Giinter Ludyk Einstein in Matrix Form Exact Derivation of the Theory of Special and General Relativity without Tensors ^ Springer
Contents 1 Special Relativity 1 1.1 Galilei Transformation 1 1.1.1 Relativity Principle of Galilei 1 1.1.2 General Galilei Transformation 5 1.1.3 Maxwell's Equations and Galilei Transformation 6 1.2 Lorentz Transformation 7 1.2.1 Introduction 7 1.2.2 Determining the Components of the Transformation Matrix 8 1.2.3 Simultaneity at Different Places 12 1.2.4 Length Contraction of Moving Bodies 13 1.2.5 Time Dilation 15 1.3 Invariance of the Quadratic Form 15 1.3.1 Invariance with Respect to Lorentz Transformation 17 1.3.2 Light Cone 17 1.3.3 Proper Time 19 1.4 Relativistic Velocity Addition 20 1.4.1 Galilean Addition of Velocities 20 1.5 Lorentz Transformation of the Velocity 22 1.6 Momentum and Its Lorentz Transformation 25 1.7 Acceleration and Force 26 1.7.1 Acceleration 26 1.7.2 Equation of Motion and Force 28 1.7.3 Energy and Rest Mass 30 1.7.4 Emission of Energy 31 1.8 Relativistic Electrodynamics 32... 1.8.1 Maxwell's Equations 32 1.8.2 Lorentz Transformation of the Maxwell's Equations 34 1.8.3 Electromagnetic Invariants 37 1.8.4 Electromagnetic Forces 39 ix
x Contents 1.9 The Energy-Momentum Matrix 41 1.9.1 The Electromagnetic Energy-Momentum Matrix 41 1.9.2 The Mechanical Energy-Momentum Matrix 43 1.9.3 The Total Energy-Momentum Matrix 47 1.10 The Most Important Definitions and Formulas in Special Relativity 48 2 Theory of General Relativity 51 2.1 General Relativity and Riemannian Geometry 51 2.2 Motion in a Gravitational Field 53 2.2.1 First Solution 54 2.2.2 Second Solution 55 2.2.3 Relation Between t and G 56 2.3 Geodesic Lines and Equations of Motion 57 2.3.1 Alternative Geodesic Equation of Motion 62 2.4 Example: Uniformly Rotating Systems 64 2.5 General Coordinate Transformations 67 2.5.1 Absolute Derivatives 67 2.5.2 Transformation of the Christoffel Matrix t 69 2.5.3 Transformation of the Christoffel Matrix T 71 2.5.4 Coordinate Transformation and Covariant Derivative... 72 2.6 Incidental Remark 76 2.7 Parallel Transport 78 2.8 Riemannian Curvature Matrix 80 2.9 Properties of the Riemannian Curvature Matrix 81 2.9.1 Composition of R and R 81 2.10 The Ricci Matrix and Its Properties 88 2.10.1 Symmetry of the Ricci Matrix/?Ric 90 2.10.2 The Divergence of the Ricci Matrix 92 2.11 General Theory of Gravitation 93 2.11.1 The Einstein's Matrix & 93 2.11.2 Newton's Theory of Gravity 94 2.11.3 The Einstein's Equation with 65 97 2.12 Summary 99 2.12.1 Covariance Principle 99 2.12.2 Einstein's Field Equation and Momentum 101 2.13 Hilbert's Action Functional 101 2.13.1 Effects of Matter 105 2.14 Most Important Definitions and Formulas 106 3 Gravitation of a Spherical Mass 109 3.1 Schwarzschild's Solution 109 3.1.1 Christoffel Matrix T 110.3.1.2 Ricci Matrix RRic 112 3.1.3 The Factors A(r) and B(r) 114 3.2 Influence of a Massive Body on the Environment 116 3.2.1 Introduction 116
Contents xi 3.2.2 Changes to Length and Time 117 3.2.3 Redshift of Spectral Lines 118 3.2.4 Deflection of Light 120 3.3 Schwarzschild's Inner Solution 124 3.4 Black Holes 127 3.4.1 Astrophysics 127 3.4.2 Further Details about "Black Holes" 129 3.4.3 Singularities 131 3.4.4 Eddington's Coordinates 135 3.5 Rotating Masses 138 3.5.1 Ansatz for the Metric Matrix G 138 3.5.2 Kerr's Solution in Boyer-Lindquist Coordinates 139 3.5.3 The Lense-Thirring Effect 139 3.6 Summary of Results for the Gravitation of a Spherical Mass... 141 3.7 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.. 161 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
xii Contents Appendix C Geodesic Deviation 179 Appendix D Another Ricci-Matrix 183 References 189 Index 191