Periodic Reporting for period 4 - EPGR (The Evolution Problem in General Relativity)
Okres sprawozdawczy: 2021-11-01 do 2022-10-31
General relativity has been introduced by A. Einstein in 1915. It is a major theory of modern physics and at the same time has led to fascinating mathematical problems. The present proposal focusses on two aspects of the evolution problem for the Einstein equations which has been initiated by the pioneering work of Y. Choquet-Bruhat in 1952.
The Einstein equations form a nonlinear system of partial differential equations of hyperbolic type whose complexity raises significant challenges to its mathematical analysis. The goal of this project is to strengthen our understanding of two important themes concerning the evolution problem in general relativity. On the one hand, the control of low regularity solutions of the Einstein equations, a topic which is intimately linked with the celebrated cosmic censorship conjectures of R. Penrose, a major open problem in the field. On the other hand, the question of the stability of particular solutions of the Einstein equations in the wake of the groundbreaking proof of the stability of the Minkowski space-time due to D. Christodoulou and S. Klainerman. These directions are extremely active and have recently led to impressive results. More specifically, this project proposes to consider the following two work packages:
- Going beyond the bounded L2 curvature theorem. This result has been recently obtained by the PI in collaboration with S. Klainerman and I. Rodnianski and is the sharpest result in so far as low regularity solutions of the Einstein equations are concerned. Yet, the fundamental quest towards a scale invariant well-posedness criterion for the Einstein equations remains wide open.
- The black hole stability problem. This problem concerns the stability of the Kerr metrics which form a 2-parameter family of solutions to the Einstein vacuum equations. Many results have been obtained concerning various versions of linear stability, but significant challenges remain in order to tackle the nonlinear stability result.
The Einstein equations form a nonlinear system of partial differential equations of hyperbolic type whose complexity raises significant challenges to its mathematical analysis. The goal of this project is to strengthen our understanding of two important themes concerning the evolution problem in general relativity. On the one hand, the control of low regularity solutions of the Einstein equations, a topic which is intimately linked with the celebrated cosmic censorship conjectures of R. Penrose, a major open problem in the field. On the other hand, the question of the stability of particular solutions of the Einstein equations in the wake of the groundbreaking proof of the stability of the Minkowski space-time due to D. Christodoulou and S. Klainerman. These directions are extremely active and have recently led to impressive results. More specifically, this project proposes to consider the following two work packages:
- Going beyond the bounded L2 curvature theorem. This result has been recently obtained by the PI in collaboration with S. Klainerman and I. Rodnianski and is the sharpest result in so far as low regularity solutions of the Einstein equations are concerned. Yet, the fundamental quest towards a scale invariant well-posedness criterion for the Einstein equations remains wide open.
- The black hole stability problem. This problem concerns the stability of the Kerr metrics which form a 2-parameter family of solutions to the Einstein vacuum equations. Many results have been obtained concerning various versions of linear stability, but significant challenges remain in order to tackle the nonlinear stability result.
Concerning the first work package mentioned above, on going Beyond the bounded L2 curvature theorem, Olivier Graf (funded in his last year of PhD by the project) has proved, in collaboration with Stefan Czimek, an analog of the bounded L2 curvature theorem for the spacelike-characteristic Cauchy problem. The importance of Olivier's result is to formulate the bounded L2 curvature theorem in a context in which showing its optimality should be easier. After this initial breakthrough, the project was supposed to benefit from international research collaborations that were canceled due to Covid, and have only recently resumed. This has significantly impacted advances of the remainder of the first work package. In view of the important advances on the second work package (see below), I have decided to concentrate ressources on it for the remainder of the project.
Concerning the second work package mentioned above, on the black hole stability problem, I have fully completed the first and second tasks of this project. More precisely:
-Concerning the first task: I have obtained, in collaboration with Sergiu Klainerman, the first nonlinear result, concerning the non linear stability of Schwarzschild under a certain symmetry class. This 850 pages long manuscript have been published in Annals of Mathematics Studies. This result has been instrumental for the rest of the project as it has provided us with a blueprint for tackling the general case of Kerr (see the second task below).
-Concerning the second task: I have obtained, in collaboration with Sergiu Klainerman, and partly with Elena Giorgi and Dawei Shen, the resolution of the Kerr stability conjecture for small angular momentum (i.e. slowly rotating Kerr black holes). The proof consists of a series of five papers spanning over 2000 pages. So far, two papers have appeared in Annals of PDE and the three other papers are under review.
Still concerning the second work package, important works have been done by other members of the project:
-Siyuan Ma (postdoc 2), together with collaborators, has obtained in a work currently under review the stability of the linearized Einstein equations around Kerr for small angular momentum, extending previous work on the corresponding problem for Schwarzschild, and has also extended the gauge used in his linear work to the nonlinear setting in a work published in Communications in Mathematical Physics. Moreover, he has obtained a series of five works (three published respectively in Annals of PDE, Journal of Functional Analysis, and Journal of Differential Equations, and two being under review) on the sharp decay of various important hyperbolic equations (scalar wave, Teukolsky, Dirac) on Schwarzschild and Kerr black holes.
-Volker Schlue (postdoc 1) has considered the stability problem in the context of a positive cosmological constant and has obtained two results, published respectively in Annals of PDE and Journal of Mathematical Physics, in connection with the nonlinear stability of Schwarzschild-De-Sitter in the so-called expanding region.
-Allen Fang has also considered the stability problem in the context of a positive cosmological constant and has obtained a new proof of the nonlinear stability of Kerr de Sitter in a series of two papers currently under review.
Finally, in a joint work with Frank Merle, Pierre Raphaël and Igor Rodnianski, we have constructed blow up solutions to compressible fluids and supercritical nonlinear Schrödinger equations in a series of three papers, one published in Inventiones and the two others in Annals of Math. While not directly connected to the stability of black holes, these works fit in the general topic of stability as blow up amounts to the stability of a profile in rescaled variables.
Concerning the second work package mentioned above, on the black hole stability problem, I have fully completed the first and second tasks of this project. More precisely:
-Concerning the first task: I have obtained, in collaboration with Sergiu Klainerman, the first nonlinear result, concerning the non linear stability of Schwarzschild under a certain symmetry class. This 850 pages long manuscript have been published in Annals of Mathematics Studies. This result has been instrumental for the rest of the project as it has provided us with a blueprint for tackling the general case of Kerr (see the second task below).
-Concerning the second task: I have obtained, in collaboration with Sergiu Klainerman, and partly with Elena Giorgi and Dawei Shen, the resolution of the Kerr stability conjecture for small angular momentum (i.e. slowly rotating Kerr black holes). The proof consists of a series of five papers spanning over 2000 pages. So far, two papers have appeared in Annals of PDE and the three other papers are under review.
Still concerning the second work package, important works have been done by other members of the project:
-Siyuan Ma (postdoc 2), together with collaborators, has obtained in a work currently under review the stability of the linearized Einstein equations around Kerr for small angular momentum, extending previous work on the corresponding problem for Schwarzschild, and has also extended the gauge used in his linear work to the nonlinear setting in a work published in Communications in Mathematical Physics. Moreover, he has obtained a series of five works (three published respectively in Annals of PDE, Journal of Functional Analysis, and Journal of Differential Equations, and two being under review) on the sharp decay of various important hyperbolic equations (scalar wave, Teukolsky, Dirac) on Schwarzschild and Kerr black holes.
-Volker Schlue (postdoc 1) has considered the stability problem in the context of a positive cosmological constant and has obtained two results, published respectively in Annals of PDE and Journal of Mathematical Physics, in connection with the nonlinear stability of Schwarzschild-De-Sitter in the so-called expanding region.
-Allen Fang has also considered the stability problem in the context of a positive cosmological constant and has obtained a new proof of the nonlinear stability of Kerr de Sitter in a series of two papers currently under review.
Finally, in a joint work with Frank Merle, Pierre Raphaël and Igor Rodnianski, we have constructed blow up solutions to compressible fluids and supercritical nonlinear Schrödinger equations in a series of three papers, one published in Inventiones and the two others in Annals of Math. While not directly connected to the stability of black holes, these works fit in the general topic of stability as blow up amounts to the stability of a profile in rescaled variables.
The resolution of the first and second task of the second work package goes significantly beyond the state of the art. Indeed:
-the non linear stability of Schwarzschild under a certain symmetry class is the first proof of the nonlinear stability of a black hole spacetime in the asymptotically flat setting which is both the hardest case and the one that is physically relevant.
-the resolution of the Kerr stability conjecture for small angular momentum (i.e. slowly rotating Kerr black holes) answers a major open problem ever since the discovery of the Kerr metric in 1963 - problem which has generated an intense activity both by physicists and mathematicians.
-the non linear stability of Schwarzschild under a certain symmetry class is the first proof of the nonlinear stability of a black hole spacetime in the asymptotically flat setting which is both the hardest case and the one that is physically relevant.
-the resolution of the Kerr stability conjecture for small angular momentum (i.e. slowly rotating Kerr black holes) answers a major open problem ever since the discovery of the Kerr metric in 1963 - problem which has generated an intense activity both by physicists and mathematicians.