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El. knyga: Einstein in Matrix Form: Exact Derivation of the Theory of Special and General Relativity without Tensors

5.00/5 (4 ratings by Goodreads)
  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Physics
  • Išleidimo metai: 12-Jun-2013
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642357985
Kitos knygos pagal šią temą:
Einstein in Matrix Form: Exact Derivation of the Theory of Special and General Relativity without TensorsDidesnis paveikslėlis
  • Formatas: PDF+DRM
  • Serija: Graduate Texts in Physics
  • Išleidimo metai: 12-Jun-2013
  • Leidėjas: Springer-Verlag Berlin and Heidelberg GmbH & Co. K
  • Kalba: eng
  • ISBN-13: 9783642357985
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This book derives the fundamental equations of Einstein's theory of special and general relativity using matrix calculus, without the help of tensors. It provides the necessary mathematical tools in a user-friendly way.

This book is an introduction to the theories of Special and General Relativity. The target audience are physicists, engineers and applied scientists who are looking for an understandable introduction to the topic - without too much new mathematics. The fundamental equations of Einstein's theory of Special and General Relativity are derived using matrix calculus, without the help of tensors. This feature makes the book special and a valuable tool for scientists and engineers with no experience in the field of tensor calculus. In part I the foundations of Special Relativity are developed, part II describes the structure and principle of General Relativity. Part III explains the Schwarzschild solution of spherical body gravity and examines the "Black Hole" phenomenon. Any necessary mathematical tools are user friendly provided, either directly in the text or in the appendices.

Recenzijos

From the book reviews:

Addressing physicists, applied scientists and engineers with no previous knowledge of tensor calculus, Ludyk presents in a well-written manner a rather easy introduction to special as well as general relativity by solely using matrix calculus and elementary differential geometry. It is more than certain that Ludyks treatment will make it possible for a larger number of students to get acquainted with special and general relativity theory at an introductory, undergraduate level. (Theophanes Grammenos, Mathematical Reviews, April, 2014)

The main audience for this book can be those who have experience with practical calculations by use of matrices and vectors, but who have not the time to become familiar with tensor calculus. this book presents essentially all ingredients what one expects from a book aimed to introduce special and general relativity . (Hans-Jürgen Schmidt, zbMATH, Vol. 1272, 2013)

1 Special Relativity
1(50)
1.1 Galilei Transformation
1(6)
1.1.1 Relativity Principle of Galilei
1(4)
1.1.2 General Galilei Transformation
5(1)
1.1.3 Maxwell's Equations and Galilei Transformation
6(1)
1.2 Lorentz Transformation
7(8)
1.2.1 Introduction
7(1)
1.2.2 Determining the Components of the Transformation Matrix
8(4)
1.2.3 Simultaneity at Different Places
12(1)
1.2.4 Length Contraction of Moving Bodies
13(2)
1.2.5 Time Dilation
15(1)
1.3 Invariance of the Quadratic Form
15(5)
1.3.1 Invariance with Respect to Lorentz Transformation
17(1)
1.3.2 Light Cone
17(2)
1.3.3 Proper Time
19(1)
1.4 Relativistic Velocity Addition
20(2)
1.4.1 Galilean Addition of Velocities
20(2)
1.5 Lorentz Transformation of the Velocity
22(3)
1.6 Momentum and Its Lorentz Transformation
25(1)
1.7 Acceleration and Force
26(6)
1.7.1 Acceleration
26(2)
1.7.2 Equation of Motion and Force
28(2)
1.7.3 Energy and Rest Mass
30(1)
1.7.4 Emission of Energy
31(1)
1.8 Relativistic Electrodynamics
32(9)
1.8.1 Maxwell's Equations
32(2)
1.8.2 Lorentz Transformation of the Maxwell's Equations
34(3)
1.8.3 Electromagnetic Invariants
37(2)
1.8.4 Electromagnetic Forces
39(2)
1.9 The Energy-Momentum Matrix
41(7)
1.9.1 The Electromagnetic Energy-Momentum Matrix
41(2)
1.9.2 The Mechanical Energy-Momentum Matrix
43(4)
1.9.3 The Total Energy-Momentum Matrix
47(1)
1.10 The Most Important Definitions and Formulas in Special Relativity
48(3)
2 Theory of General Relativity
51(58)
2.1 General Relativity and Riemannian Geometry
51(2)
2.2 Motion in a Gravitational Field
53(4)
2.2.1 First Solution
54(1)
2.2.2 Second Solution
55(1)
2.2.3 Relation Between Γ and G
56(1)
2.3 Geodesic Lines and Equations of Motion
57(7)
2.3.1 Alternative Geodesic Equation of Motion
62(2)
2.4 Example: Uniformly Rotating Systems
64(3)
2.5 General Coordinate Transformations
67(9)
2.5.1 Absolute Derivatives
67(2)
2.5.2 Transformation of the Christoffel Matrix Γ
69(2)
2.5.3 Transformation of the Christoffel Matrix Γ
71(1)
2.5.4 Coordinate Transformation and Covariant Derivative
72(4)
2.6 Incidental Remark
76(2)
2.7 Parallel Transport
78(2)
2.8 Riemannian Curvature Matrix
80(1)
2.9 Properties of the Riemannian Curvature Matrix
81(7)
2.9.1 Composition of R and R
81(7)
2.10 The Ricci Matrix and Its Properties
88(5)
2.10.1 Symmetry of the Ricci Matrix RRic
90(2)
2.10.2 The Divergence of the Ricci Matrix
92(1)
2.11 General Theory of Gravitation
93(6)
2.11.1 The Einstein's Matrix
93(1)
2.11.2 Newton's Theory of Gravity
94(3)
2.11.3 The Einstein's Equation with
97(2)
2.12 Summary
99(2)
2.12.1 Covariance Principle
99(2)
2.12.2 Einstein's Field Equation and Momentum
101(1)
2.13 Hilbert's Action Functional
101(5)
2.13.1 Effects of Matter
105(1)
2.14 Most Important Definitions and Formulas
106(3)
3 Gravitation of a Spherical Mass
109(36)
3.1 Schwarzschild's Solution
109(7)
3.1.1 Christoffel Matrix Γ
110(2)
3.1.2 Ricci Matrix RRic
112(2)
3.1.3 The Factors A(r) and B(r)
114(2)
3.2 Influence of a Massive Body on the Environment
116(8)
3.2.1 Introduction
116(1)
3.2.2 Changes to Length and Time
117(1)
3.2.3 Redshift of Spectral Lines
118(2)
3.2.4 Deflection of Light
120(4)
3.3 Schwarzschild's Inner Solution
124(3)
3.4 Black Holes
127(11)
3.4.1 Astrophysics
127(2)
3.4.2 Further Details about "Black Holes"
129(2)
3.4.3 Singularities
131(4)
3.4.4 Eddington's Coordinates
135(3)
3.5 Rotating Masses
138(3)
3.5.1 Ansatz for the Metric Matrix G
138(1)
3.5.2 Kerr's Solution in Boyer-Lindquist Coordinates
139(1)
3.5.3 The Lense-Thirring Effect
139(2)
3.6 Summary of Results for the Gravitation of a Spherical Mass
141(2)
3.7 Concluding Remark
143(2)
Appendix A Vectors and Matrices
145(20)
A.1 Vectors and Matrices
145(2)
A.2 Matrices
147(7)
A.2.1 Types of Matrices
147(1)
A.2.2 Matrix Operations
148(4)
A.2.3 Block Matrices
152(2)
A.3 The Kronecker-Product
154(3)
A.3.1 Definitions
154(1)
A.3.2 Some Theorems
154(2)
A.3.3 The Permutation Matrix U p×q
156(1)
A.3.4 More Properties of the Kronecker-Product
157(1)
A.4 Derivatives of Vectors/Matrices with Respect to Vectors/Matrices
157(2)
A.4.1 Definitions
157(1)
A.4.2 Product Rule
158(1)
A.4.3 Chain Rule
159(1)
A.5 Differentiation with Respect to Time
159(3)
A.5.1 Differentiation of a Function with Respect to Time
159(1)
A.5.2 Differentiation of a Vector with Respect to Time
160(1)
A.5.3 Differentiation of a 2 × 3-Matrix with Respect to Time
161(1)
A.5.4 Differentiation of an n × m-Matrix with Respect to Time
161(1)
A.6 Supplements to Differentiation with Respect to a Matrix
162(3)
Appendix B Some Differential Geometry
165(14)
B.1 Curvature of a Curved Line in Three Dimensions
165(1)
B.2 Curvature of a Surface in Three Dimensions
166(13)
B.2.1 Vectors in the Tangent Plane
166(2)
B.2.2 Curvature and Normal Vectors
168(1)
B.2.3 Theorema Egregium and the Inner Values gij
169(10)
Appendix C Geodesic Deviation
179(4)
Appendix D Another Ricci-Matrix
183(6)
References 189(2)
Index 191
After receiving his PhD in 1967, Günter Ludyk habilitated and has been appointed Scientific Advisor and Professor (associate professor) of the Technical University of Berlin in 1970. In  1971 he has been a visiting professor at the Technical University of Graz/Austrial. Since 1972 he is a Full Professor at the Physics/Electrical Engineering Faculty of the University of Bremen. His area of research includes the theory of dynamical systems and the application of interval mathematics to generate high-precision results. He published various books on these topics both in German and English, e. g.  "Time-Variant Discrete-Time-Systems in 1981 and "Stability of Time-Variant Discrete-Time Systems in 1985.
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