Abstract: The study of static vacuum Riemannian metrics arises naturally in general relativity and differential geometry. A static vacuum metric produces a static spacetime by a warped product, and it is related to scalar curvature deformation and gluing. The well-known Uniqueness Theorem of Static Black Holes says that an asymptotically flat, static vacuum metric with black hole boundary must belong to the Schwarzschild family. In contrast to the rigidity phenomenon, R. Bartnik conjectured that there are asymptotically flat, static vacuum metric realizing certain arbitrarily specified boundary data. I will discuss recent progress toward this conjecture. It is based on joint work with Zhongshan An.
In the second part of the talk (if time permits), I will discuss related topics in the non-time-symmetric case. In many ways, a stationary vacuum initial data set is a generalization of a static vacuum metric. However, from the point of view of deforming the dominant energy condition, we discover that a “ground state” initial data set need not be stationary vacuum but can sit in a null dust spacetime with a global Killing vector field, such as the pp-waves. This part is based on joint work with Dan Lee.
Title: New results on compact Cauchy horizons of smooth vacuum spacetimes
Abstract: Cauchy horizons are rather unique and peculiar objects that have been studied for decades. In this talk I will first review old and new breakthroughs on the subject by Isenberg and Moncrief, and by Petersen and Rácz, and then discuss joint recent work together with I. Bustamante where it is proved that non-degenerate Compact Cauchy horizons on smooth vacuum spacetimes (shortly CHs) have indeed constant non-zero temperature. It then follows by the work of Petersen and Petersen-Rácz that CHs are Killing horizons, and by the work of Beig-Chruściel-Schoen that such objects are non-generic on the initial data, (in agreement with the Cosmic Censorship conjecture). A null-orbital and topological classification of CHs will also be commented.
Title: On naked singularities in Einstein equations
Abstract: I will describe recent and ongoing work with Y. Shlapentokh-Rothman on a construction of solutions to the Einstein vacuum equations corresponding to a naked singularity forming from a regular past. The talk will focus on a new geometric phenomenon, corresponding to twisting of null geodesics, new type of self-similarity, dynamical approach to the problem, and on the comparisons with the naked singularity solutions constructed by Christodoulou for the spherically symmetric Einstein-scalar field model.
Title: Well-posed formulation of Lovelock and Horndeski theories
Abstract: Lovelock theories are the most general diffeomorphism invariant theories of gravity in higher dimensions with second order equations of motion. Horndeski theories are the most general diffeomorphism invariant theories of gravity coupled to a scalar field in four dimensions with second order equations of motion. In this talk I will discuss well-posedness of the initial value problem for these theories. Previous work has shown that (generalised) harmonic gauge does not give a well-posed initial value problem. I will describe recent work with Aron Kovacs in which we introdued a modification of harmonic gauge that does give a well-posed initial value problem provided that the theory remains “weakly coupled”. Our modified harmonic gauge may also have applications in conventional GR.
Abstract: Gravitational waves are transporting information from faraway regions of the Universe. A new era began with the first detection of gravitational waves by Advanced LIGO in September 2015, and since then several events have been recorded by the LIGO/VIRGO collaboration. New challenges await us to unravel the interesting interplay between physics, astrophysics and mathematics. Most studies so far have been devoted to sources like binary black hole mergers or neutron star mergers, or generally to sources that are stationary outside of a compact set. We describe these systems by asymptotically-flat manifolds solving the Einstein equations. These sources have in common that far away their gravitational field decays fast enough towards Minkowski spacetime. In particular, far away from the source, the decay behavior can be described by a term that is homogeneous of degree -1 and lower order terms. I will present new results on gravitational radiation for sources that are not stationary outside of a compact set, but whose gravitational fields decay more slowly towards infinity. A panorama of new gravitational effects opens up when delving deeper into these more general spacetimes. In particular, whereas the former sources produce memory effects that are of purely electric parity (permanent displacement only), the latter in addition generate memory of magnetic type, thus allowing for rotation in the system. These new effects emerge naturally from the Einstein equations.
Abstract: We present several rigidity results for initial data sets motivated by the positive mass theorem. An important step in our proofs is to establish conditions that ensure that a marginally outer trapped surface is “weakly outermost”. A rigidity result for Riemannian manifolds with a lower bound on their scalar curvature is included as a special case. Relevant background on marginally outer trapped surfaces will be discussed. This talk is based on joint work with Michael Eichmair and Abraão Mendes.
Title: On highly anisotropic big bang singularities.
Abstract: In cosmology, the universe is typically modelled by spatially homogeneous and isotropic solutions to Einstein’s equations. However, for large classes of matter models, such solutions are unstable in the direction of the singularity. For this reason, it is of interest to study the anisotropic setting.
The purpose of the talk is to describe a framework for studying highly anisotropic singularities. In particular, for analysing the asymptotics of solutions to linear systems of wave equations on the corresponding backgrounds and deducing information concerning the geometry.
The talk will begin with an overview of existing results. This will serve as a background and motivation for the problem considered, but also as a justification for the assumptions defining the framework we develop.
Following this overview, the talk will conclude with a rough description of the results.
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