An Overview of General Relativity and Space-Time (Series in Astronomy and Astrophysics)

Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
List of Figures
List of Boxes
Part I: From Forces to Curvature
	Chapter 1: Space and Time: The Classical View
		1.1. Introduction
		1.2. Metric Space
		1.3. Homogeneity and Isotropy of Space
		1.4. Covariant or Contravariant Vector Components
		1.5. Motion of a Free Test-Particle
		1.6. Inertial Frames and Galileo’s Relativity Principle
		1.7. The Derivative of a Vector
		1.8. Velocity of Interactions and Second Newton’s Law
		1.9. Homogeneity of Time: Energy Conservation
		1.10. Homogeneity of Space: The Third Newton’s Law
		1.11. Planetary Motions
		1.12. Non-Inertial Reference Frames
	Chapter 2: From Space and Time to Space-Time
		2.1. Introduction
		2.2. A Metric Space-Time
		2.3. Homogeneity and Isotropy of the Minkowski Space-Time
		2.4. Ordinary vs. Hyperbolic Rotations
		2.5. Hyperbolic Rotations vs. Drag Velocities
		2.6. Lorentz Transformations: A Graphical Approach
		2.7. Proper Time and Proper Length
		2.8. Motion of a Free Test-Particle
		2.9. Four-Velocity and Four-Acceleration Vectors
		2.10. Geodesics in Minkowski Space-Time
		2.11. Four-Momentum
		2.12. Relativistic Velocity Composition Law
	Chapter 3: From Inertial to Non-Inertial Reference Frames
		3.1. Introduction
		3.2. Linear vs. Non-Linear Coordinate Transformations
		3.3. Motion of a Free Test-Particle in a Rotating Frame
		3.4. Geodesics in a Generic Space-Time
		3.5. Geodesic Motion in the Rotating Reference Frame
		3.6. Something More on Proper Time
		3.7. Clock Synchronization
		3.8. Proper Spatial Distances
	Chapter 4: Pseudo-Riemannian Spaces
		4.1. Introduction
	4.2. Manifolds
		4.3. How to Move to The Tangent Plane?
	4.4. Vector Fields
		4.5. Tensors
		4.6. How to Recognize Tensors
		4.7. General Covariance
	Chapter 5: The Riemann-Christoffel Curvature Tensor
		5.1. Introduction
		5.2. Parallel Transport in Curved Space: The 2D Case
		5.3. More About the Christoffel Symbols
		5.4. Parallel Transport in Curved Manifolds
		5.5. Covariant Derivative and Covariant Differential
		5.6. Covariant Derivatives are Tensors
		5.7. More on Geodesics
		5.8. More on Covariant Derivative
		5.9. Covariant Derivative of a Tensor
		5.10. The Riemann-Christoffel Curvature Tensor
		5.11. Second Covariant Derivatives
		5.12. Symmetries of the Riemann-Christoffel Tensor
	Chapter 6: From Non-inertial Frames to Gravity: The Equivalence Principle
		6.1. Introduction
		6.2. The Equivalence Principle
		6.3. Switching Off Gravity: The Free-Fall
		6.4. “Creating Gravity”: Non-Inertial Frames
		6.5. The Gravity Case: A Metric Space
	6.6. The Motion of a Test-Particle in a Gravitational Field
		6.7. Geodesic Deviation
		6.8. Link Between Geometry and Dynamics
		6.9. Riemann-Christoffel Tensor in the Rotating Frame
		6.10. Riemann-Christoffel Tensor and Gravity
Part II: From Curvature to Observations
	Chapter 7: Observational Test of the Equivalence Principle
	7.1. Introduction
		7.2. Inertial vs. Gravitational Masses
		7.3. Gravitational Time Dilation
		7.4. The Global Positioning System – GPS
		7.5. Gravitational Redshift
		7.6. The Long Gravitational Redshift Hunt
		7.7. The Nordtvedt Rffect
	Chapter 8: Field Equations in the “vacuum”
		8.1. Introduction
		8.2. Field Equations in the “Vacuum”: Requirements
		8.3. The Ricci Tensor
		8.4. Gravitational Field Equations in the “Vacuum”
		8.5. The Einstein Tensor
		8.6. Hilbert’s Action
		8.7. The Action for a Cosmological Constant
		8.8. The Geometry of Space-Time in the “Vacuum”
	Chapter 9: Test-Particles in the Schwarzschild Space-Time
		9.1. Introduction
		9.2. The Schwarzschild Solution for a Point Mass
		9.3. The “Embedding” Procedure
		9.4. The Jebsen-Birkhoff Theorem
		9.5. First Integrals in the Schwarzschild Space-Time
		9.6. Energy Conservation in GR
		9.7. The Radial Infall
		9.8. Orbits in a Schwarzschild Geometry
		9.9. Stable Circular Orbits: α > √3
		9.10. The Case of a Non-Radial Infall: α > √3
		9.11. Photons in the Schwarzschild Space-Time
	Chapter 10: The Classical Tests of General Relativity
		10.1. Introduction
		10.2. Planetary Motion
		10.3. The Perihelion Shift of Mercury
		10.4. Light Ray’s Deflection
		10.5. Gravitational Lensing
		10.6. The Schwarzschild Metric in Totally Isotropic Form
		10.7. Light Travel Time in a Schwarzschild Geometry
	Chapter 11: Gravitational Waves in the “Vacuum”
		11.1. Introduction
		11.2. Linearized Gravity
		11.3. Gauge Transformations
		11.4. The Lorentz Gauge
		11.5. Gravitational Waves
		11.6. TT Gauge
		11.7. Gauge Invariant Approach
		11.8. Field Equations
		11.9. Effects of a Gravitational Wave
		11.10. Interferometers
		11.11. The Direct Detection of Gravitational Waves
		11.12. The Indirect Evidence for Gravitational Waves
Part III: From Singularities to Cosmological Scales
	Chapter 12: Schwarzschild Black Holes
		12.1. Introduction
		12.2. Singularities of the Schwarzschild Metric
		12.3. Conformally-Flat Coordinates
		12.4. Kruskal-Szekeres vs. Schwarzschild Coordinates
		12.5. The Kruskal-Szekeres Plane
		12.6. The Schwarzschild Black Hole
	Chapter 13: Field Equations in Non-“Empty” Space-Times
		13.1. Introduction
		13.2. Field Equations: Requirements
		13.3. Conservation Laws for a Relativistic Fluid
		13.4. The Matter Energy-Momentum Tensor
		13.5. The EM Energy-Momentum Tensor
		13.6. An Isotropic Radiation Field
		13.7. Field Equations Inside a Matter/Energy Distribution
		13.8. The Reissner-Nordstrom Solution
		13.9. Orbits in a Reissner-Nordstrom Geometry
		13.10. Free-Fall on the Reissner-Nordstrom Black Hole
		13.11. The Kerr Solution
	Chapter 14: Further Applications of the Field Equations
		14.1. Introduction
		14.2. The Inner Schwarzschild Space-Time
		14.3. Proper vs. Observable Mass
		14.4. Spherical Relativistic Hydrodynamics
		14.5. The Gravitational Collapse
	Chapter 15: Theoretical Cosmology
		15.1. Introduction
		15.2. An Isotropic and Homogeneous Matter Distribution
		15.3. The FLRW Metric
		15.4. The Spatial Sector of the FLRW Space-Time
		15.5. The Friedmann Equations
		15.6. Equation of Motions
		15.7. Cosmological Parameters
		15.8. The De Sitter Model
		15.9. The Closed Friedmann Universe
		15.10. The Open Friedmann Universe
		15.11. The Einstein-De Sitter Universe
		15.12. A Flat, ⋀-Dominated Universe
		15.13. H0 and the Age of the Universe
		15.14. The Cosmological Redshift
		15.15. Comoving Distances
		15.16. The Proper Angular Diameter Distance
		15.17. The Proper Luminosity Distance
		15.18. SNe La and Dark Energy
	Chapter 16: The Hot Big-Bang
		16.1. Introduction
		16.2. The CMB
		16.3. A Radiation-Dominated Universe
		16.4. The Neutron-to-Baryon Ratio
		16.5. A v’s Cosmic Background
		16.6. Primordial Nucleosynthesis
		16.7. Primordial Abundance of Light Nuclei
		16.8. The Recombination of the Primordial Plasma
		16.9. The Puzzles of the Standard Model
		16.10. An Early Accelerated Phase?
		16.11. Cosmic Inflation
Appendix A: Exercises
References
Index