Previous General Relativity Seminars

Spring 2020

Date Speaker Title/Abstract
2/7/2020 Lan-Hsuan Huang (University of Connecticut) Title: Improvability of the dominant energy scalar and Bartnik’s stationary conjecture  

 

Abstract: In this talk, we will introduce the concept of improvabilty of the dominant energy scalar and discuss strong consequences of non-improvability. We employ new, large families of deformations of the modified Einstein constraint operator and show that, generically, their adjoint linearizations are either injective, or else one can prove that kernel elements satisfy a “null-vector equation”. Combined with a conformal argument, we make significant progress toward Bartnik’s stationary conjecture. More specifically, we prove that a Bartnik minimizing initial data set can be developed into a spacetime that both satisfies the dominant energy condition and carries a global Killing field. We also show that this spacetime is vacuum near spatial infinity. This talk is based on the joint work with Dan Lee.

2/14/2020 Yuewen Chen (CMSA) Title: Solutions of Jang’s Equation Inside Black Holes

 

Abstract: Jang’s equation is a degenerate elliptic differential equation which plays an important role in the positive mass theorem. In this talk, we describe a high order WENO (Weighted Essentially Non-Oscillatory) scheme for the Jang’s equation. Some special solutions will be shown, such as those possessing spherical symmetry and axial symmetry.

2/21/2020

 

 

CMSA G10

Alex Lupsasca (Harvard) Title: The Kerr 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 itself composed of an infinite sequence of self-similar subrings. Each subring is a lensed image of the main emission, indexed by the number of photon orbits executed around the black hole. I will review recent theoretical advances in our understanding of lensing by Kerr black holes, based on arXiv:1907.04329, 1910.12873, and 1910.12881. In particular, I will describe the critical parameters γ, δ, and τ that respectively control the demagnification, rotation, and time delay of successive lensed images of a source. These observable parameters encode universal effects of general relativity, which are independent of the details of the emitting matter and also produce strong, 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 such as a future extension of the EHT to space.

2/28/2020 Po-Ning Chen (University of California, Riverside) Title: A quasilocal charged Penrose inequality

 

Abstract: In this talk, we will discuss a quasi-local Penrose inequality with charges for time-symmetric initial data of the Einstein-Maxwell equation. Namely, we derive a lower bound for Brown-York type quasi-local mass in terms of the horizon area and the electric charge. The inequality we obtained is sharp in the sense that equality holds for surfaces in the Reissner-Nordström manifold. This talk is based on joint work with Stephen McCormick.

3/6/2020

 

CMSA G02

Nikolaos Athanasiou (University of Oxford) Title: A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

 

Abstract: Few notions within the realm of mathematical physics succeed in capturing the imagination and inspiring awe as well as that of a black hole. First encountered in the Schwarzschild solution, discovered a few months after the presentation of the Field Equations of General Relativity at the Prussian Academy of Sciences, the black hole as a mathematical phenomenon accompanies and prominently features within the history of General Relativity since its inception. In this talk we will lay out a brief history of the question of dynamical black hole formation in General Relativity and discuss a recent result, in collaboration with Xinliang An, on a scale-critical trapped surface formation criterion for the Einstein-Maxwell system.

3/13/2020 Hsin-Yu Chen (BHI) This meeting will be taking place virtually on Zoom.
3/20/2020 Cancelled  
3/27/2020 Sven Hirsch (Duke University) This meeting will be taking place virtually on Zoom.

 

Title: The spacetime positive mass theorem and path connectedness of initial data sets.

 

Abstract: The purpose of this talk is twofold: First we present a new proof of the spacetime positive mass theorem (joint with Demetre Kazaras and Marcus Khuri); second we discuss some new results about the topology of initial data sets (joint with Martin Lesourd).The spacetime positive mass theorem that the mass of an initial data set is non-negative with equality if and only if the initial data set arises as subset of Minkowski space. This result has first been proven by Schoen and Yau using Jang’s equation. There are further proofs by Witten using spinors and by Eichmair, Huang, Lee and Schoen using MOTS. Our proof uses Stern’s integral formula technique and also leads to a new explicit lower bound of the mass which is even valid when the dominant energy condition is not satisfied. A central conjecture in mathematical relativity is the final state conjecture which states that initial data sets will eventually approach Kerr black holes. In particular, this would imply that the space of initial data sets is path connected. Building upon the work of Marques and using deep and beautiful results of Carlotto and Li, we show that indeed the space of initial data set with compact trapped interior boundary is path connected.

4/3/2020 Hyun Chul Jang (University of Connecticut) This meeting will be taking place virtually on Zoom.

 

Title: Mass rigidity of asymptotically hyperbolic spaces and some splitting theorems

 

Abstract: In this talk, we will discuss the rigidity of positive mass theorem for asymptotically hyperbolic manifolds. That is, if the mass equality holds, then the manifold is isometric to hyperbolic space. The proof used a variational approach with the scalar curvature constraint. It also involves an investigation on a type of Obata’s equations, which leads to recent splitting results with Galloway. This talk is based on the joint works with L.-H. Huang and D. Martin, and with G. J. Galloway.

4/10/2020 Chao Li (Princeton) This meeting will be taking place virtually on Zoom.  

 

Title: Positive scalar curvature and the dihedral rigidity conjecture

 

Abstract: In 2013, Gromov proposed a dihedral rigidity conjecture, aiming at establishing a geometric comparison theory for metrics with positive scalar curvature. The conjecture states that if a Riemannian polyhedron has nonnegative scalar curvature in the interior, and weakly mean convex faces, then the dihedral angle between adjacent faces cannot be everywhere less than the corresponding Euclidean model. I will prove this conjecture for a large collection of polytopes. The strategy is to relate this conjecture with a geometric variational problem of capillary type, and apply the Schoen-Yau minimal slicing technique for manifolds with boundary. Our result is a localization of the positive mass theorem.

4/17/2020 Brian Allen (University of Hartford)

 

 

Video

This meeting will be taking place virtually on Zoom.

 

Title: Null distance and convergence of warped product spacetimes.

 

Abstract: The null distance was introduced by Christina Sormani and Carlos Vega as a way of turning a spacetime into a metric space. This is particularly important for geometric stability questions relating to spacetimes such as the stability of the positive mass theorem. In this talk, we will describe the null distance, present new properties of the metric space structure, and examine the convergence of sequences of warped product spacetimes equipped with the null distance. This is joint work with Annegret Burtscher.

4/24/2020 Daniel Stern (University of Toronto)

 

 

Video

This meeting will be taking place virtually on Zoom.

 

Title: Scalar curvature and harmonic functions

 

Abstract: We’ll discuss a new technique for relating scalar curvature bounds to the global structure of 3-dimensional manifolds, exploiting a relationship between the scalar curvature and the topology of level sets of harmonic functions. We will describe several geometric applications in both the compact and asymptotically flat settings, including a simple and effective new proof (joint with Bray, Kazaras, and Khuri) of the three-dimensional Riemannian positive mass theorem.

5/1/2020 Demetre Kazaras (Stony Brook University)

 

 

Video

This meeting will be taking place virtually on Zoom.

 

Title: Desingularizing 4-manifolds with positive scalar curvature

 

Abstract: We study 4-manifolds of positive scalar curvature (psc) with severe metric singularities along points and embedded circles, establishing a desingularization process. To carry this out, we show that the bordism group of closed 3-manifolds with psc metrics is trivial, using scalar-flat K{\”a}hler ALE surfaces recently discovered by Lock-Viaclovsky. This allows us to prove a non-existence result for singular psc metrics on enlargeable 4-manifolds, partially confirming a conjecture of Schoen.

5/8/2020

Anna Sakovich (Uppsala University)


Video

This meeting will be taking place virtually on Zoom.  

 

Title: The Jang equation and the positive mass theorem in the asymptotically hyperbolic setting

 

Abstract:  We will be concerned with asymptotically hyperbolic ‘hyperboloidal’ initial data for the Einstein equations. Such initial data is modeled on the upper unit hyperboloid in Minkowski spacetime and consists of a Riemannian manifold (M, g) whose geometry at infinity approaches that of hyperbolic space, and a symmetric 2-tensor K representing the second fundamental form of the embedding into spacetime, such that K -> g at infinity. There is a notion of mass in this setting and a positive mass conjecture can be proven by spinor techniques. Other important results concern the case K = g, where the conjecture states that an asymptotically hyperbolic manifold whose scalar curvature is greater than or equal to that of hyperbolic space must have positive mass unless it is a hyperbolic space. In this talk, we will discuss how the method of Jang equation reduction, originally devised by Schoen and Yau to prove the positive mass conjecture for asymptotically Euclidean initial data sets, can be adapted to the asymptotically hyperbolic setting yielding a non-spinor proof of the respective positive mass conjecture. We will primarily focus on the case dim M = 3.

5/14/2020

 

Thursday

9:30am

Xinliang An (National University of Singapore) This meeting will be taking place virtually on Zoom.

 

Title: On curvature blow-up rates in gravitational collapse

Abstract: In this talk, I will present two new results on the gravitational collapse of the spherically symmetric Einstein-scalar field system. i) With Ruixiang Zhang we show that even in the most singular scenario, along the singular boundary $r=0$, the Kretschmann scalar would obey polynomial blow-up upper bounds $O(1/r^N)$. This improves previously best-known double-exponential upper bounds $O\big(\exp\exp(1/r)\big)$. Our result is sharp in the sense that there are known examples showing that no sub-polynomial upper bound could hold. ii) With Dejan Gajic, we extend the aforementioned result to a global one and calculate the precise polynomial rate-$N$. We find that, when it is close to the timelike infinity, the blow-up rates of Kretschmann scalar could be different from the Schwarzschild value. In particular, the blow-up rates are not limited to discrete finite choices and they are related to the Price’s law along the event horizon. This indicates a new blow-up phenomenon, driven by a PDE mechanism, rather than an ODE mechanism.

5/15/2020 Dejan Gajic (University of Cambridge) This meeting will be taking place virtually on Zoom.  

 

Title: Polynomial tails and conservation laws of waves on black holes

Abstract: In 1972, Price suggested that inverse polynomial tails should be present in the late-time behaviour of scalar fields on Schwarzschild black holes. In the decades since, many features of these tails have been explored both numerically and heuristically in more general settings. The presence of polynomial tails in the context of the Einstein equations has important implications for the nature of singularities inside dynamical black holes and the late-time behaviour of gravitational waves observed at infinity. In this talk I will discuss recent work in collaboration with Y. Angelopoulos and S. Aretakis that establishes rigorously the existence of Price’s polynomial late-time tails in the context of scalar fields on black holes. I will moreover describe how late-time tails are connected to the existence of conservation laws for scalar fields in asymptotically flat spacetimes.

5/22/2020
11:00am
Grigorios Fournodavlos (Sorbonne)  This meeting will be taking place virtually on Zoom.  

 

Title: Construction and (partial) stability of spacelike singularities

Abstract: The presence of singularities in solutions to the Einstein equations is related to profound conjectures in the field, like strong cosmic censorship. They are typically found in the interior of black holes or Big Bang models. Their nature is an often debated topic, with various rivaling scenarios arising, such as null vs spacelike or Kasner-like vs oscillatory. In the first part of the talk, we will discuss the black hole interior problem and present a recent partial stability result of the Schwarzschild singularity (joint with Spyros Alexakis). In the second part of the talk, we will move on to the cosmological setting and present a general method of constructing Kasner-like singularities without symmetries or analyticity (joint with Jonathan Luk).spacetimes.

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.

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