Engineering Physics

Special Relativity
90 Pages
Nicholas M.J. Woodhouse

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Forward:

The mathematical content of special relativity is not hard to understand. The basic principles can be written down simply and concisely, and, with a little practice and some not very involved calculations, it is easy to derive from them the main predictions of the theory. What is less easy is to appreciate that these predictions are correct, even though they imply that space and time have properties which are contrary to intuition and to the established framework of Newtonian mechanics: within the limits of modern experimental physics, the theory of relativity is an accurate model of the real world.

Before launching into the formal development, therefore, it is helpful to reconsider the problems that originally forced theoretical physicists to abandon the classical view of space and time. These arose from attempts to fit Newton’s mechanics and Maxwell’s electrodynamics into a single consistent physical theory. In particular, there was the problem of describing electromagnetic processes in uniformly moving frames of reference.

TOC

Chapter1
Space, Time and Maxwell’s Equations . 1
Introduction . 1
Maxwell’s Equations and Galilean Transformations . 1
The Newtonian View of Space and Time . 3

Chapter 2
Inertial Coordinates . 5
Basic Principles . 5
Lorentz Transformations . 8
Time Dilatation . 11
The Standard Lorentz Transformation . 12
The Lorentz Contraction . 14
Transformation of Volumes . 15
Addition of Velocities . 16
The Michelson-Morley Experiment . 17

Chapter 3
Vectors in Space-Time . 19
Vectors . 19
The Metric . 20
The Causal Structure of Minkowski Space . 21
Orthonormal Frames . 22
Decomposition of 4-Vectors . 24
Vector Fields . 25
Velocity and Proper Time . 25
Acceleration . 26
Relative Speed . 29
The 4-Gradient . 29
The Frequency 4-Vector . 30
The Appearance of Moving Objects. 31
Examples and Exercises . 33

Chapter 4
Relativistic Particles . 37
Conservation of 4-Momentum . 37
Equivalence of Mass and Energy . 39
The 4-Momentum of a Photon . 39
Force . 40
Examples and Exercises . 40

Chapter 5
Tensors in Space-Time .'. 45
Covectors . 45
Raising and Lowering of Indices . 46
Tensors . 46
Operations on Tensors . 47
The Alternating Tensor . 49
Tensor Fields . 49
Example and Exercises . 50

Chapter 6 Electrodynamics . 53
Tensor Formulation of Maxwell’s Equations . 53
Gauge Transformations .. . . . 55
The Dual Electromagnetic Field Tensor . 55
Advanced and Retarded Solutions .x. 56
The Field of a Uniformly Moving Charge . 60
The Equation of Motion of a Test Charge . 61
Plane Waves . 62
Geometric Optics . 63
Examples and Exercises .. 65

Chapter 7
Energy-Momentum Tensors .. 69
Dust . 69
The Ideal Gas . 70
Conservation of Energy . 71
Another Form of Energy Conservation . 72
Fluids . 72
The Electromagnetic Energy-Momentum Tensor . 74

Chapter 8
Symmetries and Conservation Laws . 77
Isometries . 77
Killing Vectors . 77
Translations and Lorentz Rotations . 79
Active and Passive Transformations . 81
Conservation of Momentum and Angular Momentum . 81
Exercises . 82