Learning outcomes

The physical concepts of special and general relativity. The elements of differential calculus and tensorial calculus that are required by general relativity. The relativistic conceptions of time, space, mass, momentum and energy. The covariant formulation of electromagnetism. The geodesics. Einstein's equation. The metric of Schwarzschild. The "Newton's force" as outcome of general relativity. The slowing down of time by gravity. The gravitational lensing effect. The Python programming language for solving exercises.

Goals

To acquire the physical concepts that lead to the theory of relativity. To be able to demonstrate the main results. To integrate the mathematical tools presented in this course. To be able to apply the concepts of this course to classical problems.

Content

This is a first course of relativity. We update in the context of special relativity the concepts of space, time, mass, momentum and energy. 
These concepts are illustrated with space-time diagrams, the famous twin paradox and various numerical exercises. We show how Lorentz's transformations make the laws of mechanics consistent with those of electromagnetism. We recall some mathematical tools of general relativity (vectors, differential forms, tensors, covariant derivatives). We then address the geodesics in order to determine trajectories in a space-time that is curved by gravitation. We introduce the Riemann tensor, the Ricci tensor and the Einstein tensor in order to describe the curvature of space-time. The energy-momentum tensor is introduced in order to describe the densities and fluxes of momentum and energy. These different concepts are finally related by the equation of Einstein. We then determine Schwarzschild's metric in order to describe space-time around a central mass. We can then establish "Newton's force" as an outcome of general relativity. We demonstrate also the gravitational time dilation effect. We get briefly through different applications of relativity (advance of the perihelion of Mercury, gravitational deflection of light, gravitational waves, black holes).

Table of contents

I. The concepts of time and space in special relativity

  • Principles of special relativity

  • Experiment of Michelson-Morley

  • Time dilation

  • Length contraction

  • Illustration : the muons that cross the atmosphere

  • Invariance of ds²

  • Metric of Minkowski

  • Transformations of Lorentz

  • Speed transformation laws

  • Simultaneity

  • Space-time diagrams

II. Mass, momentum and energy in special relativity

  • Relativistic expression of momentum

  • Relativistic expression of energy

  • Rest-mass energy, kinetic energy & total energy

  • Connection with Dirac's equation & anti-matter

  • Mass as a source of energy

III. The principle of least action in special relativity

  • Construction of the relativistic Lagrangian

  • Application of the Lagrangian formalism

  • Application of the Hamiltonian formalism 

  • The twins paradox

  • Time travel

IV. Introduction to Python

  • Working with Anaconda (Python 3.7)

  • Basic commands

  • Installation of librairies

  • The numpy library

  • The matplotlib library

V. Programming exercises

VI. Mathematical tools for relativity

  • Coordinates, natural basis, general basis, vectors

  • Change of coordinates

  • Commutators

  • Tensors, laws of transformation, metric tensor

  • Covariant derivative

  • Torsion

  • Formula for Gamma^k_ij

VII. The equation of geodesics

VIII. Electromagnetism

  • Covariant formulation of electromagnetism

  • Invariance of Maxwell's equations under Lorentz transformations

IX. General relativity

  • Curvature operator

  • Tensor of Riemann, tensor of Ricci, scalar curvature

  • Energy-impulsion tensor

  • Einstein's equation

  • Metric of Schwarzschild

  • Gravitational time dilation

  • Newton's force

  • Applications of relativity

Teaching methods

The course is given using a video-projector (PowerPoint). The board is used for some developments. There is a syllabus with this course.

Assessment method

A written exam is on the material presented in class (PowerPoint to be found on WebCampus). A list of questions will be provided. The ponderation of the different parts of the final note is the following : written exam (15 points), programming exercises (5 points). Failure in the written exam is absorbing.

Sources, references and any support material

Charles W. Misner, Kip S. Thorne and John Archibald Wheeler, "Gravitation" (W.H. Freeman and Company, New York, 1973).

Language of instruction

French
Training Block Credits Mandatory
Bachelor in Physics 3 3 Yes