An analogue of non-interacting quantum field theory in Riemannian signature
2024-05-13 11:00 - 12:00 (America/New_York) | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Mikhail Molodyk, Stanford
Title: An analogue of non-interacting quantum field theory in Riemannian signature
Abstract: Recent advances using microlocal tools have led to constructions, for wave operators on various classes of spacetimes, of four distinguished Fredholm inverses which have the singular behavior required of retarded, advanced, Feynman, and anti-Feynman propagators in QFT. Vasy and Wrochna have used these to define a QFT on asymptotically Minkowski spacetimes, for which they construct Hadamard states described by asymptotic data at infinity. I will describe an analogue of this construction on Riemannian manifolds with two asymptotically conic ends, defining quantum fields satisfying the Helmholtz equation and using scattering data to construct states satisfying a wavefront mapping-property version of the Hadamard condition. The absence of a spacetime interpretation lends itself to a sharper focus on the theory’s analytic structure, from whose perspective the Feynman propagators are no less natural than the advanced/retarded ones. I will also highlight some differences between Feynman propagators defined as distinguished inverses and as time-ordered expectations. Based on joint work with András Vasy.

Compactified Hyperboloidal Evolution in Numerical Relativity
2024-08-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
Speaker: Shalabh Gautam, International Center for Theoretical Sciences, Tata Institute of Fundamental Research
Title: Compactified Hyperboloidal Evolution in Numerical Relativity
Abstract: In the field of numerical relativity, a major challenge is to incorporate future null infinity into the computational domain using a well-posed formulation. Achieving this goal would not only allow us to accurately simulate any astrophysical system, such as binary black holes, and extract the resulting signal, like gravitational waves, at future null infinity, but it would also facilitate the study of various fundamental phenomena. One proposed solution is to use hyperboloidal slices. In this presentation, I will introduce an alternative approach to numerical relativity that leverages these slices in spherical symmetry. I will also discuss our ongoing efforts to establish a well-posed formulation of the Einstein Field Equations on these slices, and propose numerical schemes to ensure stability and convergence for linear hyperbolic systems over long durations. Additionally, I will present our method for constructing initial data corresponding to an asymptotically flat spacetime. A natural extension of our work will be to generalize these methods to the Einstein Field Equations with appropriate initial data in the full 3d.

New Energy Inequality in AdS
2024-09-24 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Diandian Wang, Harvard University
Title: New Energy Inequality in AdS
Abstract: I will describe evidence for a new energy inequality in asymptotically AdS spacetimes whose conformal boundary contains a spatial circle. It is in some sense analogous but crucially different to the Penrose inequality. In the AdS4 case, this generalizes the Horowitz-Myers conjecture. I will show how static solutions play an interesting role in determining the shape of the function that bounds the gravitational energy.

Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
2024-10-01 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Daniel Kapec, Harvard
Title: Quasinormal Corrections to Near-Extremal Black Hole Thermodynamics
Abstract: Recent work on the quantum mechanics of near-extremal non-supersymmetric black holes has identified a characteristic scaling of the low temperature black hole partition function. This result has only been derived using the path integral in the near-horizon region and relies on many assumptions. We discuss how to derive the scaling for the near-extremal rotating BTZ black hole from a calculation in the full black hole background using the Denef-Hartnoll-Sachdev (DHS) formula, which expresses the 1-loop determinant of a thermal geometry in terms of a product over the quasinormal mode spectrum. We also derive the spectral measure for fields of any spin in Euclidean BTZ and use it to provide a new proof of the DHS formula and a new, direct derivation of the BTZ heat kernel. The computations suggest a path to proving the scaling for the asymptotically flat 4d Kerr black hole.

Continuation of solutions of Einstein's equations
2024-10-08 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Oswaldo Vazquez, Northeastern University
Title: Continuation of solutions of Einstein’s equations
Abstract: Klainerman-Rodnianski improved the continuation criterion for the solutions of Einstein’s equations proved by Michael Anderson using Kirchoff-Sobolev type parametrix and geometric Littlewood-Paley theory. Using their technique but a new parametrix we prove a continuation condition in the context of 3+1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended indefinitely in the future as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant H^2 Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime L^{infty} bound.

Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
2024-10-15 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Christoph Kehle, MIT
Title: Gravitational collapse to extremal Reissner-Nordström and the third law of black hole thermodynamics
Abstract: In this talk, I will present a proof that extremal Reissner-Nordström black holes can form in finite time in gravitational collapse of charged matter. In particular, this construction provides a definitive disproof of the “third law” of black hole thermodynamics. I will also discuss recent works showing that extremal black holes take on a central role in gravitational collapse, giving rise to a new conjectural picture of “extremal critical collapse.” This is joint work with Ryan Unger (Stanford).

Formation of Trapped Surfaces in Geodesic Foliation
2024-10-29 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Xuantao Chen, Johns Hopkins University
Title: Formation of Trapped Surfaces in Geodesic Foliation
Abstract: We reprove the formation of trapped surfaces for the Einstein vacuum equation using the incoming geodesic foliation. All previous results, starting with the seminal work of Christodoulou, make use of the double null foliation. In the new proof, all Ricci coefficients are estimated by transport equations in the incoming null direction, and the incoming initial data is only used in the curvature (energy) estimates. The result is based on a version of the non-integrable PT frame, which was introduced in the proof of Kerr stability with small angular momentum.

pp Waves: Quasinormal Modes & Hidden Symmetries of Black Holes
2024-11-12 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Ahmed Seta, Harvard University
Title: pp Waves: Quasinormal Modes & Hidden Symmetries of Black Holes
Abstract: The spectrum of quasinormal modes of 4D flat space black holes is not analytically tractable, but there are two asymptotic limits where the QNM spectrum is under control: weak damping and strong damping. In this talk, I will explain how these asymptotic QNMs are controlled by dynamical symmetries of the wave equation in certain kinematic limits. These two asymptotic limits are, in turn, captured by the two classes of bound null geodesics in the black hole geometry: the photon ring and the horizon. I will also discuss the Penrose limit: a scaling limit into the geometry experienced by these geodesics, which results in a plane-wave spacetime where the dynamical symmetries get enhanced into isometries.

The Einstein-Euler system with a physical vacuum boundary in spherical symmetry
2024-11-19 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Marcelo Disconzi, Vanderbilt University
Title: The Einstein-Euler system with a physical vacuum boundary in spherical symmetry
Abstract: We establish local well-posedness for the Einstein-Euler system with a physical vacuum boundary in spherical symmetry. Our proof relies on a new way of thinking about Einstein’s equations in spherical symmetry that is well-adapted to the fluid’s characteristics on the free boundary. We also exploit the Einstein constraint equations in spherical symmetry in a new way, as a tool to understand the evolution problem. This is joint work with Jared Speck.

Mass for the large and for the small
2024-12-03 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Tin Yau Tsang, Courant Institute, New York University
Title: Mass for the large and for the small

Abstract: Seeking a meaningful geometric (physical) invariant to describe a spacetime has sparked research in both mathematics and physics. In this talk, we will first see the fundamental progress made by Schoen-Yau and Witten on ADM (Arnowitt-Deser-Misner) mass. Then, we will see the progress on quasilocal masses in recent years. In particular, we would discuss the Hamiltonian formulation and the behaviour of quasilocal masses with the presence of apparent horizons.

The Einstein-Vlasov system in a large data regime
2024-12-10 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Nikolaos Athanasiou, University of Crete, Greece
Title: The Einstein-Vlasov system in a large data regime
Abstract: In this talk, our object of study is the Einstein-Vlasov system with a massless Vlasov matter field. Complementing various important works obtaining the stability of Minkowski spacetime as a solution to this system, we look at the large data regime, motivated in turn by the signature for decay rates of various Ricci, curvature and matter components, first introduced by X. An. Our work provides a semi-global existence result and a trapped surface formation result for the Einstein-Vlasov system in the absence of spherical symmetry. Our proof is based on a double null gauge. Interestingly, we give a new way of obtaining estimates for the Vlasov matter, purely by commuting with various vector fields and without the need to use Jacobi fields.

The mass angular momentum inequality
2025-02-04 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Gilbert Weinstein, Ariel University
Title: The mass angular momentum inequality
Abstract: We show that either there is a counterexample to black hole uniqueness, in the form of a regular axisymmetric stationary vacuum spacetime with an asymptotically flat end and multiple degenerate horizons which is ‘ADM stable’, or the following statement holds. Complete, simply connected, maximal initial data sets for the Einstein equations with multiple ends that are either asymptotically flat or asymptotically cylindrical, admit an ADM mass lower bound given by the square root of total angular momentum, under the assumption of nonnegative energy density and axisymmetry. Moreover, equality is achieved bound only for a constant time slice of an extreme Kerr spacetime. The proof is based on a novel flow of singular harmonic maps with hyperbolic plane target, under which the renormalized harmonic energy is monotonically nonincreasing. Relevant properties of the flow are achieved through a refined asymptotic analysis of solutions to the linearized harmonic map equations.

On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system
2025-02-11 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Liam Urban (University of Vienna)

Title: On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system
Abstract: In recent years, the formation of Big Bang singularities has been increasingly well understood in presence of scalar-field matter, which suppresses the geometric oscillations that one generically expects otherwise. From a cosmological perspective, however, one would like to understand whether this feature persists when coupled to further models that more closely resemble the matter content in our universe.
In this talk, I will discuss the past nonlinear stability of FLRW solutions to the Einstein scalar-field Vlasov system in three and four spacetime dimensions without symmetry assumptions. The latter is based on joint work with David Fajman. In both works, near-FLRW solutions are
shown to be asymptotically velocity term dominated toward the past, resulting in a quiescent Big Bang singularity with stable Kretschmann scalar blow-up. The Vlasov distribution, however, becomes highly anisotropic towards the Big Bang singularity, concentrating in preferred momentum directions associated with eigendirections of the shear. To ensure that the scalar field sufficiently mitigates this behaviour, one crucially exploits a scaling hierarchy between horizontal and vertical derivatives in the expansion normalized Vlasov equation. In four
spacetime dimensions, this analysis is paired with a Bel-Robinson energy formalism to control the spacetime and scalar field evolution, while one finds a significantly more direct proof in three dimensions since the spatial Riemann curvature is pure trace.
As a corollary, one also obtains that the Strong Cosmic Censorship conjecture holds for four-dimensional polarized $U(1)$-symmetric solutions to the Einstein vacuum equations with spatial topology $MtimesS^1$ and isotropic quotient.

General Relativity Seminar
2025-02-18 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Maximilian Ofner, University of Vienna
Title: Stability and Instability of Relativistic Fluids in Slowly Expanding Spacetimes
Abstract: Homogeneous and isotropic solutions to the relativistic Euler equations are known to be unstable on a Minkowski background. However, for FLRW models with a fast expansion rate, relativistic fluids stabilize. This scenario suggests a transition between stable and unstable behavior, somewhere along a family of spacetimes parametrized by their expansion rate. In this talk we will discuss this phase transition for various equations of state, focusing on the regime of linear and decelerated expansion. This is based on recent analytical results, complemented by numerical analysis.

BKL bounces outside homogeneity
2025-02-25 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Warren Li ( Princeton University)
Title: BKL bounces outside homogeneity
Abstract: In work spanning the late 20th century, physicists Belinski, Khalatnikov and Lifshitz (BKL) proposed a general ansatz for solutions to the Einstein (vacuum) equations near singularities. They suggest that the spacetime dynamics at different spatial points on the singularity decouple and are well-approximated by a system of autonomous nonlinear ODEs, whose orbits are governed by a chaotic cascade of “BKL bounces”. In this talk, we present recent work verifying BKL’s heuristics in a large class of symmetric, but spatially inhomogeneous, spacetimes. In particular, we prove decoupling even in the presence of (up to one) BKL bounce. The proof uses nonlinear ODE analysis coupled to hyperbolic energy estimates, and one hopes our methods may be applied more generally.

Positive scalar curvature with point singularities
2025-03-11 11:00 - 12:00 | CMSA Room G10, CMSA, 20 Garden Street, Cambridge, MA, 02138, United States
General Relativity Seminar
Speaker: Rudolf Zeidler, Mathematical Institute, University of Münster
Title: Positive scalar curvature with point singularities
Abstract: I will explain a certain topological construction of positive scalar curvature metrics with uniformly Euclidean ($L^infty$) point singularities. This provides counterexamples to a conjecture of Schoen. It also shows that there are metrics with uniformly Euclidean point singularities which cannot be smoothed via a geometric flow while simultaneously preserving non-negativity of the scalar curvature. Based on recent joint work with Simone Cecchini and Georg Frenck.

Unstable Fluids in Expanding Cosmologies
2025-03-25 16:00 - 17:00 | Virtual
General Relativity Seminar
Speaker: Elliot Marshal, School of Mathematics at Monash University
Title: Unstable Fluids in Expanding Cosmologies
Abstract: The FLRW solution is the simplest cosmological model in general relativity, describing a fluid-filled, spatially homogeneous universe. While there is extensive literature in the physics community on cosmological models with a linear equation of state , rigorous stability results have, until recently, been limited to FLRW models with accelerated expansion and small sound speeds. In this talk, I will discuss numerical work on two types of instabilities which can occur in non-linearly perturbed FLRW models outside of this regime. The first concerns a longstanding conjecture of Alan Rendall on FLRW models with positive cosmological constant and super-radiative (K>1/3) equations of state. Our numerical work, in collaboration with F. Beyer and T.A. Oliynyk, supports Rendall’s conjecture and highlights the underlying mechanism for the instability. In the second case, I will discuss evidence that perturbations of slowly expanding (decelerated) FLRW models generically form shocks in finite time. This contrasts with known results for accelerated models where shock formation is suppressed for suitably small perturbations.

Low-regularity Local Well-posedness of the Elastic Wave System
2025-04-01 11:00 - 12:00 | Virtual
General Relativity Seminar
Speaker: Sifan Yu, National University of Singapore
Title: Low-regularity Local Well-posedness of the Elastic Wave System
Abstract: In this talk, I will present a recent work on the elastic wave system in three spatial dimensions. For admissible harmonic elastic materials, we prove a low-regularity local well-posedness result for the corresponding elastic wave equations. For such materials, we can split the dynamics into the “divergence-part” and the “curl-part,” and each part satisfies a distinct coupled quasilinear wave system with respect to different acoustical metrics. Our main result is that the Sobolev norm H^{3+} of the “divergence-part” (the “faster-wave part”) and the H^{4+} of the “curl-part” (the “slower-wave part”) can be controlled in terms of initial data for short times. We note that the Sobolev norm assumption H^{3+} is optimal for the “divergence-part.” This is a joint work with Xinliang An and Haoyang Chen.

Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
2025-04-15 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Aghil Alaee, Clark University
Title: Positive mass theorem for ALE(AE) and ALF(AF) Toric 4-Manifolds
Abstract: One of the fundamental conjectures in mathematical relativity is the positivity of total mass (if it is defined!) for complete non-compact Riemannian manifolds assuming appropriate lower bounds on scalar curvature. This conjecture has been proved for AE manifolds using several techniques, starting with the celebrated results of Schoen-Yau and Witten. There are counter-examples to this conjecture in the AF, ALF, and ALE cases. In this talk, we will refine this conjecture and prove it for toric 4-manifolds. The proof is robust and can be extended to higher dimensions if additional assumptions are added. This is a joint work with Marcus Khuri and Hari Kunduri.

Hyperbolic equations in a double null gauge
2025-04-22 11:00 - 12:00 | CMSA G102, 20 Garden Street, Cambridge, MA, 02138
General Relativity Seminar
Speaker: Christopher Stith, University of Michigan
Title: Hyperbolic equations in a double null gauge
Abstract: The hyperbolic nature of the Einstein equations is well-known and has been used in many different contexts. More recently, the double null gauge has proven to be a powerful tool for quantitative analysis of the Einstein equations. It has the advantage of reducing the equations for many dynamical quantities to ODEs along null curves, and the Bianchi equations to a first-order hyperbolic system. The double null gauge has been used extensively and to great effect in analyzing the structure of spacetime for many purposes, including (for instance) stability problems and trapped surface formation. However, the local existence problem for the Einstein equations in a double null gauge has never been treated in full in its own right. In this talk, we discuss how to formulate a general procedure for solving the linearized problem, namely, the local existence theory for systems of first-order hyperbolic equations in a double null gauge.
