Relativity Made Relatively Easy

Cover

S Title

Relativity Made Relatively Easy

Copyright

     Andrew M. Steane 2012

     ISBN 978-0-19-966285-2 (hbk)

     ISBN 978-0-19-966286-9 (pbk)

Dedication

Preface

Acknowledgements

Contents


Part I  The relativistic world

     1  Basic ideas

          1.1 Newtonian physics

          1.2 Special Relativity

               1.2.1 The Postulates of Special Relativity

               1.2.2 Central ideas about spacetime

          1.3 Matrix methods

          1.4 Spacetime diagrams

          exercises

     2  The Lorentz transformation

          2.1 Introducing the Lorentz transformation

               2.1.1 Derivation of Lorentz transformation

          2.2 Velocities

          2.3 Lorentz invariance and 4-vectors

               2.3.1 Rapidity

          2.4 Lorentz-invariant quantities

          2.5 Basic 4-vectors

               2.5.1 Proper time

               2.5.2 Velocity, acceleration

               2.5.3 Momentum, energy

               2.5.4 The direction change of a 4-vector under a boost

               2.5.5 Force

               2.5.6 Wave vector

          2.6 The joy of invariants

          2.7 Summary

          Exercises

     3  Moving light sources

          3.1 The Doppler effect

          3.2 Aberration and the headlight effect

               3.2.1 Stellar aberration

          3.3 Visual appearances

          Exercises

     4  Dynamics

          4.1 Force

               4.1.1 Transformation of force

          4.2 Motion under a pure force

               4.2.1 Linear motion and rapidity

               4.2.2 Hyperbolic motion: the `relativistic rocket'

               4.2.3 4-vector treatment of hyperbolic motion

               4.2.4 Motion under a constant force

               4.2.5 Circular motion

               4.2.6 Motion under a central force

               4.2.7 (An) harmonic motion

          Exercises

     5  The conservation of energy-momentum

          5.1 Elastic collision, following Lewis and Tolman

          5.2 Energy-momentum conservation using 4-vectors

               5.2.1 Mass-energy equivalence

          5.3 Collisions

               5.3.1 'Isolate and square

          5.4 Elastic collisions

               5.4.1 Billiards

               5.4.2 Compton scattering

               5.4.3 More general treatment of elastic collisions

          5.5 Composite systems

          5.6 Energy flux, momentum density, and force

          Exercises

     6  Further kinematics

          6.1 The Principle of Most Proper Time

          6.2 Four-dimensional gradient

          6.3 Current density, continuity

          6.4 Wave motion

               6.4.1 Wave equation

               6.4.2 Particles and waves

               6.4.3 Group velocity and particle velocity

          6.5 Acceleration and rigidity

               6.5.1 The great train disaster

               6.5.2 Lorentz contraction and internal stress

          6.6 General Lorentz boost

          6.7 Lorentz boosts and rotations

               6.7.1 Two boosts at right angles

               6.7.2 The Thomas precession

               6.7.3 Analysis of circular motion

          6.8 Generators of boosts and rotations

          6.9 The Lorentz group*

               6.9.1 Further group terminology

          Exercises

     7  Relativity and electromagnetism

          7.1 Definition of electric and magnetic fields

               7.1.1 Transformation of the fields (first look)

          7.2 Maxwell's equations

               7.2.1 Moving capacitor plates

          7.3 The fields due to a moving point charge

          7.4 Covariance of Maxwell's equations

               7.4.1 Transformation of the fields: 4-vector method*

          7.5 Introducing the Faraday tensor

               7.5.1 Tensors

               7.5.2 Application to electromagnetism

          Exercises

     8  Electromagnetic radiation

          8.1 Plane waves in vacuum

          8.2 Solution of Maxwell's equations for a given charge distribution

               8.2.1 The 4-vector potential of a uniformly moving point charge

               8.2.2 The general solution

               8.2.3 The Lienard-Wierhert potentials

               8.2.4 The field of an arbitrarily moving charge

               8.2.5 Two example fields

          8.3 Radiated power

               8.3.1 Linear and circular motion

               8.3.2 Angular distribution

          Exercises


Part II  An Introduction to General Relativity

     9  The Principle of Equivalence

          9.1 Flee fall

               9.1.1 Free fall or free float?

               9.1.2 weak Principle of Equivalence

               9.1.3 The Eotvas-Pekar-Fekete experiment

               9.1.4 The Strong Equivalence Principle

               9.1.5 Falling light and gravitational time dilation

          9.2 The uniformly accelerating reference frame

               9.2.1 Accelerated rigid motion

               9.2.2 Rigid constantly accelerating frame

          9.3 Newtonian gravity from the Principle of Most Proper Time

          9.4 Gravitational redshift and energy conservation

               9.4.1 Equation of motion

          Exercises

     10  Warped spacetime

          10.1 Two-dimensional spatial surfaces

               10.1.1 Conformal flatness

          10.2 Three spatial dimensions

          10.3 Time and space together

          10.4 Gravity and curved spacetime

          Exercises

     11  Physics from the metric

          11.1 Example exact solutions

               11.1.1 The acceleration due to gravity

          11.2 Schwarzschild metric: basic properties

          11.3 Geometry of Schwarzschild solution

               11.3.1 Radial motion

               11.3.2 Circular orbits

               11.3.4 Photon orbits

               11.3.5 Shapiro time delay

          11.4 Gravitational lensing

          11.5 Black holes

               1.5.1 Horison

               11.5.2 Energy near an horizon

          11.6 What next?

               11.6.1 Black-hole thermodynamics

          Exercises


Part III  Further Special Relativity

     12  Tensors and index notation

          12.1 Index notation in a nutshell

          12.2 Tensor analysis

               12.2.1 Rules for tensor algeb

               12.2.2 Contravariant and covariant

               12.2.3 Useful methods and ideas

               12.2.4 Parity inversion and the vector product

               12.2.5 Differentiation

          12.3 Antisymmetric tensors and the dual

          Exercises

     13  Rediscovering electromagnetism

          13.1 Fundamental equations

          13.2 Invariants of the electromagnetic field

               13.2.1 Motion of particles in an electromagnetic field

          Exercises

     14  Lagrangian mechanics

          14.1 Classical Lagrangian mechanics

          14.2 Relativistic motion

               14.2.1 From classical Euler-Lagrange

               14.2.2 Manifestly covaria

          14.3 Conservation

          14.4 Equation of motion in General Relativity

          Exercises

     15  Angular momentum

          15.1 Conservation of angular momentum

          15.2 Spin

               15.2.1 Introducing spin

               15.2.2 Paull-Lubanski vector

               15.2.3 Thomas precesion revisited

               15.2.4 Precession of the spin of a charged particle

          Exercises

     16  Energy density

          16.1 Introducing the stress-energy tensor

               16.1.1 Transport of energy and momentum

               16.1.2 Ideal fluid

          16.2 Stress-energy tensor for an arbitrary system

               16.2.1 Interpreting the terms

          16.3 Conservation of energy and momentum for a fluid

          16.4 Electromagnetic energy and momentum

               16.4.1 Examples of energy density and energy flow

               16.4.2 Field momentum

               16.4.3 Stress-energy tensor of the electromagnetic field

          16.5 Field and matter pushing on one another

               16.5.1 Resolution of the `4/3 problem' and the origin of mass

          Exercises

     17  What is spacetime?


A  Some basic arguments

     A.1 Early experiments

     A.2 Simultaneity and radar coordinates

     A.3 Proper time and time dilation

     A.4 Lorentz contraction

     A.5 Doppler effect, addition of velocities


B  Constants and length scales


C  Derivatives and index notation


D  The field of an arbitrarily moving charge

     D.1 Light-cone volume element

     D.2 The field tensor


Bibliography

Index