Periodic Reporting for period 4 - BHSandAADS (The Black Hole Stability Problem and the Analysis of asymptotically anti-de Sitter spacetimes)
Reporting period: 2022-05-01 to 2024-04-30
Black holes are one of the most fascinating predictions of Einstein's theory of general relativity developed in 1915. The recent detection of gravitational waves produced by the merger of two black holes (leading to the Nobel Prize 2017), the first "photo" of a black hole by the Event Horizon Telescope in 2019, and the recent award of the Nobel Prize 2020 to Roger Penrose "for the discovery that black hole formation is a robust prediction of the general theory of relativity" are jut three very recent examples of the fundamental role that the analysis of black holes plays in modern science. The study of black holes, or more generally, general relativity, unites the mathematical areas of geometry and non-linear partial differential equation with both theoretical and experimental physics.
Mathematically, black holes are solutions of the Einstein equations, which themselves form a complicated non-linear system of partial differential equations -- so complicated that Einstein was not sure whether (non-trivial) solutions of his equations could ever be found. It came as a big surprise when the astronomer Karl Schwarzschild published an exact solution only a year after Einstein had published his equations. This solution, the Schwarzschild solution, turns out to describe the simplest (spherically symmetric) black hole geometry, a fact that was however only understood decades later! Over the years, more exact solutions have been found. Most remarkable is the Kerr family of solutions, describing rotating black holes, discovered in 1963. It is believed that all stationary asymptotically flat black holes are described by the Kerr family.
The main objectives of the project is to understand the stability of these stationary solutions. Roughly speaking: If the (Kerr)black hole is slightly perturbed, will the geometry remain close to the geometry one started with or will it "run away" and for instance form new black holes? The correct mathematical framework to address this question is the initial value problem (or Cauchy problem). Initial data are perturbed on a spacelike hypersurface and the development of the data is then analysed using modern techniques of partial differential equations and geometry. While the final goal is to understand the stability of the entire family of Kerr solutions, there are several smaller scale problems that have to be addressed on the way. One is to prove linear stability before non-iinear stability, another is to first prove stability for the simpler Schwarzschild family of black holes. As explained in the next paragraph, some of these steps have been successfully carried out during the first phase of the project.
A second objective of the proposal is the analysis of so-called asymptotically anti-de Sitter spacetimes. These spacetimes are solutions of the Einstein equations with a negative cosmological constant, whose distinguished feature in the presence of a conformal boundary at infinity. Such spacetimes appear frequently in high energy physics as models for superconductors and also feature as models for the overarching idea of "holography" in theoretical physics. The project looks at understanding such spacetimes both from the dynamical point of view, by proving, for instance, the non-linear stability of the maximally symmetric solution, anti-de Sitter space if dissipative boundary conditions are imposed at the conformal boundary. Another class of objectives is to prove unique continuation theorems for the Einstein equations at the conformal boundary to provide rigorous formulations of the principle of holography.
To conclude, understanding the Einstein equations in a mathematically rigorous fashion is essential for our understanding of the theory (e.g. the question whether black holes are "stable" predictions of theory) and complements experimental results from astrophysics. From an even broader perspective, a complete understanding of the theory is necessary to eventually move beyond the classical Einstein equations and understand settings in which quantum effects might play a role.
Mathematically, black holes are solutions of the Einstein equations, which themselves form a complicated non-linear system of partial differential equations -- so complicated that Einstein was not sure whether (non-trivial) solutions of his equations could ever be found. It came as a big surprise when the astronomer Karl Schwarzschild published an exact solution only a year after Einstein had published his equations. This solution, the Schwarzschild solution, turns out to describe the simplest (spherically symmetric) black hole geometry, a fact that was however only understood decades later! Over the years, more exact solutions have been found. Most remarkable is the Kerr family of solutions, describing rotating black holes, discovered in 1963. It is believed that all stationary asymptotically flat black holes are described by the Kerr family.
The main objectives of the project is to understand the stability of these stationary solutions. Roughly speaking: If the (Kerr)black hole is slightly perturbed, will the geometry remain close to the geometry one started with or will it "run away" and for instance form new black holes? The correct mathematical framework to address this question is the initial value problem (or Cauchy problem). Initial data are perturbed on a spacelike hypersurface and the development of the data is then analysed using modern techniques of partial differential equations and geometry. While the final goal is to understand the stability of the entire family of Kerr solutions, there are several smaller scale problems that have to be addressed on the way. One is to prove linear stability before non-iinear stability, another is to first prove stability for the simpler Schwarzschild family of black holes. As explained in the next paragraph, some of these steps have been successfully carried out during the first phase of the project.
A second objective of the proposal is the analysis of so-called asymptotically anti-de Sitter spacetimes. These spacetimes are solutions of the Einstein equations with a negative cosmological constant, whose distinguished feature in the presence of a conformal boundary at infinity. Such spacetimes appear frequently in high energy physics as models for superconductors and also feature as models for the overarching idea of "holography" in theoretical physics. The project looks at understanding such spacetimes both from the dynamical point of view, by proving, for instance, the non-linear stability of the maximally symmetric solution, anti-de Sitter space if dissipative boundary conditions are imposed at the conformal boundary. Another class of objectives is to prove unique continuation theorems for the Einstein equations at the conformal boundary to provide rigorous formulations of the principle of holography.
To conclude, understanding the Einstein equations in a mathematically rigorous fashion is essential for our understanding of the theory (e.g. the question whether black holes are "stable" predictions of theory) and complements experimental results from astrophysics. From an even broader perspective, a complete understanding of the theory is necessary to eventually move beyond the classical Einstein equations and understand settings in which quantum effects might play a role.
The main achievements of the project are:
(1) A complete proof of "The Non-Linear Stability of the Schwarzschild family of black holes" joint work with Dafermos, Rodnianski and Taylor. Our 500-page preprint is a key step on the way to a full proof of the stability of the entire family of Kerr solutions. This achieves one of the main objectives of the project.
(2) A thorough understanding of the covariant linear wave equation on black hole spacetimes in the presence of small first order perturbations. In joint work Chris Kauffman we studied such equations on the exterior of Schwarzschild an Kerr black holes and were able to prove decay of solutions for small enough perturbations.
(3) A novel robust understanding of quasilinear wave equation on black hole spacetimes. In joint work with Dafermos-Rodnianski-Taylor we developed a new scheme to prove small data global existence and decay for quasi-linear wave equations. The proof relies on a novel top order highly degenerate physical space estimate. Having dealt with the slowly rotating Kerr case, we are currently preparing paper on the full sub-extremal case, which is expected to resolve the quasilinear difficulties in the proof of the stability of the Kerr family.
(4) The conclusion of a program to formulate and prove a holographic principle for the Einstein equations on asymptotically anti-de Sitter spacetimes. In joint work with Arick Shao we were able to prove unique continuation results for the Einstein equations near the conformal boundary.
(1) A complete proof of "The Non-Linear Stability of the Schwarzschild family of black holes" joint work with Dafermos, Rodnianski and Taylor. Our 500-page preprint is a key step on the way to a full proof of the stability of the entire family of Kerr solutions. This achieves one of the main objectives of the project.
(2) A thorough understanding of the covariant linear wave equation on black hole spacetimes in the presence of small first order perturbations. In joint work Chris Kauffman we studied such equations on the exterior of Schwarzschild an Kerr black holes and were able to prove decay of solutions for small enough perturbations.
(3) A novel robust understanding of quasilinear wave equation on black hole spacetimes. In joint work with Dafermos-Rodnianski-Taylor we developed a new scheme to prove small data global existence and decay for quasi-linear wave equations. The proof relies on a novel top order highly degenerate physical space estimate. Having dealt with the slowly rotating Kerr case, we are currently preparing paper on the full sub-extremal case, which is expected to resolve the quasilinear difficulties in the proof of the stability of the Kerr family.
(4) The conclusion of a program to formulate and prove a holographic principle for the Einstein equations on asymptotically anti-de Sitter spacetimes. In joint work with Arick Shao we were able to prove unique continuation results for the Einstein equations near the conformal boundary.
(1) Our paper on "The Non-Linear Stability of the Schwarzschild family of black holes" contains several innovations that move the subject beyond the state of the art: Among others these are (1) The construction of (two) carefully chosen teleological double null foliations and the estimation of their dynamical interaction (2) Novel estimates for the Teukolsky equation (3) A modulation theory for the Einstein equations (required to prove the co-dimension 3 statement). Furthermore, we expect that our framework generalises to prove the stability of the Kerr family of solutions.
(2) The work with Kauffman introduces a novel commutator vectorfield (both a physical space version for Schwarzschild, and a pseudo-differential version for Kerr) which has many applications. It has already been applied in the cosmological setting (by Mavrogiannis) to prove exponential decay estimates on asymptotically de Sitter black holes and it also provides a fruitful alternative approach to non-linear problems.
(3) This project moved beyong the state of the art by clarifying that the quasi-linear aspects of the black hole stability problem (and the associated loss of derivatives) are not as difficult to handle as commonly thought. The proof introduces a novel, very robust multiplier that can be used for a broad range of spacetimes and in addition to this a new dyadic scheme that provides additional flexibility when proving decay for non-linear wave equations. With the anticipated results for the sull subextremal range, which we are currently writing up. this resolves one of the key difficulties in the Kerr stability problem.
(4) For this project, the state of the art were similar results in the class of analytic solutions. We pushed this to the class of smooth (or sufficiently differentiable) solutions which is a much more natural class in general relativity
(2) The work with Kauffman introduces a novel commutator vectorfield (both a physical space version for Schwarzschild, and a pseudo-differential version for Kerr) which has many applications. It has already been applied in the cosmological setting (by Mavrogiannis) to prove exponential decay estimates on asymptotically de Sitter black holes and it also provides a fruitful alternative approach to non-linear problems.
(3) This project moved beyong the state of the art by clarifying that the quasi-linear aspects of the black hole stability problem (and the associated loss of derivatives) are not as difficult to handle as commonly thought. The proof introduces a novel, very robust multiplier that can be used for a broad range of spacetimes and in addition to this a new dyadic scheme that provides additional flexibility when proving decay for non-linear wave equations. With the anticipated results for the sull subextremal range, which we are currently writing up. this resolves one of the key difficulties in the Kerr stability problem.
(4) For this project, the state of the art were similar results in the class of analytic solutions. We pushed this to the class of smooth (or sufficiently differentiable) solutions which is a much more natural class in general relativity