General Relativity Seminar
2025-02-18 11:00 - 12:00 (America/New_York) | 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.
