| A Brief Note on the Publication of This Work |
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xi | |
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| Foreword |
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xiii | |
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| Preface |
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xvii | |
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The Charm of a Manuscript |
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1 | (6) |
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Einstein's Intellectual Odyssey to General Relativity |
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7 | (30) |
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37 | (122) |
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The titles set in italics are Einstein's original titles featured in the manuscript; the page numbers in the titles relate to the annotated pages facing the original facsimile |
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The Foundation of the General Theory of Relativity |
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A Fundamental Considerations on the Postulate of Relativity |
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§1 Observations on the Special Theory of Relativity |
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P. 1 Why did Einstein go beyond special relativity? |
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39 | (2) |
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P. 2 What was wrong with the classical notions of space and time? |
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41 | (2) |
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§2 The Need for an Extension of the Postulate of Relativity |
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P. 3 Why did Einstein see difficulties that others ignored? |
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43 | (2) |
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P. 4 What was Einstein's happiest thought and how did it come about? |
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45 | (2) |
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§3 The Spacetime Continuum. Requirement of General Covariance for the Equations Expressing General Laws of Nature |
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P. 5 Why does Einstein's theory of gravitation require non-Euclidean geometry? |
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47 | (2) |
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P. 6 What is the role of coordinates in the new theory of gravitation? |
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49 | (2) |
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P. 7 What is the meaning of general covariance? |
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51 | (2) |
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§4 The Relation of the Four Coordinates to Measurement in Space and Time |
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P. 8 What is the geometry of spacetime? |
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53 | (2) |
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P. 9 When did Einstein realize that gravitation has to be described by a complex mathematical expression? |
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55 | (2) |
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B Mathematical Aids to the Formulation of Generally Covariant Equations |
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§5 Contravariant and Covariant Four-vectors |
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P. 10 Why tensors, vectors, scalars? |
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57 | (2) |
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P. 11 When did Einstein realize that he needed more sophisticated mathematical methods? |
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59 | (2) |
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§6 Tensors of the Second and Higher Ranks |
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P. 12 What lessons could be learned from the theory of electromagnetism? |
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61 | (2) |
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§7 Multiplication of Tensors |
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P. 13 How can tensors be manipulated to produce new tensors by different tensor operations? |
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63 | (1) |
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P. 13 What were Einstein's heuristic guidelines in his search for a relativistic theory of gravitation? |
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63 | (2) |
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P. 14 What was Einstein's strategy in constructing a gravitational field equation? |
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65 | (2) |
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§8 Some Aspects of the Fundamental Tensor g |
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P. 15 Why is the metric tensor so fundamental? |
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67 | (2) |
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P. 16 Why is the Zurich Notebook a unique document in the history of physics? |
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69 | (1) |
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P. 16 How are volumes measured in curved spacetime? |
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69 | (2) |
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P. 17 How can a convenient choice of coordinates simplify the theory? |
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71 | (1) |
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P. 17 What is the difference between a coordinate condition and a coordinate restriction'? |
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71 | (2) |
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§9 The Equation of the Geodetic Line. The Motion of a Particle |
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P. 18 What is the meaning of a "straight line" in curved space, and how does a particle move under the influence of gravitation? |
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73 | (2) |
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P. 19 What is the geometric and physical meaning of "Christoffel symbols"? |
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75 | (1) |
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P. 19 What was Einstein's "fatal prejudice" in the early identification of the gravitational field components? |
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75 | (2) |
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§10 The Formation of Tensors by Differentiation |
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P. 20 The geodetic line as the "straightest" possible line and its relation to the concept of "affine connection" |
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77 | (2) |
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P. 21 How do tensors change between neighboring points, or how can one produce new tensors from given tensors by differentiation? |
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79 | (2) |
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P. 22 What is the geometric context of Einstein's mathematical formulation of general relativity? |
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81 | (2) |
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§11 Some Cases of Special Importance |
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P. 23 The Entwurf theory as an intermediate step toward the general theory of relativity |
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83 | (2) |
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P. 24 What is the divergence of a vector field? What are other vector field concepts? |
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85 | (2) |
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P. 25 What is the mathematical formulation of energy-momentum conservation in general relativity? |
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87 | (2) |
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§12 The Riemann-Christoffel Tensor |
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P. 26 What is the geometric meaning of the Riemann-Christoffel tensor? |
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89 | (2) |
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P. 27 What was the "presumed gravitational tensor" and why was it abandoned? |
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91 | (2) |
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C Theory of the Gravitational Field |
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§13 Equations of Motion of a Material Point in the Gravitational Field. Expression for the Field-components of Gravitation |
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P. 28 When did Einstein lose faith in the Entwurf theory? |
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93 | (1) |
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P. 28 How does a particle move in a gravitational field? |
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93 | (2) |
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§14 The Field Equations of Gravitation in the Absence of Matter |
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P. 29 What was Einstein's greatest challenge? |
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95 | (2) |
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§15 The Hamiltonian Function for the Gravitational Field. Laws of Momentum and Energy |
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P. 30 What is the Lagrangian formalism, and what was its role in the genesis of general relativity? |
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97 | (2) |
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P. 31 What happens to the energy-momentum conservation principle in the absence of matter, or can the gravitational field be a source of itself? |
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99 | (1) |
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P. 31 Who was Einstein's main competitor? |
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99 | (2) |
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P. 32 How can the field equation without matter be generalized to include matter? |
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101 | (2) |
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§16 The General Form of the Field Equations of Gravitation |
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P. 33 The gravitational field equation---at last! |
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103 | (2) |
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§17 The Laws of Conservation in the General Case |
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P. 34 How is the conservation principle satisfied in a way that Einstein did not expect in the early stages of development of his theory? |
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105 | (2) |
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§18 The Laws of Momentum and Energy for Matter, as a Consequence of the Field Equations |
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P. 35 Do physical conservation laws follow from symmetries in nature? |
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107 | (2) |
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§19 Euler's Equations for a Frictionless Adiabatic Fluid |
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P. 36 How do established theories in physics, like hydrodynamics and electromagnetism, fit into the new theory of gravitation? |
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109 | (2) |
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§20 Maxwell's Electromagnetic Field Equations for Free Space |
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P. 37 How did Maxwell represent the laws of electromagnetism by mathematical equations and how are these equations affected by gravitation? |
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111 | (2) |
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P. 38 What was the role of "ether" in prerelativity physics, and why did Einstein eventually think that space without ether is unthinkable? |
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113 | (2) |
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P. 39 What was von Laue's crucial role? |
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115 | (2) |
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§21 Newton's Theory as a First Approximation |
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P. 40 How can the validity of the theory be tested experimentally? |
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117 | (2) |
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P. 40a What did Einstein wish to clarify and emphasize as an afterthought? |
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119 | (2) |
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P. 41 What does the metric tensor look like in the Newtonian limit? |
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121 | (1) |
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P. 41 Why was Einstein pleasantly surprised? |
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121 | (2) |
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P. 42 How could astronomers help confirm certain predictions of the theory? |
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123 | (2) |
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§22 Behavior of Rods and Clocks in the Static Gravitational Field. Bending of Light-rays. Motion of the Perihelion of a Planetary Orbit |
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P. 43 What is the length of rods and the pace of clocks in a gravitational field? |
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125 | (1) |
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P. 43 Is there a "viable" alternative theory to general relativity? |
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125 | (2) |
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P. 44 What observation catapulted Einstein to world celebrity status? |
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127 | (2) |
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P. 45 Explanation of the motion of perihelion of planet Mercury: From disappointment to triumph |
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129 | (2) |
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Appendix: Presentation of the Theory on the Basis of a Variational Principle |
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§1 The Field Equations of Gravitation and Matter |
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P. A1 Why did Einstein decide not to include this "Appendix" in the printed version of the manuscript "Foundation of General Relativity"? |
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131 | (1) |
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P. A1 Einstein applies a Hamiltonian (Lagrangian) formulation---different from Hilbert's and different from his own previous one |
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131 | (2) |
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P. A2 Why did Einstein decide to publish a modified version of this appendix after all? What were the roles of Lorentz and Hilbert? |
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133 | (2) |
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§2 Formal Consequences of the Requirement of General Covariance |
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P. A3 Is the conservation principle satisfied without any restrictions? |
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135 | (1) |
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§3 Properties of Hamilton's Function G |
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P. A3 1916: A year of hard work and new beginnings |
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135 | (2) |
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P. A4 Einstein acts as a missionary of science |
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137 | (2) |
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P. A5 Scientific creativity in the midst of personal hardships and national disaster |
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139 | (2) |
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Notes on the Annotation Pages |
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141 | (8) |
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Postscript: The Drama Continues |
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149 | (10) |
| A Chronology of the Genesis of General Relativity and Its Formative Years |
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159 | (6) |
| Physicists, Mathematicians, and Philosophers Relevant to Einstein's Thinking |
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165 | (14) |
| Further Reading |
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179 | (4) |
| English Translation of "The Foundation of the General Theory of Relativity" |
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183 | (44) |
| English Translation of "Hamilton's Principle and the General Theory of Relativity" |
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227 | (6) |
| Index |
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233 | |
