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