By the end of this course unit, students will be able to:
1. explain the role of the foundational principles of special relativity and relativistic gravitation: the principle of relativity, symmetries and covariance, the equivalence principles, and the cosmological principle;
2. set out the conceptual progression from Newtonian mechanics and classical electromagnetism to special relativity, and then to a relativistic description of gravitation;
3. use the concepts of pseudo-Riemannian geometry required to formulate problems in general relativity;
4. examine the theory's main experimental tests in relation to its foundational principles;
5. use symbolic computation to carry out or verify calculations in general relativity;
6. conduct a literature review on an advanced topic in relativistic gravitation or cosmology, produce a structured synthesis, and present it orally.
The scope of contemporary fundamental physics spans the full range of scales over which matter is structured, connecting the infinitely small with the infinitely large. This course focuses on the physics of the latter infinity: it aims to introduce the modern formulation of relativistic gravitation through special relativity, general relativity, and modified-gravity theories, using the mathematical framework of pseudo-Riemannian differential geometry.
Throughout the course, foundational physical principles provide the guiding thread for the mathematical construction of relativistic gravitation, leading to its experimental tests and contemporary research questions such as theories that go beyond Einstein's general relativity.
The course also aims to develop students' skills in using symbolic computation, using AI to assist with the solution of research problems, and carrying out a literature review of an advanced topic.
This course constitutes a first module in which general relativity is constructed from the difficulties encountered by Newtonian mechanics and classical electromagnetism. It first introduces special relativity and then its generalisation to a relativistic description of gravitation.
At each stage, the mathematical tools are introduced in connection with the physical principles under study. Elementary concepts of geometry and differential calculus are thus progressively extended to pseudo-Riemannian geometry.
The course concludes with a derivation and discussion of the Einstein field equations. It then opens onto the experimental tests of special and general relativity and possible extensions of Einstein's theory through modified-gravity approaches.
1. Limitations of Newtonian mechanics and classical electromagnetism
2. Special relativity: principle of relativity, symmetries, isometries and covariance; Minkowski spacetime and pseudo-Euclidean geometry
3. From differential geometry to pseudo-Riemannian geometry (also covered in tutorials)
4. Langevin's traveller, or Minkowski spacetime viewed from an accelerated frame: time dilation, light deflection, and gravitational redshift
5. The weak, Einstein, and strong equivalence principles
6. Construction of general relativity: Einstein's heuristic method and the Einstein-Hilbert variational principle
7. Outlook: experimental tests of relativity, extensions of general relativity, and modified gravity
Lectures delivered at the blackboard, centred on the progressive construction of physical and mathematical reasoning.
Illustration of the course's mathematical developments through symbolic computation.
Tutorial sessions providing practical training in symbolic computation and in the use of AI to assist with solving formal problems in general relativity.
Use of advanced textbooks and lecture notes to explore selected topics in greater depth and prepare an independent study topic.
Assessment consists of an in-depth study of a topic not necessarily covered in class. Typical topics include a particular approach to Einstein's equations, a fundamental result (Lovelock's, Noether's or Birkhoff's theorem; the existence of a locally geodesic reference frame; etc.), or the presentation of an alternative theory of gravity.
The topic is chosen by the student in consultation with the lecturer, who may suggest themes. Initial bibliographic references are provided, but the student is encouraged to conduct an independent literature search.
The work is presented orally during the examination. Students may refer to their own handwritten notes. Students may work in pairs when appropriate; the level of difficulty will then be adjusted accordingly.
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Master in Physics | Standard | 1 | 3 | No |
| Master in Mathematics | Finalité approfondie | 1 | 3 | No |
| Master in Physics | Finalité didactique | 1 | 3 | No |
| Master in Mathematics | Standard | 1 | 3 | No |
| Master in Mathematics | Finalité didactique | 1 | 3 | No |
| Master in Physics | Finalité spécialisée en physique et data | 1 | 3 | No |
| Master in Mathematics | Finalité spécialisée en data science | 1 | 3 | No |
| Master in Mathematics | Finalité spécialisée en en Project Engineering | 1 | 3 | No |
| Master in Physics | Finalité spécialisée en physique du vivant | 1 | 3 | No |
| Master in Physics | Finalité approfondie | 1 | 3 | No |
| Master in Physics | Finalité approfondie | 2 | 3 | No |
| Master in Physics | Finalité didactique | 2 | 3 | No |
| Master in Physics | Finalité spécialisée en physique et data | 2 | 3 | No |
| Master in Physics | Finalité spécialisée en physique du vivant | 2 | 3 | No |