The evolution problem in general relativity

Interview ---------

What field is your thesis in?

My research concerns solutions to Einstein's equations of general relativity. Einstein's equations are partial differential equations—more specifically, nonlinear wave equations—and my field of mathematics is mathematical analysis. General relativity is formulated using concepts from geometry, and a number of the ideas and tools in my thesis also come from differential, Riemannian and Lorentzian geometry.

Why did you choose this subject?

I enjoy understanding physics through mathematics: it is wonderful to be able to "see" how an electromagnetic field or a fluid behaves purely from simple equations, such as the Maxwell or Navier–Stokes equations, and very general mathematical tools (Fourier transforms, differential calculus, and so on). During my first years in a post-secondary preparatory programme in Lyon, I also wanted to understand the major concepts in geometry that I kept hearing about: "manifolds", "curvature"... I have always been fascinated by the fact that these geometric concepts could be connected with physics. In practice, however, what I have always enjoyed doing is analysis: manipulating epsilons, determining asymptotic behavior and studying solutions to differential equations. It really feels like getting your hands dirty! Under the supervision of Jérémie Szeftel at the Laboratoire Jacques-Louis-Lions of Sorbonne Université in Paris, the mathematical analysis of Einstein's equations enabled me to combine these three aspects—mathematical physics, geometry and analysis—and that is why I chose this thesis topic.

How does this thesis fit into your plans?

I am currently a postdoctoral researcher at the Westfälische-Wilhelms Universität Münster in Germany. I am now working on solutions to Einstein's equations near Kerr–anti-de Sitter black holes.

In 1915, physicist Albert Einstein (1879–1955) proposed a new theory of gravitation, which he called the theory of general relativity. Gravitation is one of the four fundamental interactions in physics, alongside the weak, strong and electromagnetic interactions.

It is responsible for the attraction between massive bodies: it is why an apple hanging from an apple tree is pulled toward Earth, and why Earth orbits the Sun (see Mathématiques et Physique, Bibliothèque Tangente 69, 2019).

Until Einstein's revolutionary work, gravitation was described by Newton's laws (see box).

Newtonian theory ----------------------

In Newton's theory, a point mass m generates a gravitational field − that is, a vector field assigning a vector to every point in space − defined by the following well-known formula:

\overrightarrow{g} = - \dfrac{\text{G}m}{r^3} \; \overrightarrow{r},

where G = 6.67430 × 10⁻¹¹ m³kg⁻¹s⁻² ± 0.00015 × 10⁻¹¹ m³kg⁻¹s⁻2 is a universal constant called the gravitational constant, and

\overrightarrow{r}

denotes the vector from the point mass m to the point in space under consideration.

More generally, an extended distribution of matter generates a gravitational field

\overrightarrow{g}

that is the gradient of the gravitational potential Φ satisfying the equation:

\dfrac{\partial^2 \Phi}{\partial x^2} + \dfrac{\partial^2 \Phi}{\partial y^2} + \dfrac{\partial^2 \Phi}{\partial z^2} = 4 \pi \text{G}\rho

, where ρ is the mass density of the matter distribution.

Mathematically, this partial differential equation is elliptic: time does not enter into it, and any change in the matter distribution ρ immediately affects the gravitational potential Φ at every point in space.

In a gravitational field