field Equations
2.6 Lense–Thirring Effect
2.6.1 Metric of the Rotating Earth
2.6.2 Gravitomagnetic Forces
2.7 Summary
2.8 Problems
References
3 Gravitational Waves and Post-Newtonian Approximation
3.1 Introduction
3.1.1 Electromagnetic Waves
3.2 Gravitational Waves
3.2.1 Helicity of Gravitational Waves
3.2.2 Particles in the Field of a Gravitational Wave
3.2.3 Energy and Momentum of a Gravitational Wave
3.2.4 Quadrupole Radiation
3.3 Sources of Gravitational Waves
3.3.1 Rigid Rotator
3.3.2 Example: The Binary Star System
3.4 Binaries on Elliptic Orbits
3.4.1 Elliptic Keplerian Orbits
3.4.2 Radiated Power
3.4.3 Parabolic Orbits
3.4.4 Frequency Spectrum of Gravitational Waves
3.4.5 Evolution of the Orbit Due to Gravitational Waves Emission
3.4.6 The PSR 1913+16 System
3.5 Waveform
3.5.1 Quadrupole Radiation from a Mass in Circular Orbit
3.6 The Post-Newtonian Approximation
3.6.1 The 1PN Approximation
3.6.2 Equations of Motion of Test Particles in the 1PN Metric
3.6.3 Two Body Problem in the 1PN Approximation
3.7 Summary
3.8 Problems
References
4 Black Holes
4.1 The Kerr Solution
4.1.1 Interpretation of the Parameters a and m
4.1.2 Kerr–Newman Solution
4.1.3 Equations of Motion for Test Particles
4.2 Tetrad Formalism
4.2.1 Introduction
4.2.2 Some Differential Geometry
4.2.3 The Schwarzschild Metric Revisited
4.3 Spinors in Curved Spacetime
4.3.1 Representations of the Lorentz Group
4.4 Dirac Gamma Matrices
4.4.1 Dirac Equation in Flat Space
4.5 Dirac Equation in Curved Spacetime
4.5.1 Dirac Equation in Kerr–Newman Geometry
4.5.2 Dirac Equation in Schwarzschild Geometry
4.6 The Four Laws of Black Hole Dynamics
4.6.1 The Zeroth Law
4.6.2 The First Law
4.6.3 The Second Law
4.6.4 The Third Law
4.7 Hawking Radiation
4.7.1 Expected Number of Outgoing Particles
4.7.2 Black Hole Evaporation
4.8 Summary
4.9 Problems
References
5 Tests of General Relativity
5.1 The Weak Equivalence Principle
5.2 The Einstein Equivalence Principle
5.2.1 Tests of the Weak Equivalence Principle
5.2.2 Tests of Local Lorentz Invariance
5.2.3 Tests of Local Position Invariance
5.3 Schiff's Conjecture
5.3.1 Quantitative Relationship Between Violations of the Weak Equivalence Principle and Local Position Invariance
5.4 The Strong Equivalence Principle
5.5 Summary
References
Appendix Solutions
Index
