Relativity and Geometry

Introduction


1. Newtonian Principles


1.1. The Task of Natural Philosophy


1.2. Absolute Space


1.3. Absolute Time


1.4. Rigid Frames and Coordinates


1.5. Inertial Frames and Newtonian Relativity


1.6. Newtonian Spacetime


1.7. Gravitation


2. Electrodynamics and the Aether


2.1. Nineteenth-Century Views on Electromagnetic Action


2.2. The Relative Motion of the Earth and the Aether


3. Einstein's 'Electrodynamics of Moving Bodies'


3.1. Motivation


3.2. The Definition of Time in an Inertial Frame


3.3. The Principles of Special Relativity


3.4. The Lorentz Transformation. Einstein's Derivation of 1905


3.5. The Lorentz Transformation. Some Corollaries and Applications


3.6. The Lorentz Transformation. Linearity


3.7. The Lorentz Transformation. Ignatowsky's Approach


3.8. The "Relativity Theory of Poincaré and Lorentz"


4. Minkowski Spacetime


4.1. The Geometry of the Lorentz Group


4.2. Minkowski Spacetime as an Affine Metric Space and as a Riemannian Manifold


4.3. Geometrical Objects


4.4. Concept Mutation at the Birth of Relativistic Dynamics


4.5. A Glance at Spacetime Physics


4.6. The Causal Structure of Minkowski Spacetime


5. Einstein's Quest for a Theory of Gravity


5.1. Gravitation and Relativity


5.2. The Principle of Equivalence


5.3. Gravitation and Geometry circa 1912


5.4. Departure from Flatness


5.5. General Covariance and the Einstein-Grossmann Theory


5.6. Einstein's Arguments against General Covariance: 1913-14


5.7. Einstein's Papers of November 1915


5.8. Einstein's Field Equations and the Geodesic Law of Motion


6. Gravitational Geometry


6.1. Structures of Spacetime


6.2. Mach's Principle and the Advent of Relativistic Cosmology


6.3. The Friedmann Worlds


6.4. Singularities


7. Disputed Questions


7.1. The Concept of Simultaneity


7.2. Geometric Conventionalism


7.3. Remarks on Time and Causality

Appendix

A. Differentiable Manifolds

B. Fiber Bundles

C. Linear Connections


1. Vector-valued Differential Forms

2. The Lie Algebra of a Lie Group

3. Connections in a Principal Fiber Bundle

4. Linear Connections

5. Covariant Differentiation

6. The Torsion and Curvature of a Linear Connection

7. Geodesies

8. Metric Connections in Riemannian Manifolds

D. Useful Formula

Notes

References

Index