Remembering Yvonne Choquet-Bruhat

TL;DR

This paper discusses the significant influence of Yvonne Choquet-Bruhat's mathematical contributions on gravitational physics and reflects on the author's personal interactions with her over time.

Contribution

It highlights the impact of Yvonne Choquet-Bruhat's mathematical work on the development of gravitational physics and personal academic relationships.

Findings

Yvonne Choquet-Bruhat's results advanced gravitational theory.

Her work influenced subsequent research in mathematical physics.

The author's interactions illustrate her role in the scientific community.

Abstract

I describe the impact of some of the mathematical results of Yvonne Choquet-Bruhat on gravitational physics, as well as the evolution of my interactions with her over the years.

Full text

1 First encounter

My earliest memory of Yvonne Choquet-Bruhat (whom I will simply call Yvonne) is her smiling face when we first met on a train platform in Paris. Was it July 1975, on our way to the first Marcel Grossmann Meeting in Trieste? I don’t know, but I do clearly remember her warm, welcoming smile toward a junior scientist just starting out in gravitational physics. That smile never faded throughout all the years I had the pleasure of interacting with her – particularly between 2003 and 2018, when she was a regular visitor at IHES.

2 First interactions linked to science

The Comptes Rendus of the (French) Academy of Sciences were originally created to announce, briefly and rapidly, significant new results. In particular, they allowed junior scientists to publish a short account of novel results under the aegis of a member of the Academy of Sciences who served as a “referee”, vouching for the (a priori) validity of their results. For instance, Yvonne’s first announcement of her breakthrough result on Einstein’s field equations, at the early age of 26, namely her “Théorème d’existence pour les équations de la gravitation einsteinienne dans le cas non analytique”, was published in early 1950, a couple of weeks after the presentation of her Note to the Academy, on February 6, 1950, by Jacques Hadamard (one of the greatest living mathematicians at the time). See [1] for an English translation, and republication, of Yvonne’s Note as a Golden Oldie, and [2] for a beautiful accompanying editorial comment .

I had become aware of this possibility of “recording the date” (“prendre date”, in the words of Lichnerowicz) in the fall of 1974 when Remo Ruffini introduced me to André Lichnerowicz in Princeton. [And this eventually led to the fast publication (December 1974) of my first published paper, a Comptes Rendus Note with Remo on the newly discovered Hulse-Taylor binary pulsar [3].] The reason for my mentioning this way of helping junior scientists to publish rapidly novel results (and also to bring their work to the attention of senior experts in their field) is that, around 1980, when Nathalie Deruelle and I obtained novel results on the two body problem in General Relativity we asked Yvonne to present to the Academy two short Notes [4, 5] on the Lagrangian dynamics of two point masses at the second post-Newtonian approximation, soon followed by another Note by me [6] also presented by Yvonne. Nathalie and I were quite grateful to Yvonne for this triple opportunity of expeditiously announcing our new results at a moment where there was an international, competitive effort to clarify the so-called “quadrupole formula controversy”, i.e. the issue of understanding the back reaction of gravitational radiation on the dynamics of binary systems, such as the Hulse-Taylor binary pulsar.

3 Deeper scientific interactions with Yvonne

Though, as mentioned above, I had met Yvonne early in my career, and I continued meeting her, in an increasingly friendly way, in many subsequent scientific meetings (Marcel Grossmann meetings, Journées Relativistes [7], GRG meetings, etc.), I must confess that, for many years, I had no real scientific interactions with her. Seen from the perspective of a junior theoretical physicist, interested in exploring the observable consequences of Einstein’s gravitation theory in our real Universe, most of her work appeared to me as being too mathematically-oriented to attract more than a rather far-away contemplation. It took me years to understand that I was wrong, and that Yvonne’s works contained many results of direct importance for gravitational physics.

Let me mention just a few of the physics-relevant results of Yvonne (besides her pioneering non-analytic existence theorem [1, 8] whose physics importance I also only slowly appreciated, and her many results on the constraints, which, I assume, are discussed in other contributions to this homage):

• •

First, her pioneering work [9] on the $3+1$ decomposition of Einstein’s equations, usually attributed only to Arnowitt-Deser-Misner [10]. As we know, the $3+1$ decomposition of Einsteinian gravity has become one of the key elements of modern Numerical Relativity. More about this below.

• •

Second, her work on strong high-frequency gravitational waves [11]. It took me time to grasp what made this work significantly superior to the famous, slightly earlier, work of Rich Isaacson [12]. Besides a mathematically clear way of defining (by the two-timing technique) and justifying (by a boundedness condition) the averaging leading to the appearance of the effective stress-energy tensor of gravitational waves, the most interesting result of [11] is the realization that the nonlinear structure of Einstein’s equations ensures the lack of steepening of strong gravitational waves during their propagation. In other words, nonlinear gravitational waves satisfy the exceptionality condition of Lax and Boillat (see, e.g., [13] and references therein). This exceptional property of nonlinear gravitational waves is directly related to the weak form of the Christodoulou-Klainerman “null condition” satisfied by Einstein’s equations (see [14] and references therein). For recent progress in the construction of exact solutions containing strong high-frequency gravitational waves see [15] and references therein.

• •

Then I would like to mention Yvonne’s work on the positive mass issue. Most importantly, her 1976 work with Jerrold Marsden [16] gave the first rigourous proof of the positivity of energy for vacuum space times near flat space using a critical point analysis in infinite dimensions. See below for a later contribution of Yvonne, and [17] for a recent survey of positive(-mass)-energy theorems.

• •

Of particular importance is the work of Yvonne on hyperbolic formulations of 3+1 versions of the evolution part of Einstein’s equations. In particular, Yvonne and Tommaso Ruggeri obtained (by combining the evolution equations with the constraints) a 3+1 hyperbolic system with zero shift $\beta$. This was then generalized by Yvonne (in collaboration with Jimmy York and Arlen Anderson) to the construction of hyperbolic 3+1 systems with non-zero shift. These results were part of an international effort to construct 3+1 evolution systems having sufficient stability to be solved on a computer. Let me note in particular that the crucial time-hyperbolicity condition used by Yvonne and Tommaso Ruggeri, which yields, in 3+1 variables, the following evolution equation for the lapse $\alpha$, $(\partial_{t}-{\mathcal{L}}_{\beta})=-\alpha^{2}K$, was later generalized [19] in the so-called “1+log” slicing condition, $(\partial_{t}-{\mathcal{L}}_{\beta})=-2\alpha K$ which has become an important ingredient of many modern Numerical Relativity codes.

• •

Let me finally mention the important contributions of Yvonne to understanding the causality properties of several of the extensions of Einstein’s theory suggested by modern theoretical physics. I have notably in mind here the works of Yvonne on Supergravity [20, 21, 22] and on Gauss-Bonnet-type gravity [23].

4 Discussions with Yvonne at IHES

From 2003 to 2018, Yvonne regularly visited IHES (at least once per week), and I had the pleasure to discuss many times with her, about either mathematics or physics. This also allowed me to have many enlightening discussions with several of the collaborators of Yvonne who visited IHES in those years, notably Vince Moncrief and Piotr Chrusciel. Let me first recall that I was always amazed by the clearness and accuracy of her answers to any of my math questions. She would either immediately answer a math question with a precise answer, or give me the name of the best person to contact.

During her stay at IHES, Yvonne wrote some twenty-six research papers, two highest-level scientific books, [24] and [25], and her autobiogaphy [26]. I had uncountable discussions with her about the content of all her books. She kindly asked me to write a chapter for her monumental, testamentary book [24] on the Belinsky-Khalatnikov-Lifshitz conjecture concerning the behavior of generic solutions of Einstein’s equations near a spacelike singularity. I was happy to do so, and I tried to formulate, in an as mathematically precise way as I could, the conjecture suggested in the pioneering work of Belinsky, Khalatnikov and Lifshitz [27], and refined in the many physics works triggered by it. Let me note in this respect the not so well-known fact that the main results of [27], and notably the coupled second-order nonlinear differential equations for the three local scale factors $a(\tau),b(\tau),c(\tau)$, were first publically presented by Isaak Khalatnikov in January 1968 at the Institut Henri Poincaré in Paris [28]. The audience comprised in particular, John Archibald Wheeler and Demetrios Christodoulou. When Isaak presented the $a-b-c$ ODEs, Wheeler made the public remark that this looked like a mechanical system where the Universe was described by the three coordinates $a,b,c$. One year later (14 April 1969), Charles Misner published his famous Lagrangian description of the complex dynamics of Bianchi IX universes (called by him “Mixmaster Universe”), mentioning at the end a private communication from J.A.W. in which he “suggested that studies of singularities by Belinsky and Khalatnikov had also found alternating Kasner-like epochs but with very simple description in terms of a related parameter ($u$)”.

Her second scientific book [25] led also to many interesting discussions with Yvonne, who wanted to be kept abreast of the points of contact between General Relativity and experimental or observational facts. Let us recall in this respect that Yvonne (whose father, Georges Bruhat, was a physicist, well known in France for his work on optics and his textbooks) considered herself as a “failed” physicist who constantly aimed at understanding the real universe through its theoretical physics description, by using, and perfecting, mathematical tools.

During Yvonne’s stay at IHES, two conferences were organized in her honor: one in March 2004 for her 80th birthday, and one in January 2014 for her 90th birthday. Another special moment happened on February 11, 2016: Yvonne and a group of scientists of IHES (including me) followed the live announcement (from Washington, DC, USA) of the discovery of the first gravitational wave signal by the two LIGO interferometers. After the end of her regular visits at IHES, a one-day conference was organized at IHES in December 2023 to celebrate her 100th birthday (unfortunately in her absence).

Let me finally mention that I had detailed technical discussions with Yvonne about one of her later research papers, namely her streamlined, complete proof (valid in arbitrary space dimension $n$, and using only spinors on some (oriented) spacelike section $\Sigma_{n}$) of the (strong) positive energy theorem ($E\geq|{\bf P}|$) in General Relativity. She kindly offered me to co-sign her paper. But I felt I had not significantly contributed to her proof, and I declined this honor, and this token of friendship.

5 Epilogue

To end this tribute to the memory of Yvonne, let me mention one of my last discussions (by phone) with her. One day in the spring, or early summer, of 2018 (at a time when she had essentially stopped visiting regularly IHES), Yvonne called me on my cell phone (I remember that I was walking in the countryside) to tell me the good news that the Italian Society of General Relativity and Gravitation (SIGRAV) had contacted her to inform her that she had been awarded the 2018 Amaldi Medal, to be received in person at the next SIGRAV National conference, due to take place in September in Cagliari (Sardegna, Italy). She was very happy to receive this distinction because she always had had close links with Italy and very friendly relations with many Italian scientists. However, she told me that she did not feel she had the energy to go there in person (she was 95 !). I offered to collect the medal on her behalf, and to make a small presentation of her life work. So it happened. It gave me the opportunity to study in detail some of her most important contributions to gravitational physics, which I greatly enjoyed. [However, as I had to travel after the SIGRAV meeting to a summer school in Ravello, I had the slight practical problem to be cautious in carrying all over Italy this very valuable Amaldi medal, made of solid gold!]