A Mathematical Journey to Relativity : Deriving Special and General Relativity with Basic Mathematics

• Autore: Boskoff Wladimir-Georges
• Pubblicazione: Cham : , : Springer International Publishing AG, , 2024; ©2024
• Edizione: 2nd ed.
• Descrizione fisica: 1 online resource (556 pages)
• Altri autori: CapozzielloSalvatore
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: A Mathematical Journey to Relativity
• ISBN: 3-031-54823-X
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 9910855391603321
• Lo trovi qui: Univ. Federico II
• Serie: UNITEXT for Physics Series