Mathematical GR and Hyperbolic PDE Seminar
Welcome to the website of our Monthly Online Mathematical GR and Hyperbolic PDE seminar!
In each meeting, we will have one or two one-hour talks and an informal 1/2-hour discussion, from 10 am to 11 am EST time on the third Friday of the month.
If you are interested in this seminar, please fill this google form, and we will add you to our email list.
- Arthur Touati (École Polytechnique), Construction of high-frequency spacetimes
In this talk, I will present recent work on high-frequency solutions to the Einstein vacuum equations. From a physical point of view, these solutions model high-frequency gravitational waves and describe how waves travel on a fixed background metric. There are also interesting when studying the Burnett conjecture, which adresses the lack of compactness of the family of vacuum spacetimes. These high-frequency spacetimes are singular and require to work under the regime of well-posedness for the Einstein vacuum equations. I will review the literature on the subject and then show how one can construct them in generalised wave gauge by defining high-frequency ansatz.
Here is a list of our past speakers:
- Elena Giorgi (Princeton University), Electromagnetic-gravitational perturbations of Kerr-Newman spacetime
The Kerr-Newman spacetime is the most general explicit black hole solution, and represents a stationary rotating charged black hole. Its stability to gravitational and electromagnetic perturbations has eluded a proof since the 80s in the black hole perturbation community, because of "the apparent indissolubility of the coupling between the spin-1 and spin-2 fields in the perturbed spacetime", as put by Chandrasekhar. We will present a derivation of the Teukolsky and Regge-Wheeler equations in Kerr-Newman in physical space and use it to obtain a quantitative proof of stability.
- John Anderson (Princeton University), Stability results for anisotropic systems of wave equations
In this talk, I hope to show a global stability result for a nonlinear anisotropic system of wave equations. This is motivated by studying phenomena involving characteristics with multiple sheets. For the proof, I will describe a strategy for controlling the solution based on bilinear energy estimates. Through a duality argument, this will allow us to prove decay in physical space using decay estimates for the homogeneous wave equation as a black box. The final proof will also require us to exploit a certain null condition that is present when the anisotropic system of wave equations satisfies a structural property involving the light cones of the equations.
- Arick Shao (Queen Mary University of London), Extension of symmetries from conformal boundaries of vacuum
asymptotically Anti-de Sitter spacetimes
In this talk, we investigate whether correspondences exist between vacuum asymptotically Anti-de Sitter spacetimes and properties of their conformal boundaries. Here, we focus primarily on the question of whether a symmetry on such a conformal boundary must be inherited by the spacetime itself. In particular, we present the first positive results in this direction for non-stationary settings, and we discuss recent developments in unique continuation theory that lead to these results. The presentation involves joint works with Gustav Holzegel and Alex McGill (as well as work in progress with Athanasios Chatzikaleas).
- Maxime van de Moortel (Princeton University), The breakdown of weak null singularities inside black holes
What singularities lie inside a black hole formed in gravitational collapse? Locally near time-like infinity, it is known for various models that a generic black hole has a weakly singular Cauchy horizon. The global structure of the black hole interior, however, has largely remained unexplored. I will present my recent proof that, in the spherical collapse of a charged scalar field, the weakly singular Cauchy horizon breaks down and gives way to either a “crushing type” singularity, or a locally naked singularity emanating from the center.
- Christoph Kehle (ETH), Diophantine Approximation as Cosmic Censor for AdS Black Holes
The statement that general relativity is deterministic finds its mathematical formulation in the celebrated ‘Strong Cosmic Censorship Conjecture due to Roger Penrose. I will present my recent results on this conjecture in the case of negative cosmological constant and in the context of black holes. It turns out that this is intimately tied to Diophantine properties of a suitable ratio of mass and angular momentum of the black hole and that the validity of the conjecture depends in an unexpected way on the notion of genericity imposed.
- Rita Teixeira da Costa (University of Cambridge), Energy, spin-reversal, and other quirks of the Teukolsky equation on Kerr black holes
The Teukolsky equation is one of the fundamental equations governing linear gravitational, electromagnetic and other perturbations of the Kerr black hole family. In this talk, we discuss the surprisingly rich structure concealed in this equation: spin-reversal through the Teukolsky--Starobinsky identities and the existence of a strange new conserved energy, among other characteristics. Though often neglected, these features are fundamental to understanding boundedness, decay and scattering properties of solutions to the Teukolsky equation in the full subextremal black hole parameter range. This talk includes joint work with Yakov Shlapentokh-Rothman (Princeton) and Marc Casals (CBPF / University College Dublin).
- Fred Alford (University of Cambridge), A Mathematical Discussion of Hawking Radiation
In the first half of this talk, we will introduce the physical phenomenon of Hawking Radiation, discussing the basics of quantum field theory to arrive at the mathematical expression for the number of particles emitted by a black hole. In the second half of this talk, we will discuss this calculation in the case of Reissner-Nordstrom black holes. By evolving a solution to the linear wave equation backwards from future null infinity, we will outline how to perform this calculation in a rigorous manner. We will then discuss the result obtained, in a physical context.
- Sam Collingbourne (University of Cambridge), Conservation Laws in Double Null Gauge on Schwarzschild From Canonical Energy
In this talk I will discuss conservation laws for linearised gravity on the Schwarzschild spacetime. In particular, I will discuss the connection of the canonical energy of Hollands and Wald (https://link.springer.com/article/10.1007/s00220-012-1638-1) with the energy conservation laws of Holzegel in double null gauge on the Schwarzschild background (https://iopscience.iop.org/article/10.1088/0264-9381/33/20/205004/pdf).
In the paper of Hollands and Wald they define the notion of canonical energy as a symplectic form on the space of linear perturbations which is conserved. They establish a criterion for linear stability of general static or stationary, axisymmetric, vacuum black holes under (axisymmetric in the rotating, stationary, axisymmetric case) perturbations, based on the whether this canonical energy is non-negative. Due to its complexity, evaluation of the canonical energy directly is difficult; even for perturbations of Schwarzschild, the canonical energy has not yet been shown to produce non-negative energies on a foliation of spacelike slices connecting the event horizon and null infinity. A solution to this issue is to establish a connection to Holzegel’s work on linearised gravity on Schwarzschild. In Holzegel's paper, it is proven that the flux of the linearised shear on the event horizon and the weighted linearised shear on null infinity is bounded by initial data. Interpreting these fluxes as a measure of the total energy radiated to null infinity and the horizon, one may view boundedness of these fluxes as a weak form of stability. A key ingredient to producing these bounds is a conservation law for the linearised metric and connection coefficients. I will show that this conservation law of Holzegel is precisely the conservation law for the canonical energy.
- Cecile Huneau (Ecole Polytechnique), High frequency limit for Einstein equations with a U(1) symmetry
Due to the nonlinear character of Einstein equations, a sequence of metrics, solutions in vacuum, which oscillate with higher and higher frequency may converge to a solution to Einstein equations coupled to some effective energy momentum tensor. This effect is called backreaction, and has been studied by physicists (Isaacson, Burnett, Green and Wald). It has been conjectured by Burnett, under some definition of the high frequency limit, that the only effective energy momentum tensor that could appear corresponds to a massless Vlasov field, and that reciprocally all solutions to Einstein equations coupled to a massless Vlasov field can be approached by a sequence of solutions to Einstein vacuum equations (with higher and higher frequencies oscillations). I will present a work in collaboration with Jonathan Luk around Burnett conjecture and its reverse.
- Federico Pasqualotto (UC Berkeley), The interaction of nonlinear waves from several localized sources
In this talk I will describe a research program aimed at understanding the interaction of gravitational waves originating from multiple localized and distant sources. This draws motivation from the problem of interaction of several localized bodies in general relativity. I will focus on the simplified interaction problem of multiple nonlinear waves, restricting attention to a model problem. I will show how one can take advantage of the specific geometric configuration to control the interaction. I will then describe the relevant directions in the context of general relativity. This is joint work with John Anderson (Princeton).
- Jan Sbierski (University of Oxford), On the structure and strength of singularities: Inextendibility results for Lorentzian manifolds
Given a solution of the Einstein equations a fundamental question is whether one can extend the solution or whether the solution is maximal. If the solution is inextendible in a certain regularity class due to the geometry becoming singular, a further question is whether the strength of the singularity is such that it terminates classical time-evolution.
In this talk we give an overview of low-regularity inextendibility results for Lorentzian manifolds and then focus on the locally Lipschitz inextendibility of FLRW models with particle horizons and spherically symmetric weak null singularities. The latter in particular apply to the spherically symmetric spacetimes constructed by Luk and Oh, improving their C^2-formulation of strong cosmic censorship to a locally Lipschitz formulation.
- Sam Zbarsky (Princeton University), Exterior perturbations of self-similar wave maps blowup.
One can construct various self-similar blowup solutions for supercritical wave maps equation. The question of whether such blowups are stable is open, though there are some results if we assume symmetric perturbations. However, assuming a suitable stability result in the past light cone of the singularity, one can obtain stability all the way to the future light cone for a large family of such solutions and general localized perturbations. I will be talking about this result.
- Leonardo Abbrescia (Vanderbilt University), The boundary of maximal hyperbolic development for compressible Euler shock solutions
Christodoulou's celebrated 2007 monograph on shock formation for the irrotational and isentropic relativistic Euler equations set the stage for the dramatic advancements in the mathematical theory of compressible fluids seen in the past 15 years. Two important examples include: 1) the extension of Christodoulou's results to the compressible Euler equations for an open set of initial data with nontrivial vorticity by Luk-Speck; 2) Christodoulou's recent resolution of the restricted shock development problem. The restricted shock development problem is essentially a local well-posedness result for irrotational and isentropic weak solutions with a shock singularity. The term “restricted” means that Christodoulou considered only irrotational initial data, and he ignored the jump in entropy and vorticity across the shock hypersurface, so that the mathematical problem concerned only irrotational solutions. We stress that the “initial data" for the restricted shock development problem is provided by the state of the solution on the boundary of its maximal classical hyperbolic development (which was derived by Christodoulou in 2007 in the irrotational case, starting from smooth initial conditions).
In this talk, we will present our joint work with Speck on the construction of the maximal classical hyperbolic development launched by an open set of initial data with non-trivial vorticity and non-constant entropy. The maximal development we construct is necessary to initiate the true "unrestricted" shock development problem. The presence of vorticity and entropy necessitate elliptic estimates to prevent a derivative loss that was not present in Christodoulou's work. These estimates are known across constant-time hypersurfaces, but to the best of our knowledge, a sharp construction of the maximal development with vorticity and entropy cannot be achieved with these hypersurfaces. We will present a new quasilinear foliation of spacetime on which one can derive these elliptic estimates and provide a sharp construction of the maximal development.
- Haoyang Chen (National University of Singapore), Low regularity ill-posedness for elastic waves driven by shock formation
In this talk, we generalize a classic result of Lindblad on the scalar quasilinear wave equation and we show that the Cauchy problem for 3D elastic waves, a physical quasilinear wave system with multiple wave-speeds, is ill-posed in $H^3(R^3)$. We further prove that the ill-posedness is caused by instantaneous shock formation, which is characterized by the vanishing of the inverse foliation density. The main difficulties arise from the multiple wave-speeds and its associated non-strict hyperbolicity. We design and combine a geometric approach and an algebraic approach to overcome these difficulties. This is based on joint work with Xinliang An and Silu Yin.
- Leonhard Kehrberger (University of Cambridge), The Case Against Smooth Null Infinity: Heuristics, Counter-Examples and Consequences
In 2002, Christodoulou gave a heuristic argument stating that Penrose's proposal of smooth conformal compactification of spacetime (a.k.a. smooth null infinity) and the “peeling property” implied by it are incompatible with any physically realistic setup of $N$ infalling masses from the infinite past. Motivated by this argument, I will, in this talk, present various dynamical scattering constructions of spacetimes that model such setups and that do not have a smooth null infinity. These counter-examples will be in the context of the Einstein-Scalar field system under spherical symmetry. I will also show that these models motivate late-time asymptotics that are quite different from the usual and well-known Price's law asymptotics. In particular, they suggest that the non-smoothness of null infinity can, in principle, be measured. This talk will be based on the preprints: arXiv:2106.00035, arXiv:2105.08084, arXiv:2105.08079
- Hamed Masaood (University of Cambridge), A scattering theory for the linearised Einstein equations on the exterior of the Schwarzschild Black hole
Scattering theory is fundamental to the study of black holes in general relativity, yet determining the extent to which the scattering problem can be said to be "well-posed" has seen little in the way of systematic study. In this talk I will describe my work towards an answer in the case of the linearised Einstein equations on the Schwarzschild exterior. I will first discuss preliminary results on scattering for the system of the Teukolsky equations of spin \pm 2 together with the Teukolsky--Starobinsky identities. I will then discuss ongoing work on using these results to obtain a scattering theory for the full system of linearised gravity.
- Renato Velozo (University of Cambridge), Stability of Schwarzschild for the spherically symmetric Einstein-massless Vlasov system
The Einstein-massless Vlasov system is a relevant model in the study of collisionless many particle systems in general relativity. In this talk, I will present a stability result for the exterior of Schwarzschild as a solution of this system assuming spherical symmetry. We exploit the hyperbolicity of the geodesic flow around the black hole to obtain decay of the energy momentum tensor, despite the presence of trapped null geodesics. The main result requires a precise understanding of radial derivatives of the energy momentum tensor, which we estimate using Jacobi fields on the tangent bundle in terms of the Sasaki metric.
- Sanchit Chaturvedi (Stanford University), The Vlasov-Poisson-Landau system in the weakly collisional regime
Consider the Vlasov-Poisson-Landau system in the weakly collisional regime. We show that solutions near global Maxwellians exhibit enhanced dissipation and Landau damping, resulting in a stronger stability threshold than previously known results. Our result is analogous to an earlier result of Bedrossian for the Vlasov-Poisson-Fokker-Planck system with the same threshold. However, unlike in the Fokker-Planck case, the linear operator cannot be inverted explicitly in the Landau case. For this reason, we develop a strategy based purely on energy methods, which combines Guo's energy method with the hypocoercive energy and the commuting vector field method. Joint work with Jonathan Luk and Toan Nguyen.
- Martin Taylor (Imperial College), The nonlinear stability of the Schwarzschild family of black holes
I will present a theorem on the full finite codimension nonlinear asymptotic stability of the Schwarzschild family of black holes. The proof employs a double null gauge, is expressed entirely in physical space, and utilises the analysis of Dafermos--Holzegel--Rodnianski on the linear stability of the Schwarzschild family. This is joint work with M. Dafermos, G. Holzegel and I. Rodnianski.
- Sergiu Klainerman (Princeton University), Nonlinear stability of Kerr black holes for small angular momentum
According to a well-known conjecture, initial data sets, for the Einstein vacuum equations, sufficiently close to a Kerr solution with parameters a, m, |a|/m <1, have maximal developments with complete future null infinity and with domain of outer communication (i.e complement of a future event horizon) which approaches (globally) a nearby Kerr solution. I will describe the main ideas in my recent joint work with Jeremie Szeftel concerning the resolution of the conjecture for small angular momentum, i.e. |a|/m sufficiently small. The work, ArXiv:2104.11857v1, also depends on forthcoming work on solutions of nonlinear wave equations in realistic perturbations of Kerr, with Szeftel and Elena Giorgi, which I will also describe.
- Jacques Smulevici (Sorbonne Université), Nonlinear periodic waves on the Einstein cylinder
Motivated by the study of small amplitudes non-linear waves in the Anti-de-Sitter spacetime and in particular the conjectured existence of periodic in time solutions, we construct families of arbitrary small time-periodic solutions to the conformal cubic wave equation and the spherically-symmetric Yang-Mills equations on the Einstein cylinder. For the conformal cubic wave equation, we consider both spherically-symmetric solutions and complexed-valued aspherical solutions with an ansatz relying on the Hopf fibration of the 3-sphere. Our proof relies on a theorem of Bambusi and Paleari, suitably modified in the Yang-Mills case, for which the main assumption is the existence of a non-degenerate zero for a non-linear operator associated with the resonant system. One easily check that any mode solutions of the associated linearized equations are zeroes of these operators and the main difficulty is the proof of the non-degeneracy conditions. As is classical with this type of approach, the smallness parameter does not belong to full interval, but to a Cantor-like set with zero as an accumulation point. This is a joint work with Athanasios Chatzikaleas.
- Grigorios Fournodavlos (Princeton University), Kasner-like singularities
We will investigate the dynamics of Big Bang singularities exhibiting monotonic blow-up. These are called Kasner-like or sometimes asymptotically velocity term dominated (AVTD), asymptotically Kasner-like, of Kasner type etc, and the first related studies were inspired by the work of Kasner himself (1921), who discovered the first cosmological solutions in vacuum $Ric(g)=0$, $g_{Kasner} = -dt^2 + t^{2p_1}(dx^1)^2 + t^{2p_2}(dx^2)^2 +t^{2p_3}(dx^3)^2$ for $(t,x)\in (0,+\infty)\times\mathbb{T}^3$ with $\sum p_i = \sum p_i^2 = 1$. There is a regime, called subcritical in the physics literature, where Kasner-like singularities are expected to be stable. Outside the subcritical regime, instabilities are expected. Nonetheless, there is a stable submanifold in the moduli space of initial data, where Kasner-like behavior is exhibited. We will review recent results that provide robust frameworks in which one can study the various regimes of Kasner-like singularities, both by means of singularity construction and stability.
- Dejan Gajic (Radboud University), Late-time tails and instabilities of extremal black holes
When Kerr black holes rotate at their maximally allowed angular velocity, they are said to be extremal. Extremal black holes display a variety of interesting phenomena that are not present in more slowly rotating black holes. I will introduce upcoming work on the existence of strong asymptotic instabilities of a non-axisymmetric nature for scalar waves propagating on extremal Kerr black hole backgrounds and I will discuss their connection with previous work on axisymmetric instabilities and the precise shape of late-time power law tails in the emitted radiation. If time permits, I will discuss the mechanism for deriving late-time tails and sharp asymptotics and compare it with previous work on late-time tails on subextremal black holes ("Price’s law").
- Olivier Graf (U of Munster), An L² curvature pinching result for the Euclidean 3-disk
When studying the Cauchy problem of general relativity we typically obtain L² bounds on the (Ricci) curvature tensor of spacelike hypersurfaces and its derivatives. In many situations it is useful to deduce from these H^{k} bounds that there exists coordinates on the spacelike hypersurface with optimal H^{k+2} bounds on the components of the induced Riemannian metric. The general idea is that this can be achieved using harmonic coordinates -- in which the principal terms of the Ricci curvature tensor are the Laplace-Beltrami operators of the metric components -- and standard elliptic regularity results. In this talk, I will make this idea concrete in the case of Riemannian 3-manifolds with a 2-sphere boundary, with Ricci curvature in L² and second fundamental form of the boundary in H^{1/2} both close to their respective Euclidean unit 3-disk values. The crux of the proof is a refined Bochner identity with boundary for harmonic functions. The cherry on the cake is that this result does not require any topology assumption on the Riemannian 3-manifold (apart from its boundary), and that we obtain -- as a conclusion -- that it must be diffeomorphic to the 3-disk. This talk is based on a result that I obtained in [Global nonlinear stability of Minkowski space for spacelike-characteristic initial data, Appendix A] .
- Eric Ling (Rutgers U), On the discrete Dirac spectrum of a point electron in the zero-gravity Kerr-Newman spacetime
In relativistic quantum mechanics, the discrete spectrum of the Dirac hamiltonian with a Coulomb potential famously agrees with Sommerfeld’s fine structure formula for the hydrogen atom. In the Coulomb approximation, the proton is assumed to only have a positive electric charge. However, the physical proton also appears to have a magnetic moment which yields a hyperfine structure of the hydrogen atom that’s normally computed perturbatively. Aiming towards a non-perturbative approach, Pekeris in 1987 proposed taking the Kerr-Newman spacetime with its ring singularity as a source for the proton’s electric charge and magnetic moment. Given the proton’s mass and electric charge, the resulting Kerr-Newman spacetime lies well within the naked singularity sector which possess closed timelike loops. In 2014 Tahvildar-Zadeh showed that the zero-gravity limit of the Kerr-Newman spacetime (zGKN) produces a flat but topologically nontrivial spacetime that’s no longer plagued by closed timelike loops. In 2015 Tahvildar-Zadeh and Kiessling studied the hydrogen problem with Dirac’s equation on the zGKN spacetime and found that the hamiltonian is essentially self-adjoint and contains a nonempty discrete spectrum. In this talk, we show how their ideas can be extended to classify the discrete spectrum completely and relate it back to the known hydrogenic Dirac spectrum but yielding hyperfine-like and Lamb shift-like effects.
- Georgios Mavrogiannis (University of Cambridge), Linear and quasilinear wave equations on Schwarzschild de Sitter
We discuss a new physical space approach to proving exponential decay for the wave equation on Schwarzschild de Sitter. In this approach, exponential decay follows from a novel "relatively non-degenerate" estimate on fixed time slabs. We then apply our physical space "relatively non-degenerate" estimates of the linear theory to prove stability for solutions of the quasilinear wave equation on Schwarzschild de Sitter.
- Allen Fang (Sorbonne University), A new proof for the nonlinear stability of slowly-rotating Kerr-de Sitter
The nonlinear stability of the slowly-rotating Kerr-de Sitter family was first proven by Hintz and Vasy in 2016 using microlocal techniques. In my talk, I will present a novel proof of the nonlinear stability of slowly-rotating Kerr-de Sitter spacetimes that avoids frequency-space techniques outside of a neighborhood of the trapped set. The proof uses vectorfield techniques to uncover a spectral gap corresponding to exponential decay at the level of the linearized equation. The exponential decay of solutions to the linearized problem is then used in a bootstrap proof to conclude nonlinear stability.
- André Lisibach (Bern University of Applied Sciences), Shock Reflection and Interaction in Plane Symmetry
The reflection of a shock on a wall and the interaction of two oncoming shocks in plane symmetry are classical problems in gas dynamics. In my talk, after presenting the physical picture and reviewing the governing equations, I will give a proof of local existence of a solution to the reflection problem. The proof is based on reformulating the problem in terms of characteristic coordinates and using twice differentiable functions.
The organizers,
Xinliang An, John Anderson, Dejan Gajic, Elena Giorgi, Shadi Tahvildar-Zadeh