## Essential Relativity: Special, General, and CosmologicalThis book is an attempt to bring the full range of relativity theory within reach of advanced undergraduates, while containing enough new material and simplifications of old arguments so as not to bore the expert teacher. Roughly equal coverage is given tospecial relativity, general relativity, and cosmology. With many judicious omissions it can be taught in one semester, but it would better serve as the basis of a year's work. It is my hope, anyway, that its level and style of presentation may appeal also to wider c1asses of readers unrestricted by credit considerations. General relativity, the modern theory of gravitation in which free particles move along "straightest possible" lines in curved spacetime, and cosmology, with its dynamics for the whole possibly curved uni verse, not only seem necessary for a scientist's balanced view of the world, but offer some of the greatest intellectual thrills of modern physics. Nevertheless, considered luxuries, they are usu ally squeezed out of the graduate curriculum by the pressure of specialization. Special relativity escapes this tag with a ven geance, and tends to be taught as a pure service discipline, with too little emphasis on its startling ideas. What better time, there fore, to enjoy these subjects for their own sake than as an und- v vi PREFACE graduate? In spite of its forbidding mathematical reputation, even general relativity is accessible at that stage. |

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### Contents

EINSTEINIAN KINEMATICS 22 Basic Features of Special Relativity | 29 |

On the Nature of Physical Laws | 32 |

An Archetypal Relativistic Argument | 33 |

The Relativity of Simultaneity | 35 |

The Coordinate Lattice | 37 |

The Lorentz Transformation | 39 |

Properties of the Lorentz Transformation | 41 |

Alternative Form of the Lorentz Transformation | 45 |

Photons The Compton Effect | 115 |

The Matter Tensor of Dust 17 | 117 |

RELATIVITY AND ELECTRODYNAMICS 61 Transformation of the Field Vectors | 120 |

Magnetic Deflection of Charged Particles | 124 |

The Field of a Uniformly Moving Charge | 125 |

The Field of an Infinite Straight Current | 127 |

BASIC IDEAS OF GENERAL RELATIVITY 65 Curved Surfaces | 130 |

Curved Spaces of Higher Dimensions | 134 |

Graphical Representation of the Lorentz Transformation | 46 |

WorldPicture and WorldMap | 48 |

Length Contraction | 49 |

Length Contraction Paradoxes | 51 |

Time Dilation | 53 |

The Twin Paradox | 56 |

Velocity Transformation | 59 |

Proper Acceleration | 62 |

Special Relativity Without the Second Postulate | 64 |

EINSTEINIAN OPTICS 39 The Drag Effect | 68 |

The Doppler Effect | 69 |

Aberration and the Visual Appearance of Moving Objects | 72 |

SPACETIME AND FOURVECTORS 42 Spacetime | 78 |

ThreeVectors | 81 |

FourVectors | 84 |

FourTensors | 88 |

The Null Cone | 89 |

Wave Motion | 91 |

RELATIVISTIC PARTICLE MECHANICS 48 Domain of Sufficient Validity of Newtons Laws | 95 |

Why Gravity Does not Fit into Special Relativity | 96 |

Shortcut to Relativistic Mechanics | 97 |

Formal Approach to Relativistic Mechanics | 99 |

A Note on Galilean FourVectors | 102 |

Equivalence of Mass and Energy | 103 |

The Center of Momentum Frame | 105 |

Relativistic Billiards | 107 |

p E Diagrams and Threshold Energies | 108 |

ThreeForce and FourForce 1 11 | 111 |

De Broglie Waves 13 | 113 |

Riemannian Spaces | 136 |

A Plan for General Relativity | 141 |

The Gravitational Doppler Effect | 145 |

The Spacetime Around a Spherical Mass | 147 |

Static Fields Geodesics and Hamiltons Principle | 151 |

FORMAL DEVELOPMENT OF GENERAL RELATIVITY 72 Tensors in General Relativity | 155 |

The Vacuum Field Equations of General Relativity | 162 |

The Schwarzschild Solution | 166 |

Rays and Orbits in Schwarzschild Space | 174 |

A GeneralRelativistic Proof of E mc | 181 |

The Schwarzschild Radius | 184 |

The Laws of Physics in Curved Spacetime | 195 |

The Field Equations in the Presence of Matter | 199 |

Modified Schwarzschild Space | 207 |

COSMOLOGY 81 The Basic Facts | 213 |

Apparent Difficulties of PreRelativistic Cosmology | 221 |

The Cosmological Principle | 225 |

Milnes Model | 227 |

The RobertsonWalker Metric | 232 |

Rubber Models Red Shifts and Horizons | 237 |

Comparison with Observation | 242 |

Cosmic Dynamics According to PseudoNewtonian Theory | 248 |

Cosmic Dynamics According to General Relativity | 251 |

The Friedmann Models | 257 |

Once Again Comparison with Observation | 265 |

Machs Principle Reexamined | 271 |

EXERCISES | 275 |

313 | |

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### Common terms and phrases

absolute space acceleration analog angle arbitrary argument assume axes axis collision components cone conservation consider coordinate system corresponding cosmic cosmology curved density diagram dilation direction distance Doppler Doppler effect earth Einstein energy equivalence principle equivalent ether Euclidean example expanding fact field equations Figure finite force formula four-acceleration four-tensors four-vector four-velocity free particles galaxies Galilean transformation geodesics geometry given gravitational field Hence homogeneous implies inertial frames infinite invariant isotropy length contraction light signal Lorentz transformation Mach's principle Maxwell’s Milne's model Minkowski Minkowski space momentum motion moving Newton’s theory Newtonian observer orbit origin particle horizon photon physical plane principle radial radius rela relativistic rest frame rest mass result Riemannian rotation RW metric Schwarzschild Schwarzschild radius Section solution spacetime spatial special relativity speed of light sphere spherical standard clock stars static surface symmetry tensor three-space timelike tion universe vector velocity worldlines zero