Yvonne Choquet-Bruhat obituary: mathematician who established that Einstein’s equations mirror the real world

Nature · 2025-06-04

The stability problem for extremal black holes

General Relativity and Gravitation · 2025-03-01 · 8 citations

Abstract I present a series of conjectures aiming to describe the general dynamics of the Einstein equations of classical general relativity in the vicinity of extremal black holes. I will reflect upon how these conjectures transcend older paradigms concerning extremality and near-extremality, in particular, the so-called “third law of black hole thermodynamics”, which viewed extremality as an unattainable limit, and the “overspinning/overcharging” scenarios, which viewed extremality as a harbinger of naked singularities. Finally, I will outline some of the difficulties in proving these conjectures and speculate on what it could mean if the conjectures turn out not to be true.

The interior of dynamical vacuum black holes I: The $C^0$-stability of the Kerr Cauchy horizon

Annals of Mathematics · 2025-09-01 · 4 citations

We initiate a series of works where we study the interior of dynamical rotating vacuum black holes without symmetry. In the present paper, we take up the problem starting from appropriate Cauchy data for the Einstein vacuum equations defined on a hypersurface already within the black hole interior, representing the expected geometry just inside the event horizon. We prove that, for all such data, the maximal Cauchy evolution can be extended across a non-trivial piece of Cauchy horizon as a Lorentzian manifold with continuous metric. In subsequent work, we will retrieve our assumptions on data assuming only that the black hole event horizon geometry suitably asymptotes to a rotating Kerr solution. In particular, if the exterior region of the Kerr family is proven to be dynamically stable---as is widely expected---then it will follow that the $C^0$-inextendibility formulation of Penrose's celebrated strong cosmic censorship conjecture is in fact false. The proof suggests, however, that the $C^0$-metric Cauchy horizons thus arising are generically singular in an essential way, representing so-called "weak null singularities", and thus that a revised version of strong cosmic censorship holds.

Mathematical Aspects of General Relativity

Oberwolfach Reports · 2025-02-14

General relativity is an area at the interface of partial differential equations, differential geometry, global analysis, mathematical physics and dynamical systems. It interacts with astrophysics, cosmology, high energy physics, and numerical analysis. The field is rapidly expanding and has witnessed remarkable developments and interconnections with other fields in recent years.The workshop Mathematical Aspects of General Relativity was organised by Carla Cederbaum (Tübingen), Mihalis Dafermos (Cambridge/Princeton), Jim Isenberg (Eugene) and Hans Ringström (KTH Stockholm). There were 48 on-site and 4 online participants. There were 16 one hour talks, nine 30 minute talks and four 10 minute talks.

Quasilinear wave equations on Kerr black holes in the full subextremal range $|a|

arXiv (Cornell University) · 2024-10-04

We prove global existence, boundedness and decay for small data solutions $ψ$ to a general class of quasilinear wave equations on Kerr black hole backgrounds in the full sub-extremal range $|a|

A scattering theory construction of dynamical vacuum black holes

Journal of Differential Geometry · 2024 · 44 citations

• Physics
• Classical mechanics
• Theoretical physics

We construct a large class of dynamical vacuum black hole spacetimes whose exterior geometry asymptotically settles down to a fixed Schwarzschild or Kerr metric. The construction proceeds by solving a backwards scattering problem for the Einstein vacuum equations with characteristic data prescribed on the event horizon and (in the limit) at null infinity. The class admits the full “functional” degrees of freedom for the vacuum equations, and thus our solutions will in general possess no geometric or algebraic symmetries. It is essential, however, for the construction that the scattering data (and the resulting solution spacetime) converge to stationarity exponentially fast, in advanced and retarded time, their rate of decay intimately related to the surface gravity of the event horizon. This can be traced back to the celebrated redshift effect, which in the context of backwards evolution is seen as a blueshift.

Black Holes Inside and Out 2024: visions for the future of black hole physics

arXiv (Cornell University) · 2024-10-18 · 4 citations

The gravitational physics landscape is evolving rapidly, driven by our ability to study strong-field regions, in particular black holes. Black Holes Inside and Out gathered world experts to discuss the status of the field and prospects ahead. We hope that the ideas and perspectives are a source of inspiration. Structure: Black Hole Evaporation - 50 Years by William Unruh The Stability Problem for Extremal Black Holes by Mihalis Dafermos The Entropy of Black Holes by Robert M. Wald The Non-linear Regime of Gravity by Luis Lehner Black Holes Galore in D > 4 by Roberto Emparan Same as Ever: Looking for (In)variants in the Black Holes Landscape by Carlos A. R. Herdeiro Black Holes, Cauchy Horizons, and Mass Inflation by Matt Visser The Backreaction Problem for Black Holes in Semiclassical Gravity by Adrian del Rio Black Holes Beyond General Relativity by Enrico Barausse and Jutta Kunz Black Holes as Laboratories: Searching for Ultralight Fields by Richard Brito Primordial Black Holes from Inflation by Misao Sasaki Tests of General Relativity with Future Detectors by Emanuele Berti Black Holes as Laboratories: Tests of General Relativity by Ruth Gregory and Samaya Nissanke Simulating Black Hole Imposters by Frans Pretorius Black Hole Spectroscopy: Status Report by Gregorio Carullo VLBI as a Precision Strong Gravity Instrument by Paul Tiede Testing the nature of compact objects and the black hole paradigm by Mariafelicia De Laurentis and Paolo Pani Some Thoughts about Black Holes in Asymptotic Safety by Alessia Platania Black Hole Evaporation in Loop Quantum Gravity by Abhay Ashtekar How the Black Hole Puzzles are Resolved in String Theory by Samir D. Mathur Quantum Black Holes: From Regularization to Information Paradoxes by Niayesh Afshordi and Stefano Liberati

Mathematical Aspects of General Relativity

Oberwolfach Reports · 2022-11-25 · 2 citations

General relativity is an area that naturally combines differential geometry, partial differential equations, global analysis and dynamical systems with astrophysics, cosmology, high energy physics, and numerical analysis. It is rapidly expanding and has witnessed remarkable developments in recent years.

Quasilinear wave equations on asymptotically flat spacetimes with applications to Kerr black holes

arXiv (Cornell University) · 2022-12-28 · 2 citations

We prove global existence and decay for small-data solutions to a class of quasilinear wave equations on a wide variety of asymptotically flat spacetime backgrounds, allowing in particular for the presence of horizons, ergoregions and trapped null geodesics, and including as a special case the Schwarzschild and very slowly rotating $\vert a \vert \ll M$ Kerr family of black holes in general relativity. There are two distinguishing aspects of our approach. The first aspect is its dyadically localised nature: The nontrivial part of the analysis is reduced entirely to time-translation invariant $r^p$-weighted estimates, in the spirit of [DR09], to be applied on dyadic time-slabs which for large $r$ are outgoing. Global existence and decay then both immediately follow by elementary iteration on consecutive such time-slabs, without further global bootstrap. The second, and more fundamental, aspect is our direct use of a "blackbox" linear inhomogeneous energy estimate on exactly stationary metrics, together with a novel but elementary physical space top order identity that need not capture the structure of trapping and is robust to perturbation. In the specific example of Kerr black holes, the required linear inhomogeneous estimate can then be quoted directly from the literature [DRSR16], while the additional top order physical space identity can be shown easily in many cases (we include in the Appendix a proof for the Kerr case $\vert a \vert \ll M$, which can in fact be understood in this context simply as a perturbation of Schwarzschild). In particular, the approach circumvents the need either for producing a purely physical space identity capturing trapping or for a careful analysis of the commutation properties of frequency projections with the wave operator of time-dependent metrics.

The non-linear stability of the Schwarzschild family of black holes

arXiv (Cornell University) · 2021-04-16 · 11 citations

We prove the non-linear asymptotic stability of the Schwarzschild family as solutions to the Einstein vacuum equations in the exterior of the black hole region: general vacuum initial data, with no symmetry assumed, sufficiently close to Schwarzschild data evolve to a vacuum spacetime which (i) possesses a complete future null infinity $\mathcal{I}^+$ (whose past $J^-(\mathcal{I}^+)$ is moreover bounded by a regular future complete event horizon $\mathcal{H}^+$), (ii) remains close to Schwarzschild in its exterior, and (iii) asymptotes back to a member of the Schwarzschild family as an appropriate notion of time goes to infinity, provided that the data are themselves constrained to lie on a teleologically constructed codimension-$3$ "submanifold" of moduli space. This is the full nonlinear asymptotic stability of Schwarzschild since solutions not arising from data lying on this submanifold should by dimensional considerations approach a Kerr spacetime with rotation parameter $a\neq 0$, i.e. such solutions cannot satisfy (iii). The proof employs teleologically normalised double null gauges, is expressed entirely in physical space and makes essential use of the analysis in our previous study of the linear stability of the Kerr family around Schwarzschild [DHR], as well as techniques developed over the years to control the non-linearities of the Einstein equations. The present work, however, is entirely self-contained. In view of the recent [DHR19, TdCSR20] our approach can be applied to the full non-linear asymptotic stability of the subextremal Kerr family.