# Introduction to Tensor Calculus, Relativity and Cosmology

###### D. F. Lawden
Publisher:
Dover Publications
Publication Date:
2003
Number of Pages:
224
Format:
Paperback
Edition:
3
Price:
15.95
ISBN:
9780486425405
Category:
Monograph
BLL Rating:

The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Michael Berg
, on
10/26/2015
]

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provide his charges with a solid approach to modern relativistic physics and its methodologies, free of unnecessary and outdated scaffolding (he refers to the Michelson-Morley experiment as a “historical accident,” with special relativity being in the works regardless).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.

 Preface 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 References Bibliography Index