Yvonne Choquet-Bruhat and relativity | Tangente

Yvonne Choquet-Bruhat's work belongs to the field of the mathematics of general relativity. To understand it, and to discover who this towering mathematician was—the first woman to enter the Académie des sciences, in 1979—Tangente contacted one of her former colleagues, Piotr Chruściel, professor of gravitational physics at the University of Vienna.

Piotr Chruściel: We first met in 1983, when I was still a doctoral student. I was one of the participants at the Les Houches School of Physics\*, where Yvonne was giving a series of lectures [she was 60 at the time—Ed.]. Stephen Hawking was also there; by then he could barely speak intelligibly, and one of his doctoral students would repeat aloud what Hawking was trying to say (he did not yet have a voice synthesizer at the time). We crossed paths a few times after that. But it was in 2006, when I had a doctoral student working on Einstein's equations, that we spent more time together, at the Max Planck Institute in Potsdam. There we discussed the results obtained by her, by my student and by me, which led to our first joint work. It was not until 2008, however, that we really began working together.

(\* Founded in 1951 by physicist Cécile DeWitt-Morette in the town of Les Houches in Haute-Savoie, the Les Houches School of Physics offers advanced courses and workshops and welcomes researchers from around the world. Many Nobel laureates have passed through it.)

I was co-leading a research programme at the Mittag-Leffler Institute, in the suburbs of Stockholm, called "Geometry, Analysis and General Relativity." I naturally invited her to take part, and she accepted despite her already advanced age (85!). As an aside, this institute was originally the residence of the Swedish mathematician Gösta Mittag-Leffler, who later bequeathed it to be turned into a research centre in mathematics. It is a magnificent house overlooking one of the many lakes in the Stockholm archipelago, and, as one of the organizers, I had a very fine office; when Yvonne arrived, I could not give her an office less fine than mine, so I gave her mine and went to work in a smaller office. It seemed inconceivable to me not to welcome her as was fitting.

Albert Einstein established the equations, but the problem is to pin down their solutions. In 1952 (see box), Yvonne accomplished the immense, pioneering work of showing how these solutions could be parametrized and their mathematical properties studied. This was not a matter of approximation methods, truncation or simplifying assumptions, but genuinely rigorous mathematical work. I should say that, in physics, what had been done before is very commendable, as is what has been done since. But, unlike what Yvonne proved, these are not exact methods. Nor, for that matter, did she draw up a list of solutions—rather, she produced a tool for describing them. This gives an exact way of constructing spacetimes, and so of understanding the mathematical structure of Einstein's equations—equations whose author never claimed to be doing mathematics, only physics. Broadly speaking, her most important result, which lies at the heart of the mathematical understanding of Einstein's equations, can be summed up as follows: for any initial data for Einstein's equations, there exists a unique spacetime that is maximal (in differential geometry, maximality means roughly that the object cannot be extended any further) within the class of spacetimes with good causal properties (these "good causal properties" are linked, for instance, to the fact that time travel is not permitted in the spacetimes Yvonne constructed).

Yvonne did indeed tell me that she was deeply moved when she met Einstein, who was very kind to her. All the same, the general idea that emerged from their conversation about her 1952 theorem could roughly be summed up as "very good, but nothing more." This was because that purely mathematical aspect of Yvonne's work was absolutely not a concern of Einstein's. He built physical models: you take the equations, then find exact solutions in simplified cases, and finally extrapolate. She, by contrast, did purely mathematical work aimed at identifying properties of the set of all solutions.

Fundamental! I like to say that Yvonne is the mother of the mathematics of relativity. Einstein's equations are a rather complicated business, which physicists have dissected in very interesting ways, but as far as the mathematics goes, her article is really the first to open up the theory: it shows that all solutions of Einstein's equations can be constructed from initial data, something that was not known before. That said, it must be admitted that it took time for the theorem to be appreciated: at the time, taking an interest in the mathematics of general relativity was a niche pursuit. Today, however, it has become a respected branch of mathematics. That is no doubt thanks to three major results: Yvonne's theorem of 1952, the 1970 theorem of Yvonne and Robert Geroch showing that the domain of dependence determines the evolution of a spacetime, and the 1979 theorem of Richard Schoen and Shing-Tung Yau showing that the mass of an asymptotically flat spacetime is positive.

It was in August, in Cargèse, Corsica. Since the 1960s there has been an Institut d'études scientifiques de Cargèse, affiliated with the University of Corsica. An impressive gathering of people came together to celebrate Yvonne: Robert Geroch, Jean-Pierre Bourguignon, Jim Isenberg, Vincent Moncrief, Robert Bartnik, Sergiu Klainerman… Looking at the list of students at that summer school, it is a pleasure to see that many of them went on to become well-known researchers in our field.

This was our second joint article. We began collaborating at the Mittag-Leffler Institute on the solutions of Einstein's equations within a light cone\\. After "cleaning up" the problem, we proved a nice theorem that vividly illustrates the fact that gravity is an attractive force acting on spacetime. More precisely, the theorem states the following: take a point and send light rays out in every direction, then measure the area formed by the wavefront at each instant. If Einstein's equations hold and we are either in a vacuum or in a matter field with positive energy, then the surface is smaller than the one obtained in Minkowski space, with equality if and only if we are in Minkowski space. This shows that spatial distances shrink when matter or a gravitational field is introduced. I should add that this article required, besides mathematical questions, a certain amount of computational power to carry out some very heavy symbolic computations, which we were able to do thanks to a collaboration with José Martín García. This research at the Mittag-Leffler Institute led to several joint articles, which turn out to have been almost her last articles.

(\\ Minkowski space models three-dimensional space together with time, the fourth dimension of space. It has the distinctive feature of being devoid of gravity: it is "flat" because it represents an absolute vacuum. A light cone in Minkowski space is called a Minkowski cone. In Minkowski space, light propagates at speed c, so the wavefront has radius ct, whose area is 4πc ²t ².)

She was very simple, kind and modest. That may sound like a platitude, but it is true. It was a pleasure to be with her and with our mutual friends. My wife adored Yvonne. I would have liked to keep working with her, because she had a very original way of approaching things, a fantastic mathematical imagination. You know, when you have a mathematical problem to solve, you need to find an angle of attack: you try this or that technique, you attempt to turn the problem into an equivalent one… Everyone in the field knows plenty of tricks for digging into a problem. But Yvonne always had ideas for working around the difficulties; she knew how to put the right tricks together to reach a solution. That takes a great deal of inventiveness. From a mathematical point of view, it was one of her very great qualities.