The nonlinear stability of the Schwarzschild family of black holes

I will present a theorem on the full finite codimension 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. Martin Taylor

Spacetime Extensions of the Big Bang

In this talk I will show the big bang is a coordinate singularity for a large class of k = -1 inflationary FLRW spacetimes dubbed 'Milne-like.' This is analogous to how the r = 2m event horizon is a coordinate singularity in the Schwarzschild spacetime. The geometry of the coordinate singularity for Milne-like spacetimes is that of a lightcone in a spacetime conformal to Minkowski space. I will discuss how the mathematics of these Milne-like spacetimes may provide connections to certain problems in cosmology. Eric Ling

Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

We prove the nonlinear stability of the Schwarzschild spacetime under axially symmetric polarized perturbations, i.e. solutions of the Einstein vacuum equations for asymptotically flat 1+3 dimensional Lorentzian metrics which admit a hypersurface orthogonal spacelike Killing vectorfield with closed orbits. While building on the remarkable advances made in last 15 years on establishing quantitative linear stability, we introduce a series of new ideas among which we emphasize the general covariant modulation (GCM) procedure which allows us to construct, dynamically, the center of mass frame of the final state. The mass of the final state itself is tracked using the well known Hawking mass relative to a well adapted foliation itself connected to the center of mass frame. Our work here is the first to prove the nonlinear stability of Schwarzschild in a restricted class of nontrivial perturbations. To a large extent, the restriction to this class of perturbations is only needed to ensure that the final state of evolution is another Schwarzschild space. We are thus confident that our procedure may apply in a more general setting. This is joint work with Jeremie Szeftel. Sergiu Klainerman

The Einstein Equations and Gravitational Waves.

In mathematical general relativity (GR) the Einstein equations describe the laws of the universe. This system of hyperbolic nonlinear partial differential equations (PDE) has served as a playground for all kinds of new problems and methods in PDE analysis and geometry. Among the most interesting solutions of these equations we find spacetimes allowing for gravitational waves. The latter are fluctuations of the curvature of the spacetime. In 2015, gravitational waves were observed for the first time by Advanced LIGO (and several times since then). These waves are produced during the mergers of black holes or neutron stars in core-collapsed supernovae. Understanding gravitational radiation is tightly interwoven with the study of the Cauchy problem of GR. Many more interesting mathematical challenges are expected to emerge from recent and future observations. I will talk about the Cauchy problem for the Einstein equations, explain geometric-analytic results on gravitational radiation and the memory effect of gravitational waves, the latter being a permanent change of the spacetime. We will connet the mathematical findings to experiments. There is a significant difference between isolated systems and cosmological questions in GR that impacts gravitational radiation. I will also address my recent work on various asymptotically-flat spacetimes describing isolated systems, properties of gravitational wave memory, and joint work with David Garfinkle and Nicolas Yunes on gravitational radiation. Lydia Bieri

A quasi-local Penrose inequality for the quasi-local energy with static references.

The positive mass theorem is one of the fundamental results in general relativity. It states that the total mass of an asymptotically flat spacetime is non-negative. The Penrose inequality provides a lower bound on mass by the area of the black hole and is closely related to the cosmic censorship conjecture in general relativity. Recently, Lu and Miao proved a quasi-local Penrose inequality for the quasi-local energy with reference in the Schwarzschild manifold. In this article, we prove a quasi-local Penrose inequality for the quasi-local energy with reference in any spherically symmetric static spacetime. Po-Ning Chen

Null infinity of spacetime: the Bondi-Sachs approach

Null infinity is where the detection of gravitational waves takes place. It also determines the global structure of spacetime, in particular the location of the black hole region. I shall discuss topics such as the Bondi-Trautman mass, the mass loss formula, and the BMS (Bondi-Metzner-Sachs) group. Mu-Tao Wang

On the linear stability of higher dimensional Schwarzschild black hole

The linear stability of the four dimensional Schwarzschild black hole has been proved using a new approach that generalizes Chandrasekhar's and relies on spherical symmetry. I will discuss how this approach is applied to study the linear stability of higher dimensional Schwarzschild black hole. This is based on joint work with Pei-Ken Hung and Jordan Keller. Mu-Tao Wang

Evaluating Quasi-local Angular Momentum and Center-of-Mass at Null Infinity

We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau for a family of spacelike two-spheres approaching future null infinity in an asymptotically flat spacetime admitting a Bondi-Sachs expansion. Our result complements earlier work of Chen-Wang-Yau, where the authors calculate the quasi-local energy and linear momentum at null infinity. Finiteness of the quasi-local center-of-mass requires that the spacetime be in the so-called center-of-mass frame, a mild assumption on the mass aspect function amounting to vanishing of linear momentum at null infinity. With this condition and the assumption that the mass aspect function is non-trivial, we obtain explicit expressions for the angular momentum and center-of-mass at future null infinity in terms of the observables appearing in the Bondi-Sachs expansion of the spacetime metric. This is joint work with Ye-Kai Wang and Shing-Tung Yau. Jordan Keller

Linear stability of Reissner-Nordstrom spacetime,

We address the linear stability of the Reissner-Nordstrom family of charged black holes, subject to coupled gravitational and electromagnetic perturbations. Boundedness and decay of the gauge-invariant quantities verifying the Teukolsky equation is the first step towards controlling all the terms in the perturbation. Then, we should identify the residual gauge freedom and the Kerr-Newman parameters to obtain boundedness and decay of all the remaining gauge-dependent quantities. Elena Giorgi

Boundedness and decay for the Teukolsky system in Reissner-Nordstrom,

We prove boundedness and polynomial decay statements for solutions of the spin 2 generalized Teukolsky system on a Reissner-Nordstrom background with small charge. The first equation of the system is the generalization of the standard Teukolsky equation in Schwarzschild for the extreme component of the curvature $\alpha$. The second equation, coupled with the first one, is a new equation for a gauge-invariant quantity involving the electromagnetic curvature components. The proof is based on the use of a generalization of the higher order quantities in previous works on linear and non-linear stability of Schwarzschild, as well as in the Teukolsky equation in Kerr. Elena Giorgi