# Previous General Relativity Seminars

#### Fall 2019:

Date Speaker Title/Abstract
9/13/2019

CMSA G02

Martin Lesourd (BHI) Title: Shi-Tam’s Existence of Minimal Surfaces from Quasi-Local Mass

Abstract: Thorne’s hoop conjecture is an intuitive hypothesis intended to capture necessary and sufficient conditions for the existence of a black hole region. The first result in this direction was Schoen-Yau 83 and later Yau 01, which give sufficient conditions for the existence of an apparent horizon within a 3-dimensional initial data set. Shi-Tam 07 have done something analogous in the Riemannian context and obtained the existence of minimal surfaces from conditions on quasi-local mass. We shall review the main ideas of their proof.

9/20/2019 Martin Lesourd (BHI) Title: Existence of minimal or marginally outer trapped surfaces from quasi-local mass

Abstract: I will describe work in progress with Aghil Alaee and Shing-Tung Yau in which sufficient conditions for the existence of minimal or marginally outer trapped surfaces inside a domain are expressed in terms of the quasi-local mass of the domain.

10/4/2019 Cancelled
10/11/2019 Graham Cox (Memorial University) Title: Blowup solutions of Jang’s equation near a spacetime singularity

Abstract: Jang’s equation is a semilinear elliptic equation define on an initial data set. It was shown by Schoen and Yau that the (non)existence of global solutions is closely related to the existence of marginally outer trapped surfaces (MOTS), which are quasi-local analogues of black hole boundaries. As a result, Jang’s equation can be used to prove the existence of MOTS by imposing appropriate geometric conditions on the initial data set. These proofs proceed by contradiction: one assumes there is a global solution, then proves that its existence is not compatible with the given geometric assumptions.

In this talk I will outline a constructive approach to proving the existence of MOTS. In particular, I will consider a distinguished family of spacelike hypersurfaces in the maximally extended Schwarzschild spacetime, and prove that Jang’s equation admits no global solutions once the hypersurfaces become sufficiently close to the r=0 singularity.  This suggests a general strategy for relating spacetime singularities to the presence of MOTS. This is joint with Amir Aazami.

10/18/2019 Jordan Keller (BHI) Title: Angular Momentum and Center-of-Mass at Null Infinity

Abstract: 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. Working in the center-of-mass frame, i.e. assuming vanishing of linear momentum at null infinity, 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.

10/25/2019 Cancelled
11/1/2019 Peter Hintz (MIT) Title: Linear stability of slowly rotating Kerr black holes

Abstract: I will describe joint work with Dietrich Häfner and András Vasy in which we study the asymptotic behavior of linearized gravitational perturbations of Schwarzschild and slowly rotating Kerr black hole spacetimes. We show that solutions of the linearized Einstein equation decay at an inverse polynomial rate to a stationary solution (given by an infinitesimal variation of the mass and angular momentum of the black hole), plus a pure gauge term. Our proof uses a detailed description of the low energy resolvent of an associated wave equation on symmetric 2-tensors.

11/8/2019 Pei-Ken Hung (MIT)
Abstract: In this talk, I will discuss a wave equation for one forms in the Schwarzschild spacetime which is the linearization of a modified wave map gauge. The equation behaves like a damped wave equation and we obtain robust estimates. In particular, it allows us to show the stability of the modified wave map equation. This is on-going joint work with S. Brendle.
11/13/2019

Wednesday

1:00pm

CMSA G10

Eric Woolgar (University of Alberta)
Abstract: Curvature-dimension inequalities are modifications of a Ricci curvature bound or, in the language of relativity, an energy condition. They have proved useful in applications of Fourier analysis to diffusion processes. As tools to prove theorems in Riemannian geometry and general relativity, they are often as powerful as the usual Ricci curvature bounds and can yield new results. Applications include static Einstein metrics, near-extremal-horizon geometry, and scalar-tensor gravity. I will discuss an application of a Riemannian curvature-dimension bound to horizon topology, and use Lorentzian curvature-dimension bounds to prove some singularity theorems and splitting theorems. Parts of the talk are based on joint work with Marcus Khuri, Will Wylie, and Greg Galloway.
11/22/2019 Sahar Hadar (BHI) Title: Universal signatures of a black hole’s photon ring

Abstract: The Event Horizon Telescope image of the supermassive black hole in the galaxy M87 is dominated by a bright, unresolved ring. General relativity predicts that embedded within this image lies a thin “photon ring,” which is composed of an infinite sequence of self-similar subrings that are indexed by the number of photon orbits around the black hole. The subrings approach the edge of the black hole “shadow,” becoming exponentially narrower but weaker with increasing orbit number, with seemingly negligible contributions from high order subrings. In the talk, I will discuss the structure of the photon ring, starting with non-rotating black holes, and then proceeding to the complex patterns that emerge when rotation is taken into account. Subsequently I will argue that the subrings produce strong and universal signatures on long interferometric baselines. These signatures offer the possibility of precise measurements of black hole mass and spin, as well as tests of general relativity, using only a sparse interferometric array.

11/29/2019 Thanksgiving Holiday
12/6/2019
12/13/2019

#### 2018-2019:

 Date Speaker Title/abstract 9/7/2018 Christos Mantoulidis (MIT) Title: Capacity and quasi-local mass Abstract. This talk is based on work with P. Miao and L.-F. Tam. We derive new inequalities between the boundary capacity of an asymptotically flat 3-manifold with nonnegative scalar curvature and boundary quantities that relate to quasi-local mass; one relates to Brown-York mass and the other is new. Among other things, our work yields new variational characterizations of Riemannian Schwarzschild manifolds and new comparison results for surfaces in them. 9/12/2018 Aghil Alaee (CMSA) Title:  Mass-angular momentum inequality for black holes Abstract:  In this talk, I will review the results of mass-angular momentum inequality for four-dimensional axisymmetric black holes. Then I will establish versions of this inequality for five-dimensional black holes and in particular black ring, which is the most intriguing asymptotically flat solution of vacuum Einstein equations. Moreover, I will show these inequalities are sharp if and only if the initial data sets are isometric to the canonical slices of known extreme stationary solutions. These results are joint work with Marcus Khuri and Hari Kunduri. 9/19/2018 Pei-Ken Hung (MIT) Title: The linear stability of the Schwarzschild spacetime in the harmonic gauge: odd part Abstract: We study the odd solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of Regge-Wheeler quantities, we are able to estimate the odd part of Lichnerowicz d’Alembertian equation. In particular, we prove the solution decays at rate $\tau^{-1+\delta}$ to a linearlized Kerr solution. 9/26/2018 Jordan Keller (BHI) Title: Quasi-local Angular Momentum and Center-of-Mass at Future Null Infinity Abstract: We calculate the limits of the quasi-local angular momentum and center-of-mass defined by Chen-Wang-Yau [3] 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 [2], 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, which amounts to a mild assumption on the mass aspect function corresponding to vanishing of the quasi-local linear momentum calculated in [2].  With this condition and the assumption that the mass aspect function is non-trivial, we obtain explicit expressions for the quasi-local 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. 10/3/2018 Christos Mantoulidis (MIT) Title: The Bartnik mass of apparent horizons Abstract: We will discuss a spectral characterization of apparent horizons in three-dimensional time-symmetric initial data sets. Then, for a dense class of nondegenerate apparent horizons, we will construct sharp asymptotically flat extensions to conclude that their Bartnik mass equals their Hawking mass. This is joint work with R. Schoen. 10/10/2018 Salem Al Mosleh (CMSA) Title: Thin elastic shells and isometric embedding of surfaces in three-dimensional Euclidean space Abstract: We will first discuss the reduction of theories describing elastic bodies in three-dimensions to effective descriptions defined on embedded surfaces. Then, we describe the isometric deformations of surfaces and the key role of played by asymptotic curves, curves with zero normal curvature, in determining the local mechanical behavior of thin shells. This was joint work with C. Santangelo. 10/17/2018 Sébastien Picard (Harvard) Title: The Anomaly flow over Riemann surfaces Abstract: The Anomaly flow is a geometric flow on Calabi-Yau threefolds which is motivated by string theory. We will study the flow on certain fibrations where it reduces to a scalar evolution equation on a Riemann surface. This is joint work with T. Fei and Z. Huang. 10/31/2018 Alex Lupsasca (Harvard) Title: Polarization Whorls from M87 at the Event Horizon Telescope Abstract: The Event Horizon Telescope (EHT) is expected to soon produce polarimetric images of the supermassive black hole at the center of the neighboring galaxy M87. This black hole is believed to be very rapidly spinning, within 2% of extremality. General relativity predicts that such a high-spin black hole has an emergent conformal symmetry near its event horizon. In this talk, I will briefly review this symmetry and use it to derive an analytic prediction for the polarized near-horizon emissions to be seen at the EHT. The resulting pattern is very distinctive and consists of whorls aligned with the spin. 11/7/2018 Jordan Keller (BHI) Title: Linear Stability of Higher Dimensional Schwarzschild Black Holes Abstract: The Schwarzschild-Tangherlini black holes are higher-dimensional generalizations of the Schwarzschild spacetimes, comprising a static, spherically symmetric family of black hole solutions to higher-dimensional vacuum gravity. The physical relevance of such solutions is intimately related to their stability under gravitational perturbations. This talk will address results on the linear stability of the Schwarzschild-Tangherlini black holes, part of ongoing joint work with Pei-Ken Hung and Mu-Tao Wang. 11/14/2018 Niky Kamran (McGill) Title: Lorentzian Einstein metrics with prescribed conformal infinity Abstract: We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space and compatible boundary data prescribed at the time-like conformal boundary of space-time. This extends the fundamental result of Friedrich on the existence of anti-de Sitter space-times in 3+1 dimensions to arbitrary space-time dimensions, by a different approach that allows for generic smoothness and polyhomogeneity assumptions on the initial data. This is joint work with Alberto Enciso (ICMAT, Madrid). 12/05/2018 *room G02* Pengzi Miao (University of Miami) Title: Localization of the Penrose inequality and variation of quasi-local mass Abstract: In the study of manifolds with nonnegative scalar curvature, a fundamental result is the Riemannian Positive mass theorem. If the manifold has horizon boundary, one has the Riemannian Penrose inequality. Given a compact region with boundary in these manifolds, one wants to understand how much mass or energy is localized in such a region. This question is usually referred to as the quasi-local mass problem. In this talk, we discuss an inequality on a compact manifold with nonnegative scalar curvature, which can be thought as a body surrounding horizons. Our discussion of the rigidity case of this inequality reveals an intriguing relation between two of the most important notions of quasi-local mass, the Bartnik mass and the Wang-Yau mass. The talk is based on joint work with Siyuan Lu. 1/31/2019 SC 232 3-4pm Shahar Hadar (Harvard University) Title: Late-time behavior of near-extremal black holes from symmetry Abstract: Linear perturbations of extremal black holes exhibit the Aretakis instability, in which higher derivatives of the fields grow polynomially with time along the event horizon. Near-extremal black holes display similar behavior for some time, and eventually decay exponentially through quasinormal modes. In the talk I will show that the above behaviors are dictated by the conformal symmetry of the near-horizon region of such black holes. I will then discuss the significance of backreaction in the problem, and show how it can be simply accounted for within the near-horizon picture. 2/7/2019 SC 411 3-4pm Pei-Ken Hung (MIT) Title: The linear stability of the Schwarzschild spacetime in the harmonic gauge: even part Abstract: We study the even solution of the linearlized Einstein equation on the Schwarzschild background and in the harmonic gauge. With the aid of the Zerilli equation, we estimate the even part of Lichnerowicz d’Alembertian equation. In particular, we show that up to a one dimensional stationary mode, the solution decays to a linearlized Kerr solution. This is ongoing joint work with S. Brendle. 2/14/2019 SC 411 3-4pm Charles Marteau (Ecole Polytechnique) Title: Null hypersurfaces and ultra-relativistic physics in gravity Abstract: I will explain how the induced geometry on a null hypersurface gives rise to a particular type of structure called Carrollian geometry. The latter emerges when taking the ultra-relativistic limit of the usual pseudo-Riemannian metric. This property has strong consequences on the gravitational dynamics satisfied by the extrinsic geometry of the null hypersurface and on its symmetry group. We will see how the first one can be interpreted as ultra-relativistic conservation laws while the second corresponds to the isometries of the induced Carrollian geometry. These are very general statements for any null hypersurface but I will focus all along on a physically interesting case: the null infinity of an asymptotically flat spacetime. 2/21/2019 Hsin-Yu Chen (Black Hole Initiative) Title: Measuring the Hubble Constant with Gravitational Waves Abstract: The first detection of binary neutron star merger by Advanced LIGO-Virgo and the discovery of the optical counterpart allowed for the first independent measurement of Hubble constant with gravitational waves. In this talk, I will summarize latest cosmological measurements with gravitational waves, and discuss the future aspects of them. I will then talk about the potential challenges and how we improve the measurements. 3/7/2019 Laura Donnay (Harvard) Title: Carrollian physics at the black-hole horizon Abstract: In this talk, I will show that the near-horizon geometry of a black hole can be described as a Carrollian geometry emerging from an ultra-relativistic limit.  The laws governing the dynamic of a black hole horizon, the null Raychaudhuri and Damour equations, are shown to be Carrollian conservation laws obtained by taking the ultra-relativistic limit of the conservation of an energy-momentum tensor. Vector fields preserving the Carrollian geometry of the horizon, dubbed Carrollian Killings, include BMS-like supertranslations and superrotations, and have non-trivial associated conserved charges. If time allows, I will discuss their relation with the infinite-dimensional horizon charges of the covariant phase space formalism. 3/14/2019 3:30pm Peter Hintz (MIT) Title: Stability of Minkowski space and polyhomogeneity of the metric Abstract: I will explain a new proof of the non-linear stability of the Minkowski spacetime as a solution of the Einstein vacuum equation. The proof relies on an iteration scheme at each step of which one solves a linear wave-type equation globally. The analysis takes place on a compactification of R^4 to a manifold with corners whose boundary hypersurfaces correspond to spacelike, null, and timelike infinity. I will describe how the asymptotic behavior of the metric can be deduced from the structure of simple model operators at these boundaries. Joint work with András Vasy. 3/28/2019 TBA TBA 4/4/2019 CMSA G02 Marcus Khuri (Stony Brook) Title: Stationary Vacuum Black Holes in Higher Dimensions Abstract: A result of Galloway and Schoen asserts that horizon cross-sections must be of positive Yamabe invariant. In this talk we discuss results on a converse problem. That is, which manifolds of positive Yamabe invariant arise as horizon cross-sections in a stationary vacuum spacetime. 4/11/2019 Amir Babak Aazami (Clarks) Title: Kähler metrics via Lorentzian geometry in dimension 4 Abstract:  Given a Lorentzian -manifold with two distinguished vector fields satisfying properties determined by their shear, twist and various Lie bracket relations, a family of Kähler metrics is constructed on . Under certain conditions and share various properties, such as a Killing vector field or a vector field with geodesic flow. The Ricci and scalar curvatures of are computed in some cases in terms of data associated to ; in certain cases the Kähler manifold will be complete and Einstein.  Many classical spacetimes fit into this construction: warped products, for instance de Sitter spacetime, as well as gravitational plane waves and metrics of Petrov type , such as Kerr and NUT metrics. This work is joint with Gideon Maschler. 4/19/2019 CMSA G02 Friday @ 9:30am Lydia Bieri (University of Michigan) Title: Logarithmic or Not Logarithmic Abstract: In General Relativity, we describe isolated gravitating systems by asymptotically flat solutions of the Einstein equations. For various classes of initial data corresponding classes of solutions have been constructed in the nonlinear stability proofs when slightly moving away from Minkowski spacetime. Many of the null asymptotic results still hold when one replaces the small initial data by large initial data. Therefore, these solutions have become an interesting and important source to understand gravitational waves and memory as observed at null infinity. Lately, discussions have flared up whether logarithmic terms are present at highest order in crucial components of the Riemannian curvature and shear of the spacetime and whether such terms would give a tail effect for ordinary memory. There is a large literature (older and newer) on terms of this sort. In this talk, I will address some of my recent work that proves that for asymptotically flat solutions of the Einstein equations the crucial curvature and shear components do not have logarithmic terms at highest order, but logarithmic terms naturally show up at lower order. From thisit follows that there is no divergent memory caused by logarithmic terms. However, in my earlier work, considering spacetimes with very slow decay, the ordinary memory diverges (though not logarithmically but faster) and null memory is always finite. Last but not least, these logarithmic terms do show up at leading order of certain other curvature and geometric components for specific decay of the initial data. These mathematical results are in accordance with a physical argument that I will present as well. 4/25/2019 10:30pm CMSA G02 Pengyu Le (University of Michigan) Title: Perturbations of Null Hypersurfaces and Null Penrose Inequality Abstract: The Penrose inequality in general relativity is a conjectured inequality between the area of the horizon and the mass of a black-hole spacetime. The null Penrose inequality is the case where it concerns the area of the horizon and the Bondi mass at null infinity on a null hypersurface. An effective method to prove Penrose-type inequalities is to exploit the monotonicity of the Hawking mass along certain foliations. The constant mass aspect function foliation is such a desired foliation, but the behavior of the foliation at past null infinity is an obstacle for the proof. An idea to overcome this difficulty is to vary the null hypersurface to achieve the desired behavior of the foliation at null infinity, leading to a spacetime version of the Penrose inequality. To formalise this idea, one need to study perturbations of null hypersurfaces. I will talk about my work on the study of perturbations of null hypersurfaces and its application to the null Penrose inequality. 4/26/2019 2:30pm CMSA G02 Armando Cabrera Pacheco (Universität Tübingen) Title: Asymptotically flat extensions with charge Abstract: Inspired by the Mantoulidis and Schoen construction, we obtain time-symmetric black hole initial data sets for the Einstein–Maxwell equations satisfying the dominant energy condition, such that their horizon boundary geometry is prescribed, and their total masses and total charges are controlled. We also formulate a notion of boundary Bartnik mass in this context and compute its value for minimal Bartnik data. This talk is based on a joint work with A. Alaee and C. Cederbaum.