By the end of this course unit, students will be able to:
1. calculate relativistic effects around a point mass in Schwarzschild spacetime (time dilation, gravitational redshift, light deflection, etc.) and apply them to space physics and precision metrology (GNSS, Shapiro delay, etc.);
2. generalise the restricted Kepler two-body problem to strong gravitational fields and to light rays or ultrarelativistic particles;
3. understand the gravitational physics of black holes and its applications in astrophysics;
4. describe the construction and essential properties of Friedmann-Lemaitre-Robertson-Walker cosmological models, interpret the main cosmological tests, and understand the scope and limitations of the 'Big Bang theory';
5. use symbolic computation, numerical methods and AI to carry out or verify calculations in applications of general relativity;
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 module follows on from the first module on the foundations of general relativity and focuses on applications such as black holes and the Big Bang theory.
The course develops the classical solutions of general relativity - Schwarzschild and Friedmann-Lemaitre-Robertson-Walker spacetimes - and explores several applications in fundamental astrophysics and space metrology.
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 is the second module, devoted to applications of relativistic gravitation: on the one hand, the relativistic gravitational fields of Earth, celestial bodies and black holes; on the other, cosmological expansion since the Big Bang.
The course is divided into two parts. The first covers relativistic astrodynamics: the motion of bodies in Schwarzschild or Kerr spacetimes. We first develop the classical interior and exterior Schwarzschild solutions of Einstein's equations and describe Kerr spacetime; we then study free geodesic motion and forced motion for massive particles and light, generalising the restricted two-body problem (Kepler problem).
The second part focuses on cosmology through an in-depth study of Friedmann-Lemaitre-Robertson-Walker spacetimes. Starting from the cosmological principle and the historical models of Einstein and de Sitter, we introduce the cosmological constant and then cosmic expansion, discovered by Georges Lemaitre. We examine cosmologies sourced by a scalar field and the cosmological problems of inflation, dark matter and dark energy.
Einstein's equations, vacuum solutions, horizons, the Einstein effect and gravitational redshift, spherical symmetry
Exterior and interior Schwarzschild solutions, coordinate systems (Boyer-Lindquist, isotropic, Eddington-Finkelstein, etc.), compactness, Flamm's paraboloid, black holes and wormholes
Free geodesic motion and forced motion in Hamiltonian formalism for massive particles and light
Experimental tests and applications of general relativity in space physics and the Solar System
Static universe models: perfect cosmological principle, Einstein and de Sitter, cosmological constant
Restricted cosmological principle, synchronous reference frame and time, maximal symmetry and the geometry of the Universe
Friedmann-Lemaitre-Robertson-Walker universe models and the dynamics of cosmological expansion
Variational approach to the equations of cosmologies with a minimally coupled scalar field (inflaton or quintessence)
Outlook: cosmological tests and the problems of inflation, dark matter, dark energy and structure formation
Lectures delivered at the blackboard.
Illustration of the course results through symbolic or numerical computation, with interactive experiments (motion around a black hole, time dilation, navigation through possible universes, etc.).
Tutorial sessions providing practical training in symbolic computation and in the use of AI to assist with solving practical 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 an application chosen by the student, whether or not it was covered in class.
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 must include a technical contribution by the student: analytical calculations involving a particular solution of general relativity (by hand or using symbolic computation); numerical simulations (of trajectories, cosmological evolution, etc.) and the calculation of observables; or the modelling and analysis of space, astrophysical or cosmological data.
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 |