Astrofísica para Todos

Following the approach of Lev Landau and Evgenii Lifshitz, this book introduces the theory of special and general relativity with the Lagrangian formalism and the principle of least action. This method allows the complete theory to be constructed starting from a small number of assumptions, and is the most natural approach in modern theoretical physics. The book begins by reviewing Newtonian mechanics and Newtonian gravity with the Lagrangian formalism and the principle of least action, and then moves to special and general relativity. Most calculations are presented step by step, as is done on the board in class. The book covers recent advances in gravitational wave astronomy and provides a general overview of current lines of research in gravity. It also includes numerous examples and problems in each chapter.

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Summary

1 Introduction
1.1 Special Principle of Relativity
1.2 Euclidean Space
1.3 Scalars, Vectors, and Tensors
1.4 Galilean Transformations
1.5 Principle of Least Action
1.6 Constants of Motion
1.7 Geodesic Equations
1.8 Newton’s Gravity
1.9 Kepler’s Laws
1.10 Maxwell’s Equations
1.11 Michelson–Morley Experiment
1.12 Towards the Theory of Special Relativity

2 Special Relativity
2.1 Einstein’s Principle of Relativity
2.2 Minkowski Spacetime
2.3 Lorentz Transformations
2.4 Proper Time
2.5 Transformation Rules
2.5.1 Superluminal Motion
2.6 Example: Cosmic Ray Muons

3 Relativistic Mechanics 
3.1 Action for a Free Particle
3.2 Momentum and Energy
3.2.1 3-Dimensional Formalism
3.2.2 4-Dimensional Formalism

3.3 Massless Particles
3.4 Particle Collisions
3.5 Example: Colliders Versus Fixed-Target Accelerators
3.6 Example: The GZK Cut-Off
3.7 Multi-body Systems
3.8 Lagrangian Formalism for Fields
3.9 Energy-Momentum Tensor
3.10 Examples
3.10.1 Energy-Momentum Tensor of a Free Point-Like
Particle
3.10.2 Energy-Momentum Tensor of a Perfect Fluid

4 Electromagnetism 
4.1 Action
4.2 Motion of a Charged Particle
4.2.1 3-Dimensional Formalism
4.2.2 4-Dimensional Formalism
4.3 Maxwell’s Equations in Covariant Form
4.3.1 Homogeneous Maxwell’s Equations
4.3.2 Inhomogeneous Maxwell’s Equations
4.4 Gauge Invariance
4.5 Energy-Momentum Tensor of the Electromagnetic Field
4.6 Examples
4.6.1 Motion of a Charged Particle in a Constant Uniform Electric Field
4.6.2 Electromagnetic Field Generated by a Charged

5 Riemannian Geometry 
5.1 Motivations
5.2 Covariant Derivative
5.2.1 Definition
5.2.2 Parallel Transport
5.2.3 Properties of the Covariant Derivative
5.3 Useful Expressions
5.4 Riemann Tensor
5.4.1 Definition
5.4.2 Geometrical Interpretation
5.4.3 Ricci Tensor and Scalar Curvature
5.4.4 Bianchi Identities
6 General Relativity
6.1 General Covariance
6.2 Einstein Equivalence Principle
6.3 Connection to the Newtonian Potential
6.4 Locally Inertial Frames
6.4.1 Locally Minkowski Reference Frames
6.4.2 Locally Inertial Reference Frames
6.5 Measurements of Time Intervals
6.6 Example: GPS Satellites
6.7 Non-gravitational Phenomena in Curved Spacetimes
7 Einstein’s Gravity 
7.1 Einstein Equations
7.2 Newtonian Limit
7.3 Einstein–Hilbert Action
7.4 Matter Energy-Momentum Tensor
7.4.1 Definition
7.4.2 Examples
7.4.3 Covariant Conservation of the Matter Energy-Momentum Tensor
7.5 Pseudo-Tensor of Landau–Lifshitz
8 Schwarzschild Spacetime 
8.1 Spherically Symmetric Spacetimes
8.2 Birkhoff’s Theorem
8.3 Schwarzschild Metric
8.4 Motion in the Schwarzschild Metric
8.5 Schwarzschild Black Holes
8.6 Penrose Diagrams
8.6.1 Minkowski Spacetime
8.6.2 Schwarzschild Spacetime
9 Classical Tests of General Relativity 
9.1 Gravitational Redshift of Light
9.2 Perihelion Precession of Mercury
9.3 Deflection of Light
9.4 Shapiro’s Effect
9.5 Parametrized Post-Newtonian Formalism

10 Black Holes 
10.1 Definition
10.2 Reissner–Nordström Black Holes
10.3 Kerr Black Holes
10.3.1 Equatorial Circular Orbits
10.3.2 Fundamental Frequencies
10.3.3 Frame Dragging
10.4 No-Hair Theorem
10.5 Gravitational Collapse
10.5.1 Dust Collapse
10.5.2 Homogeneous Dust Collapse
10.6 Penrose Diagrams
10.6.1 Reissner–Nordström Spacetime
10.6.2 Kerr Spacetime
10.6.3 Oppenheimer–Snyder Spacetime
11 Cosmological Models
11.1 Friedmann–Robertson–Walker Metric
11.2 Friedmann Equations
11.3 Cosmological Models
11.3.1 Einstein Universe
11.3.2 Matter Dominated Universe
11.3.3 Radiation Dominated Universe
11.3.4 Vacuum Dominated Universe
11.4 Properties of the Friedmann–Robertson–Walker Metric
11.4.1 Cosmological Redshift
11.4.2 Particle Horizon
11.5 Primordial Plasma
11.6 Age of the Universe
11.7 Destiny of the Universe

12 Gravitational Waves 
12.1 Historical Overview
12.2 Gravitational Waves in Linearized Gravity
12.2.1 Harmonic Gauge
12.2.2 Transverse-Traceless Gauge
12.3 Quadrupole Formula
12.4 Energy of Gravitational Waves

12.5 Examples
12.5.1 Gravitational Waves from a Rotating Neutron Star
12.5.2 Gravitational Waves from a Binary System
12.6 Astrophysical Sources
12.6.1 Coalescing Black Holes
12.6.2 Extreme-Mass Ratio Inspirals
12.6.3 Neutron Stars
12.7 Gravitational Wave Detectors
12.7.1 Resonant Detectors
12.7.2 Interferometers
12.7.3 Pulsar Timing Arrays

13 Beyond Einstein’s Gravity
13.1 Spacetime Singularities
13.2 Quantization of Einstein’s Gravity
13.3 Black Hole Thermodynamics and Information Paradox
13.4 Cosmological Constant Problem
Author: Cosimo Bambi