Flat and Curved Space-Times

by Ellis, George F. R.; Williams, Ruth M.

• ISBN13: 9780198511649
• ISBN10: 0198511647
• Format: Hardcover
• Copyright: 1988-09-22
• Publisher: Oxford University Press

Summary

This text explains special relativity and the foundations of general relativity theory in terms understandable to undergraduates in mathematics and physics coming to the subject for the first time. Space-time geometry is emphasized throughout, providing a basic understanding of the special relativity effects of time dilation, length contraction, and the relativity of simultaneity. Bondi's k-calculus is introduced as a simple means of calculating the magnitudes of these effects, and this leads to a derivation of the Lorentz Transformation as a way of unifying these results. The invariant interval of flat space-time is compared to that of curved space-times, and the basic properties of simple cosmological models are presented. Appendices enable the advanced student to master the application of four tensors to the relativistic study of energy, momentum and electromagnetism.

Table of Contents

Introduction	p. 1
Space-time diagrams and the foundations of special relativity	p. 6
The concept of a space-time	p. 6
Causality and the speed of light	p. 13
Relative motion in special relativity	p. 26
Fundamentals of measurement	p. 38
Time	p. 38
Distance	p. 40
Simultaneity	p. 44
World maps, world pictures, and radar maps	p. 48
Measurements in flat space-times	p. 53
The Doppler effect	p. 53
Relative velocity	p. 62
Simultaneity	p. 72
Time dilation	p. 80
Length contraction	p. 92
The whole package of kinematic effects	p. 101
Relativistic dynamics	p. 107
The consistency of physics	p. 127
The Lorentz transformation and the invariant interval	p. 129
The Lorentz transformation	p. 129
Space-time separation invariants	p. 147
Some flat-space universes	p. 172
Curved space-times	p. 197
The general concept	p. 197
Acceleration and gravitation: the principle of equivalence	p. 200
Freely falling motion and the meaning of geodesics	p. 207
The metric form and the metric tensor	p. 213
Light rays	p. 220
Causality	p. 227
Parallel propagation along a curve	p. 229
Spherical and stellar collapse	p. 234
The Schwarzschild solution	p. 234
Spherical collapse to black holes	p. 244
Simple cosmological models	p. 254
Space-time geometry	p. 254
The evolution of the universe	p. 261
Observable quantities	p. 268
The light cone, observational limits, and horizons	p. 275
Steady state and inflationary universes	p. 283
Small universes	p. 285
Afterword	p. 290
Appendices	p. 293
Line integrals	p. 293
Four-vectors and relativistic dynamics	p. 300
Four-tensors, electromagnetism, and energy-momentum conservation	p. 317
Symbols used	p. 345
Index	p. 346
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