A Mathematical Journey to Relativity: Deriving Special and General Relativity with Basic Mathematics (UNITEXT for Physics)

Preface to€the€Second Edition
Preface I to the First Edition
Preface II to the First Edition
Contents
1 Euclidean and Non-Euclidean Geometries: How They Appear
	1.1 Absolute Geometry
	1.2 From Absolute Geometry to Euclidean Geometry Through …
	1.3 From Absolute Geometry to Non-Euclidean Geometry Through Non-Euclidean Parallelism Axiom
2 Basic Facts in Euclidean and Minkowski Plane Geometry
	2.1 Pythagoras Theorems in Euclidean Plane
	2.2 Space-Like, Time-Like, and Null Vectors in Minkowski Plane
	2.3 Minkowski–Pythagoras Theorems
3 From Projective Geometry to Poincaré Disk. How to Carry Out a Non-Euclidean Geometry Model
	3.1 Geometric Inversion and Its Properties
	3.2 Cross Ratio and Projective Geometry
	3.3 Poincaré Disk Model
4 Revisiting the Differential Geometry of Surfaces in 3D-Spaces
	4.1 Basic Notations and Definitions of the Geometry of Surfaces
	4.2 Surfaces, Tangent Planes and Gauss Frames
	4.3 The Metric of a Surface
	4.4 How Metric is Changing with Respect to Changes of Coordinates and Isometries
	4.5 Intrinsic Properties of Surfaces
	4.6 Extrinsic Properties of Surfaces. The Weingarten Equations
	4.7 The Gaussian Curvature of Surfaces
	4.8 The Geometric Interpretation of Gaussian Curvature
	4.9 Christoffel Symbols, Riemann Symbols and Gauss Formulas
	4.10 The Gauss Equations and the Theorema Egregium
	4.11 The Einstein Theorem
	4.12 Covariant Derivative, Parallel Transport and Geodesics
	4.13 Changes of Coordinates
	4.14 What if the Ambient Space is Not an Euclidean One?
	4.15 Transferring Metrics. Is Our Geometric Intuition Intrinsically …
5 Basic Differential Geometry Concepts and Their Applications
	5.1 Tensors in Differential Geometry. Definition and Examples
	5.2 Properties of Riemann and Ricci Tensors in the New Geometric Context
	5.3 Covariant Derivative for Vectors. Geodesics and Their Properties
	5.4 Covariant Derivative of Tensors and Applications
	5.5 A Step Towards General Relativity: The Bianchi Second Formula
6 Differential Geometry at Work: Two Ways of Thinking the Gravity. The Einstein Field Equations from a Geometric Point of View
	6.1 From Newtonian Gravity to the Geometry of Space-Time
	6.2 The Einstein Field Equations and the Energy–Momentum Tensor
	6.3 Including the Cosmological Constant
7 Differential Geometry at Work: Euclidean, Non-Euclidean, and Elliptic Geometric Models from Geometry and Physics
	7.1 Euclidean, Non-Euclidean, and Elliptic Geometric Models from Geometry
	7.2 Euclidean, Non-Euclidean, and Elliptic Geometric Models from Physics
	7.3 The Physical Interpretation
	7.4 Another Way to Obtain the Poincaré Disc Model Metric
8 Gravity in Newtonian Mechanics
	8.1 Gravity. The Vacuum Field Equation
	8.2 Divergence of a Vector Field in a Euclidean 3D-Space
	8.3 Covariant Divergence
	8.4 The General Newtonian Gravitational Field Equations
	8.5 Tidal Acceleration Equations
	8.6 The Kepler Laws
	8.7 Circular Motion, Centripetal Force, Deflection of Light Effect …
	8.8 The Mechanical Lagrangian
	8.9 Geometry Induced by a Lagrangian
9 Special Relativity
	9.1 Principles of Special Relativity
	9.2 Lorentz Transformations in Geometric Coordinates and Consequences
		9.2.1 The Relativity of Simultaneity
		9.2.2 The Lorentz Transformations in Geometric Coordinates
		9.2.3 The Minkowski Geometry of Inertial Frames in Geometric Coordinates and Consequences: Time Dilation and Length Contraction
		9.2.4 Relativistic Mass, Rest Mass and Energy
	9.3 Consequences of Lorentz Physical Transformations: Time …
		9.3.1 The Minkowski Geometry of Inertial Frames in Physical Coordinates and Consequences: Time Dilation and Length Contraction
		9.3.2 Relativistic Mass, Rest Mass and Rest Energy in Physical Coordinates
	9.4 The Maxwell Equations
	9.5 The Doppler Effect in Special Relativity
	9.6 Gravity in Special Relativity: The Case of the Constant Gravitational Field
		9.6.1 The Doppler Effect in Constant Gravitational Field and Consequences
		9.6.2 Bending of Light-Rays in a Constant Gravitational Field
		9.6.3 The Basic Incompatibility Between Gravity and Special Relativity
10 General Relativity and Relativistic Cosmology
	10.1  What is a Good Theory of Gravity?
		10.1.1 Metric or Connections?
		10.1.2 The Role of Equivalence Principle
	10.2 Gravity Seen Through Geometry in General Relativity
		10.2.1 The Einstein Landscape for the Constant Gravitational Field
	10.3 The Einstein–Hilbert Action and The Einstein Field Equations
	10.4 An Introduction to f left parenthesis upper R right parenthesisf(R) Gravity
	10.5 The Schwarzschild Solution of Vacuum Field Equations
		10.5.1 Orbit of a Planet in the Schwarzschild Metric
		10.5.2 Relativistic Solution of the Mercury Perihelion Drift Problem
		10.5.3 Speed of Light in a Given Metric
		10.5.4 Bending of Light in the Schwarzschild Metric
	10.6 The Einstein Metric: Einstein\'s Computations Related …
	10.7 Black Holes: A Mathematical Introduction
		10.7.1 Escape Velocity and Black Holes
		10.7.2 The Rindler Metric and Pseudo-Singularities
		10.7.3 Black Holes in the Schwarzschild Metric
		10.7.4 The Light Cone in the Schwarzschild Metric
	10.8 Cosmological Solutions of the Einstein Field Equations …
		10.8.1 More About FLRW Universes
		10.8.2 A Remarkable Universe without Matter from FLWR Conditions
		10.8.3 The Cosmological Expansion
	10.9 Measuring the Cosmos
	10.10 The Fermi Coordinates
		10.10.1 Determining the Fermi Coordinates
		10.10.2 The Fermi Viewpoint on the Einstein Field Equations in Vacuum
		10.10.3 The Gravitational Coupling in the Einstein Field Equations: K = StartFraction 8 pi upper G Over c Superscript 4 Baseline EndFraction8πGc4
	10.11 Weak Gravitational Field and the Classical Counterparts …
	10.12 The Einstein Static Universe and the Cosmological Constant
	10.13 Cosmic Strings
	10.14 Planar Gravitational Waves
	10.15 The Gödel Universe
	10.16 Is it Possible a Space-Time without Matter and Time?
	10.17 A Remarkable Universe without Time
	10.18 Another Exact Solution of Einstein Field Equations Induced …
	10.19 The Wormhole Solutions
11 A Geometric Realization of Relativity: The de Sitter Space-time
	11.1 About the Minkowski Geometric Gravitational Force
	11.2 De Sitter Spacetime and Its Cosmological Constant
	11.3 Some Physical Considerations
	11.4 A FLRW Metric for de Sitter Space-time Given …
	11.5 Deriving Cosmological Singularities in the Context of de Sitter Space-time
12 Another Geometric Realization of Relativity: The Anti-de Sitter Space–Time
	12.1 The Minkowski upper M Superscript left parenthesis 2 comma 4 right parenthesisM(2,4) Geometric Gravitational Force
	12.2 The Minkowski–Tzitzeica Surfaces
	12.3 The Geometric Nature of the Affine Radius in a Minkowski upper M Superscript left parenthesis 2 comma 3 right parenthesisM(2,3) Space
	12.4 Geometrical Considerations Related to the Affine Radius in the Minkowski upper M Superscript left parenthesis 2 comma 4 right parenthesisM(2,4) Space
	12.5 Anti-de Sitter Space–Times as Affine Hypersurfaces. Their Cosmological Constant and Its Connection with the Affine Radius
13 More Than Metric: Geometric Objects for Alternative Pictures of Gravity
	13.1 Differentiable Manifolds
	13.2 Abstract Frame for Tensors, Exterior Forms, and Differential Forms
	13.3 Vector Fields and the Structure Equations of double struck upper R Superscript nmathbbRn
	13.4 Affine Connections, Torsion, and Curvature
	13.5 Covariant Derivative, Parallel Transport, and Geodesics
	13.6 A Geometric Description of Riemann Curvature Mixed Tensor …
	13.7 The Levi-Civita Connection
	13.8 Coordinate Changes for Geometric Objects Generated …
	13.9 Some Remarks on the Mathematical Language  of Metric-Affine Gravity
		13.9.1 From Latin to Greek Indexes and Vice Versa
14 Metric-Affine Theories of Gravity
	14.1 A Survey on Theories of Gravity
	14.2 Metric-Affine Theories of Gravity
	14.3 The Geometric Trinity of Gravity
	14.4 Tetrads and Spin Connection
		14.4.1  The Tetrad Formalism
		14.4.2 The Spin Connection
	14.5 Equivalent Representations of Gravity: The Lagrangian Level
		14.5.1 Metric Formulation of Gravity: The Case of General Relativity
		14.5.2 Gauge Formulation of Gravity: The Case of Teleparallel Gravity
		14.5.3 A Discussion on Trinity Gravity at Lagrangian Level
	14.6 Field Equations in Trinity Gravity
		14.6.1 GR Field Equations
		14.6.2 TEGR Field Equations
		14.6.3 STEGR Field Equations
	14.7 Solutions in Trinity Gravity
		14.7.1 Spherically Symmetric Solutions in GR
		14.7.2 Spherically Symmetric Solutions in TEGR
		14.7.3 Spherically Symmetric Solutions in STEGR
	14.8 Discussion and Perspectives
15 Conclusions
Appendix  References
Index