Giinter Ludyk. Einstein in Matrix. Form. Exact Derivation of the Theory of Special. without Tensors. and General Relativity.

Transcription

1 Giinter Ludyk Einstein in Matrix Form Exact Derivation of the Theory of Special and General Relativity without Tensors ^ Springer

2 Contents 1 Special Relativity Galilei Transformation Relativity Principle of Galilei General Galilei Transformation Maxwell's Equations and Galilei Transformation Lorentz Transformation Introduction Determining the Components of the Transformation Matrix Simultaneity at Different Places Length Contraction of Moving Bodies Time Dilation Invariance of the Quadratic Form Invariance with Respect to Lorentz Transformation Light Cone Proper Time Relativistic Velocity Addition Galilean Addition of Velocities Lorentz Transformation of the Velocity Momentum and Its Lorentz Transformation Acceleration and Force Acceleration Equation of Motion and Force Energy and Rest Mass Emission of Energy Relativistic Electrodynamics Maxwell's Equations Lorentz Transformation of the Maxwell's Equations Electromagnetic Invariants Electromagnetic Forces 39 ix

3 x Contents 1.9 The Energy-Momentum Matrix The Electromagnetic Energy-Momentum Matrix The Mechanical Energy-Momentum Matrix The Total Energy-Momentum Matrix The Most Important Definitions and Formulas in Special Relativity 48 2 Theory of General Relativity General Relativity and Riemannian Geometry Motion in a Gravitational Field First Solution Second Solution Relation Between t and G Geodesic Lines and Equations of Motion Alternative Geodesic Equation of Motion Example: Uniformly Rotating Systems General Coordinate Transformations Absolute Derivatives Transformation of the Christoffel Matrix t Transformation of the Christoffel Matrix T Coordinate Transformation and Covariant Derivative Incidental Remark Parallel Transport Riemannian Curvature Matrix Properties of the Riemannian Curvature Matrix Composition of R and R The Ricci Matrix and Its Properties Symmetry of the Ricci Matrix/?Ric The Divergence of the Ricci Matrix General Theory of Gravitation The Einstein's Matrix & Newton's Theory of Gravity The Einstein's Equation with Summary Covariance Principle Einstein's Field Equation and Momentum Hilbert's Action Functional Effects of Matter Most Important Definitions and Formulas Gravitation of a Spherical Mass Schwarzschild's Solution Christoffel Matrix T Ricci Matrix RRic The Factors A(r) and B(r) Influence of a Massive Body on the Environment Introduction 116

4 Contents xi Changes to Length and Time Redshift of Spectral Lines Deflection of Light Schwarzschild's Inner Solution Black Holes Astrophysics Further Details about "Black Holes" Singularities Eddington's Coordinates Rotating Masses Ansatz for the Metric Matrix G Kerr's Solution in Boyer-Lindquist Coordinates The Lense-Thirring Effect Summary of Results for the Gravitation of a Spherical Mass Concluding Remark 143 Appendix A Vectors and Matrices 145 A.l Vectors and Matrices 145 A.2 Matrices 147 A.2.1 Types of Matrices 147 A.2.2 Matrix Operations 148 A.2.3 Block Matrices 152 A.3 The Kronecker-Product 154 A.3.1 Definitions 154 A.3.2 Some Theorems 154 A.3.3 The Permutation Matrix Upxq 156 A.3.4 More Properties of the Kronecker-Product 157 A.4 Derivatives of Vectors/Matrices with Respect to Vectors/Matrices. 157 A.4.1 Definitions 157 A.4.2 Product Rule 158 A.4.3 Chain Rule 159 A.5 Differentiation with Respect to Time 159 A.5.1 Differentiation of a Function with Respect to Time 159 A.5.2 Differentiation of a Vector with Respect to Time 160 A.5.3 Differentiation of a 2 x 3-Matrix with Respect to Time A. 5.4 Differentiation of an n x m-matrix with Respect to Time. 161 A. 6 Supplements to Differentiation with Respect to a Matrix 162 Appendix B Some Differential Geometry 165 B. l Curvature of a Curved Line in Three Dimensions 165 B.2 Curvature of a Surface in Three Dimensions 166 B. 2.1 Vectors in the Tangent Plane 166 B.2.2 Curvature and Normal Vectors 168 B.2.3 Theorema Egregium and the Inner Values gij 169

5 xii Contents Appendix C Geodesic Deviation 179 Appendix D Another Ricci-Matrix 183 References 189 Index 191