Geometric studies of the interplay between spin and gravity

TL;DR

This thesis explores how the spin of elementary particles influences gravitational theories, deriving covariant formulations of spin-related equations and revealing phenomena like gravitational birefringence in various spacetime contexts.

Contribution

It provides a covariant formulation of the Levy-Leblond-Newton equation and analyzes spin effects on particle trajectories in General Relativity, including new insights into gravitational birefringence.

Findings

The Schrödinger-Newton group is the symmetry group of the LLN equation.

Spin affects photon trajectories, causing gravitational birefringence.

Birefringence effects are predicted but currently undetectable in experiments.

Abstract

This PhD thesis is the conclusion of some of my works carried out at the Centre de Physique Th\'eorique, under the supervision of Serge Lazzarini. Two subjects are presented in the manuscript, both aiming at studying the effect of the spin of elementary particles on otherwise known theories. First is a study of the L\'evy-Leblond-Newton (LLN) equation, which is used to describe the evolution of a quantum system with spin 1/2 that is coupled to its own gravitational potential. After reviewing some symmetries of non relativistic Quantum Mechanics, and how to geometrize them with the help of Bargmann structures, we recall what is the L\'evy-Leblond equation. In this chapter, the LLN equation is described in a fully covariant way on Bargmann structures. This covariant formulation then helps to derive the dynamical symmetries of the equation, and conserved quantities. The symmetry group of this equation turns out to be the already known Schr\"odinger-Newton group. The second part deals with the trajectory of particles with spin in General Relativity. We first highlight Souriau's geometric method to derive the Mathisson-Papapetrou-Dixon (MPD) equations, and then discuss the different possible Spin Supplementary Conditions (SSC) that exist to close the system of MPD equations. In this chapter, we assume the Tulczyjew SSC to describe the trajectory of photons, leading to the Souriau-Saturnini equations. We then present some works where we have applied these equations, in a Schwarzschild spacetime, and in a spacetime deformed by a gravitational wave. In the first work, we found gravitational birefringence: the trajectory deviates from the geodesic plane, depending on the helicity of the photon and its wavelength. The second work presented also found birefringence in gravitational interferometry experiments, however the effect is many orders of magnitude lower than what we can detect.