A Mathematical Journey to Relativity

Preface I
Preface II
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 ...
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 Geometric Inversion, Cross Ratio, Projective Geometry and Poincaré Disk Model
	3.1 Geometric Inversion and Its Properties
	3.2 Cross Ratio and Projective Geometry
	3.3 Poincaré Disk Model
4 Surfaces in 3D-Spaces
	4.1 Geometry of Surfaces in a 3D-Euclidean Space
	4.2 Intrinsic Geometry of Surfaces
	4.3 Geometry of Surfaces in a 3D-Minkowski Space
	4.4 A Short Story of a Person Embedded in a Surface
5 Basic Differential Geometry
	5.1 Covariant and Contravariant Vectors and Tensors. The Christoffel Symbols
	5.2 Covariant Derivative for Vectors and Tensors. Geodesics
	5.3 Riemann Mixed, Riemann Curvature Covariant, Ricci and Einstein Tensors
6 Non-Euclidean Geometries and Their Physical Interpretation
	6.1 Poincaré Distance and Poincaré Metric of the Disk
	6.2 Poincaré Distance and Poincaré Metric of the Half-Plane
	6.3 Connections Between the Models of Non-Euclidean Geometries
	6.4 The Exterior Disk Model
	6.5 A Hemisphere Model for the Non-Euclidean Geometry
	6.6 A Minkowski Model for the Non-Euclidean Geometry: The Hyperboloid Model
	6.7 The Theoretical Minimum About Non-Euclidean Geometry …
	6.8 A Physical Interpretation
7 Gravity in Newtonian Mechanics
	7.1 Gravity. The Vacuum Field Equation
	7.2 Divergence of a Vector Field in an Euclidean 3D-Space
	7.3 Covariant Divergence
	7.4 The General Newtonian Gravitational Field Equations
	7.5 Tidal Acceleration Equations
	7.6 Geometric Separation of Geodesics
	7.7 Kepler's Laws
	7.8 Circular Motion, Centripetal Force and Dark Matter Problem
	7.9 The Mechanical Lagrangian
	7.10 Geometry Induced by a Lagrangian
8 Special Relativity
	8.1 Principles of Special Relativity
	8.2 Lorentz Transformations in Geometric Coordinates and Consequences
		8.2.1 The Relativity of Simultaneity
		8.2.2 The Lorentz Transformations in Geometric Coordinates
		8.2.3 The Minkowski Geometry of Inertial Frames in Geometric Coordinates and Consequences: Time Dilation and Length Contraction
		8.2.4 Relativistic Mass, Rest Mass and Energy
	8.3 Consequences of Lorentz Physical Transformations ...
		8.3.1 The Minkowski Geometry of Inertial Frames in Physical Coordinates and Consequences: Time Dilation and Length Contraction
		8.3.2 Relativistic Mass, Rest Mass and Rest Energy in Physical Coordinates
	8.4 Maxwell's Equations
	8.5 Doppler's Effect in Special Relativity
	8.6 Gravity in Special Relativity ...
		8.6.1 Doppler's Effect in Constant Gravitational Field and Consequences
		8.6.2 Bending of Light-Rays in a Constant Gravitational Field
		8.6.3 The Basic Incompatibility Between Gravity and Special Relativity
9 General Relativity and Relativistic Cosmology
	9.1 What Is a Good Theory of Gravity?
		9.1.1 Metric or Connections?
		9.1.2 The Role of Equivalence Principle
	9.2 Gravity Seen Through Geometry in General Relativity
		9.2.1 Einstein's Landscape for the Constant Gravitational Field
	9.3 The Einstein–Hilbert Action and the Einstein Field Equations
	9.4 A Short Introduction to f(R) Gravity
	9.5 The Energy-Momentum Tensor and Another Proof for Einstein's Field Equations
		9.5.1 The Covariant Derivative of the Energy-Momentum Tensor
		9.5.2 Another Proof for Einstein's Field Equations
	9.6 Introducing the Cosmological Constant
	9.7 The Schwarzschild Solution of Vacuum Field Equations
		9.7.1 Orbit of a Planet in the Schwarzschild Metric
		9.7.2 Relativistic Solution of the Mercury Perihelion Drift Problem
		9.7.3 Speed of Light in a Given Metric
		9.7.4 Bending of Light in Schwarzschild Metric
	9.8 About Einstein's Metric: Einstein's Computations Related to Perihelion's Drift and Bending of the Light Rays
	9.9 Solutions of General Einstein's Field Equations: The Friedmann–Lemaître–Robertson–Walker Models of Universe
		9.9.1 The Cosmological Expansion
	9.10 The Fermi Coordinates
		9.10.1 Determining the Fermi Coordinates
		9.10.2 The Fermi Viewpoint on Einstein's Field Equations in Vacuum
		9.10.3 The Gravitational Coupling in Einstein's Field Equations: K =8πGc4
	9.11 Weak Gravitational Field and the Classical Counterparts of the Relativistic Equations
	9.12 The Einstein Static Universe and its Cosmological Constant
	9.13 Cosmic Strings
	9.14 The Case of Planar Gravitational Waves
	9.15 The Gödel Universe
	9.16 Black Holes: A Mathematical Introduction
		9.16.1 Escape Velocity and Black Holes
		9.16.2 Rindler's Metric and Pseudo-Singularities
		9.16.3 Black Holes Studied Through the Schwarzschild Metric
		9.16.4 The Light Cone in the Schwarzschild Metric
10 A Geometric Realization of Relativity: The Affine Universe and de Sitter Spacetime
	10.1 About the Minkowski Geometric Gravitational Force
	10.2 The de Sitter Spacetime and Its Cosmological Constant
	10.3 Some Physical Considerations
	10.4 A FLRW Metric for de Sitter Spacetime Given …
	10.5 Deriving Cosmological Singularities in the Context of de Sitter Spacetime
11 Conclusions
Appendix  References
		
Index