Special Relativity

This course deals with the Minkowski spacetime and its symmetries, that are Lorentz transformations. As a prerequisite, the basic mathematical knowledge of Matrix Theory and Tensor Calculus (Orthogonal Matrices, parametric description of a curve, functionals, etc) and examples from Physics are described. Then, the notion of an inertial Galilean frame and the cause which prompted the need to introduce the notion of Minkowski spacetime is presented; that is, the experimental fact of the constancy of the speed of light, which is independent of the inertial observer. This fact leads to the need of a new mechanics based on the Lorentz transformations as its basic symmetry group. This new theory is the special theory of relativity. The geometrical objects of this theory correspond to physical entities and the various algebraic manipulations among them comprise the tensor calculus. The Maxwell's electromagnetic theory is reformulated in a way that makes explicit its invariance under Lorentz transformations. The introduction of this new symmetry group entails a new relativistic dynamics. Finally, some basic applications are analysed (time dilation, length contraction, Doppler effect, etc).

Objectives

To familiarize the students with the concepts of the inertial reference frames, the spacetime transformations, the symmetries of the physical systems The analysis and study of the Lorentz transformations, the electromagnetic field and the relativistic dynamics

Prerequisites

Classical Mechanics Matrix Theory

Syllabus

CHAPTER 1 MATHEMATICAL PREREQUISITES 1.1. Review of the matrix theory 1.2. Translations – Rotations of an inertial frame – Orthogonal matrices 1.2.1. Translations 1.2.2. Rotations 1.2.3. Orthogonal Matrices 1.3. Parametric form of a curve- Reparametrisation of a curve 1.3.1. Parametric form of a curve 1.3.2. Reparametrisation of a curve 1.4. Functionals-Dirac functional 1.4.1. Mathematics 1.4.2. The physical problem 1.4.3. Mathematical solution of the problem 1.4.4. Applications in physics CHAPTER 2 INERTIAL SYSTEMS OF REFERENCE –GALILEAN TRANSFORMATIONS 2.1. Inertial systems of frames 2.2. Galilean transformation 2.2.1. Time translations 2.2.2. Space translations 2.2.3. Spatial rotations 2.2.4. Galilean boosts 2.2.5. General Galilean transformation 2.3. Transformation law of the physical quantities 2.4. The range and the limits of the Galilean transformation 2.4.1. The range 2.4.2. The limits CHAPTER 3 LORENTZ TRANSFORMATIONS 3.1. Introduction 3.2. Lorentz boost 3.2.1. Boost along an axis 3.2.2. Spacetime coordinates –General Lorentz boost 3.3. Spacetime distance and the Minkowski spacetime 3.4. Lorentz group CHAPTER 4 THE QUANTITIES OF SPECIAL THEORY OF RELATIVITY 4.1. Introduction 4.2. Spacetime distance and proper time 4.3. Four-velocity 4.4. Four-acceleration 4.5. Four-momentum CHAPTER 5 TENSOR CALCULUS 5.1. Introduction 5.2. Tensors – Tensor fields 5.3. Constant tensors 5.4. Mathematical operations between tensors CHAPTER 6 ELECTROMAGNETISM 6.1. Introduction 6.2. The sources 6.3. Maxwell’s equations 6.4. Transformation law of the fields 6.5. Invariants of the electromagnetic field CHAPTER 7 RELATIVISTIC DYNAMICS 7.1. General considerations 7.2. Application (Lorentz four-force) CHAPTER 8 APPLICATIONS 8.1. Time dilation 8.2. Length contraction 8.3. Doppler effect (Special case) 8.4. Doppler effect (General case) 8.5. Non-central elastic collision 8.6. Photon absorption from a static particle 8.7. Photon emission from a static particle 8.8. Photon emission from a moving particle 8.9. Threshold energy 8.10. Elastic scattering CHAPTER 9 EXERCISES 9.1 Lorentz transformations 9.2 Minkowski spacetime 9.3 Four - velocity, Four - momentum 9.4 Tensor calculus 9.5 Momentum - Energy conservation 9.6 Electromagnetism