Schedule for: 25w5397 - Geometry, Analysis, and Physics in Lorentzian Signature

Black holes are among the most remarkable exact solutions to the Einstein field equations and serve as central objects in the study of Lorentzian manifolds with physical relevance. In this lecture, I will survey key aspects of black hole spacetimes, focusing on the geometry, the analysis and the physics involved in recent advances in understanding their dynamical stability.

I will introduce the class of causally-null-compactifiable spacetimes that can be canonically converted into compact timed-metric spaces using the cosmological time function of Andersson-Galloway-Howard and the null distance of Sormani-Vega. This class of space-times includes future developments of compact initial data sets and regions exhausting asymptotically flat space-times. I will discuss various intrinsic notions of distance between such space-times and show that some of them are definite in the sense that they are equal to zero if and only if there is a time-oriented Lorentzian isometry between the space-times. These definite distances allow us to define notions of convergence of space-times to limit space-times that are not necessarily smooth. This is joint work with Christina Sormani.

I give an introduction to the theory of Lorentzian metric spaces developed in collaboration with Aleksei Bykov and Stefan Suhr.

Described by Roger Penrose as "perhaps the most important unresolved problem" in classical General Relativity, the Cosmic Censorship Conjectures govern the behavior of spacetime singularities. The Weak Cosmic Censorship Conjecture proposes that all singularities are cloaked by event horizons, preventing the existence of naked singularities (at least generically). The Strong Cosmic Censorship Conjecture conversely asserts the inevitability of singularities within black holes, supporting the deterministic nature of the Einstein equations. This mini course provides an overview of recent mathematical progress on both fronts and highlights key open questions related to both conjectures.

In this mini-course I will introduce the mathematically rigorous renormalization framework of Epstein and Glaser that allows one to construct perturbative quantum field theory models directly in Lorentzian signature. It is applicable on a large class of curved spacetimes and covers gauge theories as well as effective quantum gravity.

In this talk we give an overview of several notions of curvature in non-smooth Lorentzian settings. We will address distributional curvature for spacetime metrics of regularity below $C^2$ as well as its extension using nonlinear distributional geometry. Then we will turn to synthetic (sectional and Ricci) curvature bounds in Lorentzian length spaces and discuss the interrelations between these concepts.

The Myers-Steenrod theorem states that the isometry group of a compact Riemannian manifold is a compact Lie group. In Lorentzian signature, however, there are examples of compact manifolds with non-compact isometry group. In this talk, we will instead consider the isometry groups of non-compact Lorentzian manifolds with well-behaved causal structure. That is, globally hyperbolic spacetimes with compact Cauchy surfaces, satisfying the ``no observer horizons'' condition. Our main result is that the group of time orientation-preserving isometries acts properly on the spacetime. As corollaries, we obtain the existence of an invariant Cauchy time function, and a splitting of the isometry group into two subgroups, roughly corresponding to spatial isometries and time translations. The first one is compact, and the latter is either trivial, $\mathbb{Z}$, or $\mathbb{R}$.

In the 1960's, using a Hamiltonian approach, Arnowitt, Deser, and Misner gave very satisfactory definitions of total energy, linear momentum, and mass for asymptotically Euclidean initial data sets sitting in an asymptotically Minkowskian spacetime. By work of Bartnik from the 1980's, we know that these quantities are well-defined, while by work of Chrusciel around the same time, we know that they transform equivariantly under the asymptotic symmetry group of the spacetime. In the first part of the talk, we will briefly recall these definitions and the results mentioned before, adding some details ensuring that the mentioned equivariance is indeed correct without assuming that one transforms within a given asymptotic spacetime coordinate chart. We will also explain why they conicide with the definitions given by Michel in the 2010's based on the linearization of the constraint equations. We will then shift attention to the definitions of angular momentum and center of mass, defined via a Hamiltonian approach by Regge and Teitelboim in the 1970's and by Beig and O'Murchadha in the 1980's, respectively. We will compare these definitions with the definitions given by Michel based on the linearization of the constraint equations and show that the angular momentum definitions do not coincide. We will also discuss known and unknown problems related to all of these definitions. The insights we present rely on ongoing joint work with Senthil Velu and joint work with Nerz from 2014 and with Sakovich from 2021. We will also discuss some preliminary ideas to overcome the shortcomings of these definitions.

Described by Roger Penrose as "perhaps the most important unresolved problem" in classical General Relativity, the Cosmic Censorship Conjectures govern the behavior of spacetime singularities. The Weak Cosmic Censorship Conjecture proposes that all singularities are cloaked by event horizons, preventing the existence of naked singularities (at least generically). The Strong Cosmic Censorship Conjecture conversely asserts the inevitability of singularities within black holes, supporting the deterministic nature of the Einstein equations. This mini course provides an overview of recent mathematical progress on both fronts and highlights key open questions related to both conjectures.

We will review recent results obtained with K. Melnick about the so called Lichnerowicz conjecture for 3-dimensional spacetimes. The emphasis will be put on the relationship between conformal dynamics and geometry.

Plane waves are Lorentzian manifolds which can be understood as deformations of the flat Minkowski space. In this talk, we consider homogeneous ones and describe the identity component of their isometry group, which has finite index in the full isometry group. Then we consider compact manifolds that are locally isometric to homogeneous plane waves and classify their fundamental groups. This gives a very good understanding of the topology of the compact models. This is a joint work with I. Kath, L. Mehidi and A. Zeghib. In the last part of the talk, we motivate a higher signature version of plane waves, highlighting some fundamental differences with the Lorentzian setting, and present partial results on the completeness of their compact models.

In this talk, we review recent developments in the reformulation of classical concepts such as Ricci curvature, maximality and Cauchy developments in the setting of Lorentzian (length) spaces.

We classify globally hyperbolic Cauchy-compact maximal conformally flat spacetimes whose universal cover admits a unique maximal point in its future causal boundary. We prove that they are Kleinian. More precisely, they are all obtained as a quotient of a regular domain Omega of Minkowski spacetime by a discrete group Gamma of conformal transformations. Furthermore, we show that when the conformal factor is non-trivial, the group Gamma is $P_1$-Anosov in $O(2,n)$. This is joint work with Thierry Barbot.

The classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. One drawback of these classical theorems is that they require smoothness of the metric while in many physical models the metric is less regular. In my talk I will present recent work proving Hawking's singularity theorem for spacetime metrics of local Lipschitz regularity based on a distributional version of the strong energy condition and an almost everywhere mean curvature bound along a local flowout of the initial hypersurface. The proof relies on improved Friedrichs-type estimates and volume/area estimates under lower Ricci curvature bounds based on a segment type inequality. Our results directly extend known C^1 results and complement versions of Hawking's theorem for timelike non-branching Lorentzian length spaces. This is joint work with M. Calisti, E. Hafemann, M. Kunzinger and R. Steinbauer.

The linking number (lk) is a well-known invariant of links dating back to Gauss. This notion has been generalized to non-zero-homologous (n-1) spheres in the unit spherical tangent bundle of an n-manifold by the affine linking number (alk). In 2004 Chernov and Rudyak used this to determine when points in a manifold are causally related. We'll discuss expanding this idea to detecting the number of light-rays between observers in a spacetime.

My current research is in propounding conditions on a class of observers, foliating a spacetime, that lead to the conclusion of a "nice" causal boundary: spacelike and with the topology of the leaf-space. The most subtle part is conditions that guarantee no null relationships in the boundary elements. The way I found to do this calls back to a technique I long ago encountered, that I call the method of timelike 2-sufaces. It is so ubiquitous in my research, that I must share it. In this case, I first uncovered the requisite conditions by assuming just a 2-dimensional spacetime: This has an easy-to-determine causal structure (just two foliations by null curves). Then I reduce the problem in the original spacetime to examining a timelike 2-surface: the pullback of a curve in the leaf-space via the foliation projection (i.e., a curve of observer-worldlines).

In this mini-course I will introduce the mathematically rigorous renormalization framework of Epstein and Glaser that allows one to construct perturbative quantum field theory models directly in Lorentzian signature. It is applicable on a large class of curved spacetimes and covers gauge theories as well as effective quantum gravity.

The Buchholz' scattering theory of waves in two dimensional massless models suggests a natural definition of a scattering amplitude. We compute such a scattering amplitude for charged infraparticles that live in the (irregular) vacuum representation of the 2d massless scalar free quantum field and obtain a non-trivial result. It turns out that these excitations exchange phases, depending on their charges, when they collide. This form of interaction, which coexists with a linear field equation, is effected by the exotic infrared structure of the vacuum in which the excitations move. (One can draw a distant analogy here to the gravitational interaction, which coexists with the geodesic motion and is effected by the curvature of spacetime). The talk is based on a joint work with Jens Mund: Commun. Math. Phys. 395, 1197–1210 (2022), arXiv:2109.02128.

We obtain existence and uniqueness of minimizers for the p-capacity functional defined with respect to a symmetric anisotropy for p>1, including the case of a crystalline norm in Euclidean space . The result is obtained by a characterization of the corresponding subdifferential and it applies for unbounded exterior domains under mild regularity assumptions (Lipschitz-continuous boundary) and no convexity requirements. By means of an approximation procedure introduced by Moser, as limits of these minimizers (after a change of variable), we construct weak solutions of the anisotropic inverse mean curvature flow under very few assumptions both on the anisotropy and on the initial data. Surprisingly enough, our notion of solution still recovers variational and geometric definitions similar to those introduced by Huisken-Ilmanen, but requires to work within the broader setting of BV -functions. Despite of this, we still reach classical results like the continuity and exponential growth of perimeter, as well as outward minimizing properties of the sublevel sets. This is joint work with Salvador Moll and Marcos Solera.

The inflationary scenario, which states that the early universe underwent a brief but dramatic period of accelerated spatial expansion, has become the current paradigm of early universe cosmology. Although inflationary cosmology has its many successes, it does not (as of yet) have the status of an established physical theory. In this paper, we provide mathematical support for the inflationary scenario in a class of anisotropic spacetimes. These anisotropic spacetimes satisfy certain initial conditions so that they are perfectly isotropic at the big bang but become less isotropic as time progresses. The resulting inflationary eras are a consequence of the initial conditions which force the energy-momentum tensor to be dominated by a cosmological constant at the big bang. This is joint work with Annachiara Piubello.

The positive energy theorems serve as a fundamental pillar in the geometric development of general relativity. Proved by Schoen and Yau, and later by Witten, the original positive energy theorems are stated for asymptotically flat manifolds with either time-symmetric initial data or data whose second fundamental form decays to zero at infinity. This ansatz on the metric and second fundamental form is traditionally motivated by a desire to model an "isolated gravitational system." However, actual astrophysical massive objects are not truly isolated but exist within an expanding cosmological universe. In this talk, we present a definition of energy in the context of such an expanding universe. In this approach, we take the flat expanding de Sitter model as the background spacetime, in contrast to the more conventional Minkowski spacetime, which forces our definition of energy to be quasi-local due to the presence of a cosmological horizon. We prove positivity of the energy provided the cosmological constant associated with the de Sitter background is not too large. This is joint work with Eric Ling and Rodrigo Avalos.

The theory of causal fermion systems is an approach to  fundamental physics. It gives quantum mechanics, general relativity and quantum field theory as limiting cases and is therefore a candidate for a unified physical theory. From the mathematical perspective, causal fermion systems provide a general framework for desribing and analyzing non-smooth geometries. The dynamics of causal fermion systems is described by a variational principle called the causal action principle. In the talks, I will focus on the geometric structures of a causal fermion system.

While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually break down. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions from relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures.

In this mini-course I will introduce the mathematically rigorous renormalization framework of Epstein and Glaser that allows one to construct perturbative quantum field theory models directly in Lorentzian signature. It is applicable on a large class of curved spacetimes and covers gauge theories as well as effective quantum gravity.

From recent progresses in the study of smooth and nonsmooth Lorentzian structures it emerges the need of a functional-analytic theory where, among other things, the relevant norms satisfy a reverse triangle inequality. Aim of the talk is to show that perhaps such a theory is possible.

The so-called Lipschitz-Killing curvature integrals are ubiquitous objects in riemannian geometry. They appear for instance in Weyl's tube formula, in the heat kernel of differential forms, and in several formulas of integral geometry. In the talk I will present a joint project with Andreas Bernig and Dmitry Faifman where we extend (as distributions) the Lipschitz-Killing curvature integrals to generic submanifolds of pseudo-riemannian spaces.

By using strategies and techniques from metric (measure) geometry and optimal transport one can study Lorentzian geometry and General Relativity from a synthetic point-of-view. This approach does neither rely on smoothness nor on manifolds, thereby leaving the framework of classical differential geometry altogether. This opens up the possibility to study curvature (bounds) for spacetimes of low regularity or even more general spaces. An analogous shift in perspective proved extremely fruitful in the Riemannian case (Alexandrov-, CAT(k)- and CD-spaces). I will review the basics of this approach, milestones along the way and recent progress.

We shall discuss radiative spacetimes of various types. Thereby we will present new results on their behavior at null infinity. In particular, antipodal symmetry and non-symmetry questions shall be addressed. Moreover, we will discuss angular momentum at future null infinity and related matter. We will also present details for an experiment to measure the EM analogue of gravitational wave memory.

Non-null hypersurfaces can be easily viewed as geometric objects detached from the spacetime where they are embedded. This process of “detachment” is more complicated in the case of null hypersurfaces. In this talk I will introduce the notion of “null manifold”, together with related concepts such as “ruled null manifold” or “metric hypersurface data”, as a way to study null hypersurfaces from a fully detached point of view. With this framework at hand, I will then discuss three applications related to existence of spacetimes. Specifically, I will present a detached version of the characteristic initial value problem, a general analysis of transverse expansions across a null hypersurface and an existence and asymptotic uniqueness result of Λ-vacuum spacetimes from data on a (detached) non-degenerate Killing horizon. The three applications are joint work with Gabriel Sánchez Pérez.

Described by Roger Penrose as "perhaps the most important unresolved problem" in classical General Relativity, the Cosmic Censorship Conjectures govern the behavior of spacetime singularities. The Weak Cosmic Censorship Conjecture proposes that all singularities are cloaked by event horizons, preventing the existence of naked singularities (at least generically). The Strong Cosmic Censorship Conjecture conversely asserts the inevitability of singularities within black holes, supporting the deterministic nature of the Einstein equations. This mini course provides an overview of recent mathematical progress on both fronts and highlights key open questions related to both conjectures.

Nemirovski and myself proved that for Globally Hyperbolic Spacetimes causal relation between $p$, $q$ is equivalent to Legendrian Linking of spheres of lights rays $S_p$ and $S_q$ through the points $p$ and $q$ within the contact manifolds of all light rays. Sadykov and myself conjecturally generalize these results to Borde Sorkin spacetimes.

The Hawking energy is one of the simplest quasi-local energy definitions in general relativity. Despite its simplicity, the Hawking energy has faced challenges due to ambiguities when applied to general surfaces. In this talk, I will present recent results demonstrating that the Hawking energy exhibits key physical and mathematical properties—non-negativity, rigidity, and monotonicity—when evaluated on Area-constrained critical surfaces of the Hawking energy, for short Hawking surfaces. These results establish Hawking surfaces as a useful tool for evaluating the Hawking energy and reinforce its potential as a meaningful tool for understanding gravitational phenomena.

Semi-Riemannian manifolds that satisfy homogeneous linear differential conditions of arbitrary order on the curvature are analyzed. Of special relevance is the Lorentzian case, that connects these spacetimes with Penrose limits, Gauss-Bonnet gravity and other relevant gravitational studies.

This lecture highlights recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In particular, we foreshadow a low-regularity splitting theorem obtained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity $p$-d'Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988), Galloway (1989), and Newman's (1990) confirmation of Yau's (1982) conjecture, bringing both Lorentzian splitting results into a framework closer to the Cheeger--Gromoll (1971) splitting theorem from Riemannian geometry. I will present recent work which shows how to prove the Lipschitz inextendibility of weak null singularities. Such singularities are expected to form generically in the interior of rotating black holes.

In this work we present some recent advances in the study of hyperspaces of lenght spaces. Particularly, motivated by results regarding metric Hadamard spaces, we provide a description of the hyperspace of compact diamonds of a globally hyperbolic Lorentzian length space. Joint work with W. Barrera and L. Montes de Oca.

A theorem by Labourie and Schlenker asserts that any negatively curved metric on a closed surface $\Sigma$ admits a unique isometric, equivariant embedding into the open solid timelike cone in Minkowski space of dimension 2+1. In this talk, we will focus on quantifying embaddable negatively curved metrics on $\Sigma$ for a fixed representation. More precisely, we prove that this set of metrics projects into a relatively compact subset of the Teichmüller space.

Hilbert’s 6th problem has been called ``The axiomatic foundations of physics’’ In 1900, physics still meant pre-relativistic classical physics. Newton had formulated his atomistic mechanics axiomatically already. Hilbert elaborated that he meant the rigorous derivation of the effective laws of everyday physics from the more fundamental axiomatic atomistic laws. As a particular example Hilbert mentioned the derivation of the kinetic Maxwell-Boltzmann equation for a dilute neutral gas from Newton’s mechanics of many tiny hard balls. Quite some progress has been made on this front in the past 120+ years, not only for the Maxwell-Boltzmann equation but also for other kinetic equations, in particular the Vlasov equations of a Coulomb plasma and the Jeans equation of stellar dynamics. Yet there still is room for improvement and much left to be done in order to fulfill Hilbert’s program in its original formulation. The birth of the special theory of relativity in 1905, and of the general one in 1916, and then of quantum theory in the mid 1920s, has made it plain that analogous versions of Hilbert’s 6th problem could be posed in these contexts as well, at least in principle. In this talk we will survey the significant progress that has been made on the classical special-relativistic front where the mathematical foundations of the Vlasov-Maxwell equations are now in close reach, and we emphasize the obstacles that have yet to be overcome in the general-relativistic quest to lay the foundations of the Einstein-Maxwell-Vlasov system.

Initial point singularities are of interest in general relativity because considerable evidence suggests that our universe had one. Prescribing initial data for such solutions presents some obvious problems though. Work over the last few decades has explored how to do this for a variety of reasonable matter models though, including polytropic fluids, electromagnetic radiation, and kinetic theory, with or without collisions. This talk will mostly be an overview but I will briefly discuss some recent work with Ho Lee, Ernesto Nungesser and Paul Tod.

In this talk, I hope to describe elements of proving a certain stable singularity formation result for the Einstein—Euler system, which is the topic of work in progress with Jonathan Luk. I will first describe where this fits into the big picture of the study of multidimensional shocks, and why it is appropriate to call this a shock formation result. Then, I will try to describe some of the main ideas that go into proving shock formation results. Finally, I will try to describe some of the main difficulties that arise in the case of Einstein—Euler.

My most recent monograph addresses the problem of the development of shocks in a compressible fluid past the point of their formation. The monograph solves a restricted form of this problem where the jump in entropy at the shock is neglected. In the talk I shall discuss the methods of geometric analysis used in the monograph, confining myself to the nonrelativistic theory. Geometry enters as the acoustical structure, a Lorentzian metric structure defined on the spacetime manifold by the fluid. This acoustical structure interacts with the background Galilean spacetime structure. Reformulating the equations as two coupled first order systems, the characteristic system, which is fully nonlinear, and the wave system, which is quasilinear, a complete regularization of the problem is achieved. Geometric methods also arise from the need to treat the free boundary. These methods involve the concepts of bi-variational stress and of variation fields. The main new analytic method arises from the need to handle the singular integrals appearing in the energy identities.