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Introduction to Tensor Calculus, Relativity and Cosmology

D. F. Lawden
Dover Publications
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on

First published in 1982 and now a reissue by the much-appreciated Dover Publications, the book under review was offered by its author as a supporting text for university and early graduate courses in mathematical physics at British universities. Lawden based his work on some two decades of experience teaching this material. His goal was to provide his charges with a solid approach to modern relativistic physics and its methodologies, free of unnecessary and outdated scaffolding (he refers to the Michelson-Morley experiment as a “historical accident,” with special relativity being in the works regardless). The compact book is replete with a wealth of exercises to bring the reader into the thick of things effectively and does so, pardon the play on words, relatively swiftly.

Lawden clearly wants to provide fledgling researchers in this wonderful area of mathematical physics with a very solid and relevant background for delving deeper and deeper into the material, and of course eventually to get to research. I can’t help but note that this particular boot-camp approach is idiosyncratic: I think it is something of a characteristic of the British system, and this is certainly a virtue, given that the student is given a heavy responsibility for doing his own learning — do the reading, write all over the book (or, if you’re not so inclined, all over note books and tablets), and do the exercises — the text is fast-paced and no-nonsense. After all the kids are to be prepared for sporty examinations.

The author gets to the point right away: special relativity is featured in chapters 1 and 3, with a solid start in chapter 2 on tensors (in the sense geometers and physicists — don’t go looking for \(\otimes\)s and universality properties: it’s all about doing funky things with indices. In the afterlife, Einstein and Dirac are smiling…). Lawden downshifts in chapter 4: it’s about special relativity electrodynamics; thereafter it’s on to general relativity. Chapter 5 does a load of Riemannian geometry and general tensor calculus, and in Chapter 6 the big cat is let out of the bag: we hit general relativity properly so-called with extreme prejudice.

Lawden starts with the principle of equivalence and the business of a metric in a gravitational field, does such marvelous things as analysis of particles’ behavior in a gravitational field and the effect of gravitation on light rays and on spectral lines (hubba hubba!), and ends with black holes and gravitational waves. Finally, with the stage having been set in no uncertain terms in the previous discussions, Chapter 7 is all on cosmology. We go from cosmical time and spaces of constant curvature through model universes à la Einstein and de Sitter, and then to particle and event horizons. Even though the material dates to over thirty years ago, there is, as Feynman would put it, a great deal, or even most, of “the good stuff” to be had here.

Again, the book is compact and dense (about 200 pages), laden with pretty beefy and sporty exercise sets that need to be taken very seriously, ambitious and even a bit polemical: for instance, Lawden complains about the fact that many other authors, including authors of very serious books and papers appearing in serious journals, suggest that there is still trouble about paradoxes of relativistic physics that in truth have been completely resolved. His book hits this mendacity square on, and thus intends his reader to be free of such misconceptions and therefore more effective and honest as a future scholar. Feisty stuff, and pedagogically ambitious. But I think Lawden’s book is on target: if you want to learn relativity of both flavors in the proper modern context, with boots on the ground, go for it here.

Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.


List of Constants
Chapter 1 Special Principle of Relativity. Lorentz Transformations
1. Newton's laws of motion
2. Covariance of the laws of motion
3. Special principle of relativity
4. Lorentz transformations. Minkowski space-time
5. The special Lorentz transformation
6. Fitzgerald contraction. Time dilation
7. Spacelike and timelike intervals. Light cone
  Exercises 1
Chapter 2 Orthogonal Transformations. Cartesian Tensors
8. Orthogonal transformations
9. Repeated-index summation convention
10. Rectangular Cartesian tensors
11. Invariants. Gradients. Derivatives of tensors
12. Contraction. Scalar product. Divergence
13. Pseudotensors
14. Vector products. Curl
  Exercises 2
Chapter 3 Special Relativity Mechanics
15. The velocity vector
16. Mass and momentum
17. The force vector. Energy
18. Lorentz transformation equations for force
19. Fundamental particles. Photon and neutrino
20. Lagrange's and Hamilton's equations
21. Energy-momentum tensor
22. Energy-momentum tensor for a fluid
23. Angular momentum
  Exercises 3
Chapter 4 Special Relativity Electrodynamics
24. 4-Current density
25. 4-Vector potential
26. The field tensor
27. Lorentz transformations of electric and magnetic vectors
28. The Lorentz force
29. The engery-momentum tensor for an electromagnetic field
  Exercises 4
Chapter 5 General Tensor Calculus. Riemannian Space
30. Generalized N-dimensional spaces
31. Contravariant and covariant tensors
32. The quotient theorem. Conjugate tensors
33. Covariant derivatives. Parallel displacement. Affine connection
34. Transformation of an affinity
35. Covariant derivatives of tensors
36. The Riemann-Christoffel curvature tensor
37. Metrical connection. Raising and lowering indices
38. Scalar products. Magnitudes of vectors
39. Geodesic frame. Christoffel symbols
40. Bianchi identity
41. The covariant curvature tensor
42. Divergence. The Laplacian. Einstein's tensor
43. Geodesics
  Exercises 5
Chapter 6 General Theory of Relativity
44. Principle of equivalence
45. Metric in a gravitational field
46. Motion of a free particle in a gravitational field
47. Einstein's law of gravitation
48. Acceleration of a particle in a weak gravitational field
49. Newton's law of gravit
50. Freely falling dust cloud
51. Metrics with spherical symmetry
52. Schwarzchild's solution
53. Planetary orbits
54. Gravitational deflection of a light ray
55. Gravitational displacement of spectral lines
56. Maxwell's equations in a gravitational field
57. Black holes
58. Gravitational waves
  Exercises 6
Chapter 7 Cosmology
59. Cosmological principle. Cosmical time
60. Spaces of constant curvature
61. The Robertson-Walker metric
62. Hubble's constant and the deceleration parameter
63. Red shifts of galaxies
64. Luminosity distance
65. Cosmic dynamics
66. Model universes of Einstein and de Sitter
67. Friedmann universes
68. Radiation model
69. Particle and event horizons
  Exercises 7