Introduction to Tensor Calculus, Relativity and Cosmology

by Lawden, D. F.

• ISBN13: 9780486425405
• ISBN10: 0486425401
• Edition: 3rd
• Format: Paperback
• Copyright: 2003-01-27
• Publisher: Dover Publications

Summary

This elementary introduction pays special attention to aspects of tensor calculus and relativity that students tend to find most difficult. Its use of relatively unsophisticated mathematics in the early chapters allows readers to develop their confidence within the framework of Cartesian coordinates before undertaking the theory of tensors in curved spaces and its application to general relativity theory. Additional topics include black holes, gravitational waves, and a sound background in applying the principles of general relativity to cosmology. Numerous exercises advance the theoretical developments of the main text, thus enhancing this volume's appeal to students of applied mathematics and physics at both undergraduate and postgraduate levels. 1982 ed. Solution guide available upon request.

Table of Contents

Preface	ix
List of Constants	xiii
Special Principle of Relativity. Lorentz Transformations	1	(20)
Newton's laws of motion	1	(2)
Covariance of the laws of motion	3	(1)
Special principle of relativity	4	(2)
Lorentz transformations. Minkowski space--time	6	(3)
The special Lorentz transformation	9	(3)
Fitzgerald contraction. Time dilation	12	(2)
Spacelike and timelike intervals. Light cone	14	(7)
Exercises 1	17	(4)
Orthogonal Transformations. Cartesian Tensors	21	(18)
Orthogonal transformations	21	(2)
Repeated-index summation convention	23	(1)
Rectangular Cartesian tensors	24	(3)
Invariants. Gradients. Derivatives of tensors	27	(1)
Contraction. Scalar product. Divergence	28	(1)
Pseudotensors	29	(1)
Vector products. Curl	30	(9)
Exercises 2	31	(8)
Special Relativity Mechanics	39	(34)
The velocity vector	39	(2)
Mass and momentum	41	(3)
The force vector. Energy	44	(2)
Lorentz transformation equations for force	46	(1)
Fundamental particles. Photon and neutrino	47	(1)
Lagrange's and Hamilton's equations	48	(2)
Energy--momentum tensor	50	(3)
Energy--momentum tensor for a fluid	53	(4)
Angular momentum	57	(16)
Exercises 3	59	(14)
Special Relativity Electrodynamics	73	(13)
4-Current density	73	(1)
4-Vector potential	74	(1)
The field tensor	75	(2)
Lorentz transformations of electric and magnetic vectors	77	(2)
The Lorentz force	79	(1)
The energy--momentum tensor for an electromagnetic field	79	(7)
Exercises 4	82	(4)
General Tensor Calculus. Riemannian Space	86	(41)
Generalized N-dimensional spaces	86	(3)
Contravariant and covariant tensors	89	(5)
The quotient theorem. Conjugate tensors	94	(1)
Covariant derivatives. Parallel displacement. Affine connection	95	(3)
Transformation of an affinity	98	(2)
Covariant derivatives of tensors	100	(2)
The Riemann-Christoffel curvature tensor	102	(3)
Metrical connection. Raising and lowering indices	105	(2)
Scalar products. Magnitudes of vectors	107	(1)
Geodesic frame. Christoffel symbols	108	(3)
Bianchi identity	111	(1)
The covariant curvature tensor	111	(1)
Divergence. The Laplacian. Einstein's tensor	112	(2)
Geodesics	114	(13)
Exercises 5	117	(10)
General Theory of Relativity	127	(47)
Principle of equivalence	127	(3)
Metric in a gravitational field	130	(3)
Motion of a free particle in a gravitational field	133	(2)
Einstein's law of gravitation	135	(2)
Acceleration of a particle in a weak gravitational field	137	(2)
Newton's law of gravitation	139	(1)
Freely falling dust cloud	140	(2)
Metrics with spherical symmetry	142	(3)
Schwarzschild's solution	145	(2)
Planetary orbits	147	(3)
Gravitational deflection of a light ray	150	(2)
Gravitational displacement of spectral lines	152	(2)
Maxwell's equations in a gravitational field	154	(1)
Black holes	155	(4)
Gravitational waves	159	(15)
Exercises 6	163	(11)
Cosmology	174	(25)
Cosmological principle. Cosmical time	174	(2)
Spaces of constant curvature	176	(4)
The Robertson--Walker metric	180	(1)
Hubble's constant and the deceleration parameter	181	(1)
Red shift of galaxies	182	(1)
Luminosity distance	183	(2)
Cosmic dynamics	185	(3)
Model universes of Einstein and de Sitter	188	(1)
Friedmann universes	189	(4)
Radiation model	193	(2)
Particle and event horizons	195	(4)
Exercises 7	197	(2)
References	199	(1)
Bibliography	200	(1)
Index	201