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Mathematics of Relativity

George Yuri Rainich
Publisher: 
Dover Publications
Publication Date: 
2014
Number of Pages: 
173
Format: 
Paperback
Price: 
14.95
ISBN: 
9780486783253
Category: 
Textbook
[Reviewed by
Allen Stenger
, on
04/28/2015
]

This is an old but still useful guide to the mathematics behind relativity, covering mostly tensors and metrics (i.e., curved space, Riemann tensor, general coordinates, and geodesics). It is a 2014 Dover unaltered reprint of the 1950 Wiley edition.

The math and the physics are woven together so that we see the mathematical tools being developed and then immediately applied to physical problems. The approach is from mathematics and is rigorous, with tensors being developed in the modern way, as linear forms rather than dyads of vectors. The book starts with “Old Physics” (Newtonian) and Cartesian geometry, and gradually generalizes to relativity.

The major weakness of the book is that it is formatted as a continuous narrative without much explicit structure, so it’s hard to jump around or look back to refresh your memory. For example, there is a complete development of all the properties of tensors in 12 pages (Section 10), but there is nothing there labeled Definition or Theorem.

Despite its age, the book has held up well. Its coverage of general relativity is a little skimpy (there’s no mention of cosmology), and there have been some new developments since 1950 that are not covered. A thoroughly-modern look at the same material (including curved space but omitting general relativity) is Faraoni’s 2014 book Special Relativity.


Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His mathematical interests are number theory and classical analysis.

  • 1 OLD PHYSICS
    • 1. Motion of a Particle. The Inverse Square Law
    • 2. Two Pictures of Matter
    • 3. Vectors, Tensors, Operations
    • 4. Maxwell's Equations
    • 5. The Stress-Energy Tensor
    • 6. General Equations of Motion. The Complete Tensor
  • 2 NEW GEOMETRY
    • 7. Analytic Geometry of Four Dimensions
    • 8. Transformation of Coordinates
    • 9. Postulates for Four-Dimensional Geometry
    • 10. Tensor Analysis
    • 11. Complications Resulting from Imaginary Coordinates
    • 12. Are the Equations of Physics Invariant?
    • 13. Curves in the New Geometry
  • 3 SPECIAL RELATIVITY
    • 14. Equations of Motion
    • 15. Lorentz Transformations
    • 16. Addition of Velocities
    • 17. Light Corpuscles, or Photons
    • 18. Electricity and Magnetism in Special Relativity
    • 19. The Complete Tensor
  • 4 CURVED SPACE
    • 20. Curvature of Curves and Surfaces
    • 21. Generalizations
    • 22. The Riemann Tensor
    • 23. Points and Vectors in General Coordinates
    • 24. Tensors in General Coordinates
    • 25. Differentiation in General Coordinates
    • 26. General Coordinates in Curved Spaces
    • 27. Riemann Tensor in General Coordinates
    • 28. Geodesics
    • 29. Equations of Physics in General Coordinates
  • 5 GENERAL RELATIVITY
    • 30. The Law of Geodesics
    • 31. Solar System. Symmetry Conditions
    • 32. Solution of the Field Equations
    • 33. Equations of Geodesics
    • 34. Newtonian Motion of a Planet
    • 35. Relativity Motion of a Planet
    • 36. Deflection of Light
    • 37. Shift of Spectral Lines
  • CONCLUSION
  • INDEX

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