Marco van den Beld-Serrano
The misalignment time function
Given a globally defined timelike vector field, does there exist a (unique) time function whose gradient best aligns with it? Despite the importance of time functions in Lorentzian Geometry and the existence of physical theories that assume the presence of a privileged timelike vector field, the above-mentioned question has, to the best of our knowledge, not been addressed before. We will tackle it focusing on compact subsets of smooth globally hyperbolic spacetimes. In the first place, we will introduce a functional that, on the one hand, captures the misalignment between the given timelike vector field and the gradient of suitable Sobolev functions, and, on the other hand, penalizes null gradients. Secondly, we will prove that, under suitable assumptions on the Sobolev index and the strength of the null gradient penalization, the considered functional has a unique smooth minimizer -- the misalignment time function -- with everywhere timelike gradient.
Gabriel Sánchez-Pérez
Asymptotic expansion of the metric at conformal null infinity
The study of gravitational radiation is a central problem in mathematical relativity. It is also technically difficult because the metric that defines fall-off at infinity is itself a dynamical variable on which boundary conditions must be imposed. A natural framework to address this difficulty is Penrose’s conformal approach, in which asymptotic flatness and radiation are encoded in the geometry of a null boundary (scri) attached to an unphysical spacetime satisfying the conformal Einstein equations. In this setting, infinity becomes a genuine geometric object. Most existing results on gravitational radiation focus on the four-dimensional case, typically assuming a spherical topology for null infinity and the vanishing of the Weyl tensor there. In this talk, motivated by the fact that comparatively fewer results are available beyond this setting, I analyze the conformal Einstein equations at null infinity to all orders without imposing restrictions on the spacetime dimension, the topology of scri, or fall-off conditions on the Weyl tensor. Assuming only that null infinity admits a foliation by cross-sections, not necessarily spherical, I study how the equations constrain its geometry. Our approach is fully coordinate-free and treats the conformal factor as a genuine dynamical variable. After identifying the free data at scri, I show that any two asymptotically flat spacetimes sharing the same free data are necessarily isometric to infinite order at null infinity. I also present a detached definition of null infinity and prove an existence theorem for asymptotically flat spacetimes realizing prescribed initial data to infinite order at scri. Finally, I discuss possible obstructions to the smoothness of null infinity and state a conjecture relating one such obstruction to the Fefferman-Graham ambient metric. This talk is based on joint work with Marc Mars and on the preprint arXiv:2602.05061.
Inés Vega González
Singularity theorems below Lipschitz regularity
The singularity theorems of General Relativity establish the occurrence of spacetime ``singularities", in the sense of causal geodesic incompleteness of the spacetime manifold under certain physically reasonable conditions. These results were formulated for smooth metrics, however for the theorems to make physically meaningful predictions, one needs to extend their validity to lower regularities. In recent years there have been efforts in finding low regularity versions of the Penrose and Hawking singularity theorems, getting as low as Lipschitz Lorentzian metrics. The main goal of my research is lowering the regularity threshold of the Hawking theorem to metrics of Sovolev regularity W^{1,p}_{\rm loc}, which is lower that Lipschitz regularity, with L^p bounded curvature. Our main tool is the use of RT-equations, a set of elliptic equations which allow us to raise the regularity of the metric by one derivative on a W^{2,p}_{\rm loc} related atlas. A particular issue is to find a reasonable definition of mean curvature for that specific scenario.
Miguel Ortega
Lifting Up Translators of the Mean Curvature Flow Via Submersions
There are plenty of papers using subgroups of isometries of some famous ambient manifolds to obtain hypersurfaces which are translating solitons of the mean curvature flow. This simplification of the PDE to ODE can be seen as using submersions from the ambient manifold to some interval. In this talk, somehow we make the return trip. Indeed, given a smooth submersion \pi:P\to B, and a smooth function u:B\to R, we lift it up \hat{u}:=u\circ\pi:P\to R. We construct metrics on P by considering a metric on B, an Ehresmann connection in TP, and solving under some conditions a PDE involving the gradient of u, in such a way that (a piece of) the graph of the lifted map \hat{u} is a translating soliton of the MCF in P\times R (with a warped metric if desired). In particular, given a pseudo-Riemannian submersion and a map u as before, by twisting the metric on the total space, we can construct translating solitons in P\times R. We will exhibit examples by twisting the metric in some famous manifolds. This is a joint work with Marie-Amélie Lawn (Imperial College London).
Argam Ohanyan
Singularity theorems for low regularity Lorentz-Finsler metrics
Questions of low regularity in Lorentzian and spacetime geometry have attracted a lot of attention over the last years. In order to cover many physically relevant models and to give credence to the singularity theorems of Hawking and Penrose, it is paramount to study spacetime geometries whose underlying metric is non-smooth. In this talk, motivated by the work of Graf on Lorentzian metrics of regularity C^1, we introduce and discuss foundational aspects of Lorentz-Finsler metrics which are C^1 in the manifold variable and C^3 in the vector variable. Our main results are the singularity theorems of Hawking and Penrose for such metrics, extending both the C^1-Lorentzian results of Graf as well as the classical singularity theorems from smooth Lorentz-Finsler geometry. This talk is based on ongoing joint work with Darius Erös (U. Vienna), Ettore Minguzzi (U. Pisa) and Shin-ichi Ohta (Osaka U.).
Benjamín Olea
Weakly parallel null hypersurfaces in Lorentzian manifolds
A nondegenerate submanifold in a semi-Riemannian manifold is parallel if its second fundamental form is covariantly constant. They can be though as a generalization of totally geodesic ones or as an extrinsic analogue to locally symmetric manifolds. They have been extensively studied and classified primarily within a Riemannian context. However, there are also some remarkable works on a Lorentzian ambient. We can adapt the definition of “parallel submanifold” to the case of a degenerate hypersurface in a Lorentzian manifold - that is, a null hypersurface - but it is easy to check that this definition is too rigid since they are always totally geodesic. In order to get a definition of parallel null hypersurface covering a wider family than the totally geodesic ones, O. Palmas defined and studied null hypersurfaces with screen parallel shape operator, \cite{key-2}. Nevertheless, having screen parallel shape operator is not an invariant property (it depends on some choices), which suppose a great drawback. This is why we introduce the concept of ``weakly parallel null hypersurface'', \cite{key-1} . They are null hypersurfaces in Lorentzian manifolds which hold certain condition on the covariant derivative of the null second fundamental form of the null hypersurface, but only along the null tangent direction. In this manner, we get an invariant property (it does not depend on any choice) similar, in some sense, to the definition of parallel hypersurface and which contains strictly the family of totally geodesic null hypersurfaces. We show that weakly parallel null hypersurfaces are related to the existence of some special kind of vector fields in a Riemannian manifold, called exponential vector fields. The study of this kind of vector fields allow us to get some properties of weakly parallel null hypersurfaces, as for example a classification in the Minkowski space.
B. Olea, Exponential vector fields in Riemannian manifolds and weakly parallel null hypersurfaces in Lorentzian manifolds, Math. Z. 311 (2025) Paper No. 37, 28.
O. Palmas, Null hypersurfaces with screen parallel shape operator, Mediterr. J. Math. 18 (2021) Paper No. 133, 12.
Sebastian Gieger
A synthetic Lorentzian Cartan-Hadamard theorem
The Cartan-Hadamard theorem is a classical result in Riemannian geometry stating that on simply connected complete Riemannian manifolds with nonpositive sectional curvature, the exponential map at any point is a global diffeomorphism. In this talk we prove a version of the Cartan-Hadamard theorem for Lorentzian length spaces with timelike curvature bounded above by 0 that are future one-connected. Along the way, we establish a conjugate point estimate and show that in this setting any timelike curve can be deformed into a timelike geodesic with the same endpoints and give some applications. The results presented are part of joint work with Darius Erös, Joe Barton, Tobias Beran, Mauricio Che, Felix Rott and Jona Röhrig
Miguel Ángel Javaloyes Victoria
Cone structure and metrics that depend on time
How can a metric that depends on time be studied? In General Relativity, space and time are intertwined in such a way that they cannot be understood separately, and moreover, time is relative, depending on a choice of reference frame. But our goal in this lecture will be different. We will show that it makes sense to work with an absolute time and still consider time-dependent metrics. The role of length will be played by time, and the Riemannian metric will determine the velocity of objects in each direction. Geodesics will be defined as the fastest trajectories, which, by applying the relativistic Fermat principle, can be calculated as light-like geodesics on a Finsler spacetime, most generally, from a cone structure. Finally, we will demonstrate that these structures can be applied to the study of forest fires and that we can also define a curvature that measures how geodesics diverge using the degenerate curvature introduced by Harris in Lorentzian manifolds. This curvature helps to identify focal points, which are crucial for firefighters. Moreover, we will give an interpretation of curvature in terms of Jacobi fields and will obtain some applications to compute flag curvature in Finsler manifolds.
Felix Rott
Lorentz meets Ptolemy
We consider a Lorentzian analogue of the Ptolemy inequality and we prove that in the setting of globally hyperbolic spacetimes it is equivalent to a global timelike sectional curvature bound from above by zero. We investigate the link between the Ptolemy inequality and the hyperbolic inversion and establish some applications and rigidity properties. Joint work with Zhe-Feng Xu and Matteo Zanardini.
José Luis Flores
A Lifting principle of curves under exponential-type maps
We develop a lifting theory for the exponential map of semi-Riemannian manifolds that overcomes the classical obstruction caused by its singularities. We show that every smooth path in the manifold admits, up to a nondecreasing reparametrization, a partial lift through the exponential map which is inextensible in its domain of definition. If the exponential map satisfies the path-continuation property—a natural topological condition—these lifts extend globally, yielding a general path-lifting theorem. Some relevant applications will be discussed.
Jónatan Herrera
Lightlike convexity and its role in spacetime causality
This talk aims to explore the interplay between three concepts associated with a spacetime with timelike boundary: the (light) convexity of its boundary, the structure of the interior lightlike geodesic space, and its position within the so-called causal ladder. The presentation is structured as follows. We introduce the space of lightlike geodesics and equip it with a topological structure. We then define several notions of convexity for the boundary of a spacetime with timelike boundary. Finally, we discuss the interactions among these elements. This talk is based on the following work: https://arxiv.org/pdf/2506.09032
Miguel Prados Abad
Timelike ideal boundary of a non-positively curved Lorentzian Length Spaces
We study the notion of timelike ideal boundary of a Lorentzian Length Space with non-positive curvature. We establish metric curvature bounds of such an object and consider the case of generalized cones, studying the relation between the timelike ideal boundary and the metric ideal boundary of the fiber.
Saul Burgos
Temporal functions and spacetime convergence
The first approach to the convergence of spacetimes based on the Lorentzian distance was taken by Noldus. Since then, many different approaches have been considered, most of them by considering the Lorentzian "metric" structure given by the time separation function. A cutting-edge idea, proposed by Sormani and Vega, where they transform a spacetime directly into a metric space using a time function to define the so-called null distance. In this talk we revisit the notion of time/temporal function and exploit the fact that, when one has a temporal function τ it is always possible to obtain a Riemannian metric on a spacetime which is "Wick-rotated" with respect to τ , which we compare with the null distance. Moreover, if this temporal function is good enough, the Wick-rotated metric is complete. This allows us to obtain a new characterization of global hyperbolicity and study spacetime convergence using Riemannian tools.
Miguel Vadillo Cambrón
Conformal Einstein spaces and conformally covariant operators
In this talk we give general neccessary and sufficient conditions to ensure that a pseudo-Riemannian manifold is conformal to an Einstein space. These conditions are algorithmic in the metric tensor whenever the Weyl endomorphism is invertible. Our conditions depend in an essential manner on the $\mathcal{C}$-connection. We also show how to construct conformally covariant, pseudo-differential operators which has an independent interest.
Simone Vincini
Causally convex functions and gradient flows on Lorentzian manifolds
Convex functions and their gradient flows are central concepts in nonsmooth analysis and nonsmooth geometry, often allowing to bypass smoothness assumptions. In positive signature, they play a big role in the study of Alexandrov spaces and spaces with Ricci curvature bounded from below; by contrast, in Lorentzian geometry no analogue seems to be available. We introduce a notion of convexity, causal convexity, tailored to Lorentzian signature and we study the analytical properties of causally convex functions on globally hyperbolic spacetimes. We then formulate two notions of gradient flow for these functions, one akin to the energy dissipation inequality in positive signature, the other encoding a differential inclusion, similar to the EVI notion. Finally, we prove general stability, existence and uniqueness results for these flows and clarify the relationship between the two notions. This is a work in collaboration with Mathias Braun, Nicola Gigli, Robert McCann and Matteo Zanardini.
Juan Flores Torres
On Lorentz Supertransformations
The Lorentz group of two-dimensional Minkowski spacetime, O(1,1), a one-dimensional Lie group whose Lie algebra \mathfrak{o}(1,1) is likewise one-dimensional. It is well known that there exist precisely three Lie superalgebras of dimension 1|1, denoted \mathfrak{a}, \mathfrak{s}, \mathfrak{q}. This leads to three natural extensions of \mathfrak{o}(1,1) to Lie superalgebras of dimension 1|1. In this talk, we construct and analyze the corresponding extensions of the Lorentz group O(1,1) to Lie supergroups O(1,1), \mathcal{A}_{O(1,1)}) associated with the superalgebras \mathfrak{a}, \mathfrak{s}, \mathfrak{q}. We further investigate how these supergroups act on the supermanifold \mathbb{R}^{2|2}, describing the induced fundamental supervector fields and their integral flows.
Leonardo Garcia Heveling
Conformal transformations of spacetimes without observer horizons
The "no observer horizons" (NOH) condition on a spacetime, inspired by cosmology, means that there is a single point in the future/past causal boundary. Conformal transformations of such spacetimes are quite rigid. In a recent preprint, we classified them into two types: those that send (necessarily all) points towards infinity, and those that preserve compact sets. We will discuss this result and its implications for the Einstein static universe (which has a large conformal group), and formulate a version of the Lichnerowicz conjecture for NOH spacetimes. The latter states that, up to conformal equivalence, the only NOH spacetime with conformal group strictly larger than its isometry group is the Einstein static universe. Based on joint work with Abdelghani Zeghib.
Eduardo Hafemann
A low-regularity Riemannian positive mass theorem for non-spin manifolds with distributional curvature
In this talk, we investigate whether metrics with nonnegative scalar curvature in the distributional sense share any good properties with smooth metrics that have nonnegative scalar curvature. In particular, I will present a positive mass theorem for asymptotically flat manifolds with $C^0 \cap W^{1,n}_{\mathrm{loc}}$ metrics that are smooth outside a compact set, removing the spin assumption from the main theorem of Lee-LeFloch (2015) in this setting. The proof uses smoothing of the metric and a Sobolev version of Friedrichs' Lemma, which controls the negative part of the scalar curvature for the approximations and allows the conformal method to be applied at low regularity. Rigidity for $C^0 \cap W^{1,p}_{\mathrm{loc}}$ metrics, $p>n$, follows from comparison theory of $\sf{RCD}$-spaces and a rigidity theorem of Jiang-Sheng-Zhang (2023). If time permits, the connection between this work and the coordinate isoperimetric mass will also be discussed as part of joint work with Melanie Graf.
Karim Mosani
C^0-inextendibility of static warped-product black hole spacetimes
We investigate how far Sbierski’s proof of C^0 -inextendibility for Schwarzschild spacetime can be adapted to a much broader class of static warped-product black hole spacetimes with a curvature singularity as r\to 0. This class includes models with matter fields, and is therefore not restricted to the vacuum setting, while also going beyond spherical symmetry. This is an ongoing joint work with Clemens Saemann.
Marta Sálamo Candal
Comparison theory for Lipschitz spacetimes
In this talk, I will present comparison theorems for Lipschitz spacetimes in sharp form: d’Alembert, timelike Brunn–Minkowski, timelike Bishop–Gromov, and timelike Bonnet–Myers. The three first results are obtained as a consequence of our main theorem, in which we show that a globally hyperbolic spacetime with locally Lipschitz continuous metric and timelike distributional Ricci curvature bounded from below obeys the timelike measure contraction property. The timelike Bonnet–Myers is obtained parallely using the localization technique from convex geometry. Our framework covers a remarkable class of examples of spacetimes, including impulsive gravity waves, thin shells, and matched spacetimes. This talk is based on joint work with Mathias Braun.
Giulio Sanzeni
On causality of Kerr-Newman spacetimes
Kerr–Newman spacetimes are stationary and axisymmetric solutions of the Einstein–Maxwell field equations, characterized by mass M, angular momentum a, and electric charge e. Their analytic extension beyond the horizons, into the negative radial region, contains closed causal curves for any non-zero a, leading to violations of causality. Nevertheless, it has recently been shown that no causal geodesic can be closed in the sub-extremal uncharged case e=0. When e \neq 0, the motion of charged massive particles is governed by the Lorentz Force Equation (LFE) rather than by geodesics equation. In this talk, I investigate whether the causality violations present in Kerr–Newman spacetimes can be realized by charged timelike trajectories solutions of the LFE, arising from the interaction between charged particles and a rotating charged black holes. This is a joint project with Erasmo Caponio and Stefan Suhr.
Christian Lange
Quantitative Lorentzian isoperimetric inequalities
Stability analysis of geometric inequalities investigates how close an almost optimizer is to an optimizer. A classical example of interest is the Euclidean isoperimetric inequality whose almost optimizers are close to round balls in a volume sense. In the talk I will discuss different Lorentzian isoperimetric inequalities and their stability. In particular, I will explain how the causal structure leads to a Hausdorff stability estimate measured in terms of a natural Hausdorff-type metric on the space of Cauchy hypersurfaces. The talk is based on joint work with Jonas W. Peteranderl (LMU Munich).
Mahdi Haghshenas
On the stability of decelerated FLRW spacetimes
The standard cosmological models describing a homogeneous and isotropic universe in general relativity are represented by the Friedmann-Lemaître-Robertson-Walker (FLRW) spacetimes. In this talk, after outlining the stability problem for FLRW spacetimes, we consider the wave equation—as a proxy for the Einstein equations—on decelerated FLRW spacetimes with $\mathbb{R}^3$ spatial sections. We demonstrate how dispersion and expansion affect the long-time behaviour of waves. In particular, we present uniform energy bounds and integrated local energy decay estimates across the full decelerated expansion range. Moreover, we describe a hierarchy of r^p-weighted energy estimates, in the spirit of the Dafermos-Rodnianski r^p-method, which lead to energy decay estimates.
Sjors
Einstein-Kropina metrics and their application in Finsler Gravity
Finsler gravity is the extension of general relativity obtained by enlarging the class of allowed spacetime geometries from pseudo-Riemannian manifolds to Finsler manifolds of Lorentzian signature. The notion of an Einstein metric generalizes directly to the Finsler setting as well. In contrast to the pseudo-Riemannian case, however, not that much is known about these Einstein-Finsler metrics and only few explicit examples are known, especially in Lorentzian signature. In this talk, we focus on a particular class of Finsler metrics, the so-called `Kropina metrics'. First, we give necessary and sufficient conditions for such a metric to be of Einstein type, extending Zhang and Shen's characterization to all signatures. As a result, we are able to construct new explicit examples of Lorentzian Einstein-Finsler metrics. Second, we discuss the role of such metrics in Finsler gravity. In particular, we classify all solutions of Einstein-Kropina type to Pfeifer and Wohlfarth's vacuum field equation for Finsler gravity with cosmological constant.
Moritz Reintjes
On the essential regularity of singular geometries
The question whether a singularity is removable by coordinate transformation has been of central importance in General Relativity since Schwarzschild's discovery of black hole solution in 1916. However, beyond ad-hoc coordinate constructions, General Relativity lacks a unifying theory for identifying when singularities are removable, and it lacks a general procedure for removing them. B. Temple and I recently discovered a system of elliptic partial differential equations, the RT-equations, which provides a definitive theory for identifying and removing singularities above a threshold regularity, (including cusp and shock wave singularities in GR, but not yet those at black hole horizons). That is, based on the RT-equations, we developed a necessary and sufficient condition for when a singularity in an affine connection (the basic object of a geometry) in L^p is removable by coordinate transformation, together with a computable procedure for removing the singularity by regularizing the connection all the way up to its essential (highest possible) regularity. More generally, the RT-equations apply to singular connections on vector bundles of Yang-Mills gauge theories, and imply that connections with L^p curvature can always be regularized to one derivative above L^p; based on this, we gave the first extension of Uhlenbeck compactness from Riemannian to Lorentzian geometry.
Martín de la Rosa Díaz
Dynamics of torsion-dependent particle models in three-dimensional pp-wave spacetimes
Title: Dynamics of torsion-dependent particle models in three-dimensional pp-wave spacetimes. Abstract: In this talk, we present a work-in-progress study of the dynamics of a relativistic particle model which generalizes the well-known geodesic motion model by adding a term that depends on the second Frenet curvature, or torsion, of the trajectory. The analysis of models of this kind was popularized by M. S. Plyushchay [1] and have since been investigated by both physicists [2] and mathematicians [3, 4, 5]. We will discuss the trajectories of such a model in three-dimensional pp-wave spacetimes [6], which are also of great interest due to their connection to gravitational waves. This is a joint work with Jónatan Herrera and Rafael M. Rubio.
References: [1] Plyushchay, M. S. (1989). Massive relativistic point particle with rigidity. International Journal of Modern Physics A, 4, 15, 3851–3865. [2] Nesterenko, V. V. (1995). Dynamics of relativistic particles with Lagrangians dependent on acceleration. Journal of Mathematical Physics 36, 5552. [3] Barros, M., Ferrández, A., Javaloyes, M. A., & Lucas, P. (2005). Relativistic particles with rigidity and torsion in D = 3 spacetimes. Classical and Quantum Gravity 22, 489–513. [4] Herrera, J., de la Rosa, M., & Rubio, R. M. (2022). Dynamics of relativistic particles with torsion in warped-product spacetimes. Journal of Physics A 55, 245201. [5] Herrera, J., de la Rosa, M., & Rubio, R. M. (2024). Relativistic particles in generalized Robertson–Walker spacetimes. International Journal of Geometric Methods in Modern Physics 21, 2450048. [6] Candela, J. A., Flores, J. L. & Sánchez, M. (2003). On general plane fronted waves. Geodesics. General Relativity and Gravitation 35, 631–649.
Erasmo Caponio
Massive particle surfaces and Jacobi-Randers metrics
The geometry of photon surfaces --timelike hypersurfaces trapping light rays-- is well-characterized in General Relativity by the condition of total umbilicity. However, realistic astrophysical environments around black holes also involve massive charged particles. Unlike photons, these particles are governed by the Lorentz force and depend on a fixed charge-to-mass ratio \rho. Moreover, if there exists a timelike Killing vector K field and the electromagnetic field is also K-invariant, they also have a well-defined specific energy \varepsilon. Recent literature has shown that surfaces trapping such particles, known as massive particle surfaces (MPS) satisfy an extrinsic condition of ``partial umbilicity''. In this talk, after revisiting this condition, I present a characterization of a Killing invariant MPS in a stationary spacetimes using Finsler geometry. To this end, we show that the dynamics of charged massive particles with fixed (\rho,\varepsilon) reduces to the geodesic flow of a Jacobi-Randers type metric defined on a spacelike slice. Then we obtain that an hypersurface of the type \mathbb{R} \times S_0 is a (\rho, \varepsilon)-MPS if and only if its spatial section S_0 is totally geodesic with respect to the associated Jacobi-Randers metric.
Salah Chaib
A Compact Complete Manifold Without Closed Geodesics
We present the construction of a compact complete pseudo-Riemannian manifold that does not admit closed geodesics. The example is obtained as a quotient of a Lie group endowed with a left-invariant metric and relies on specific dynamical features of the geodesic flow. This construction illustrates a striking contrast with the Riemannian setting and highlights new phenomena in pseudo-Riemannian geometry. This talk is based on joint work in progress with Souheib Allout.
Brien Nolan
Perturbations of Vaidya spacetime and the cosmic censorship hypothesis.
The Vaidya spacetime is a spherically symmetric solution of the Einstein equations with a null dust source. This can be used to model the gravitational collapse of a thick shell of radiation: a flat interior region is matched at an inner boundary to the null dust filled region, which is then matched at an outer boundary to Schwarzschild spacetime. A central singularity inevitably forms, and depending on the profile of the energy density of the null dust, this singularity can be globally naked. Motivated by the cosmic censorship hypothesis, we consider perturbations of this configuration. We review previous work, and describe recent work where the perturbation of the inner boundary - the past null cone of the central singularity - is analysed using a framework for studying perturbations of general hypersurfaces. This sets boundary conditions for perturbations at the past null cone, and we then consider the 3+1 evolutionary problem, focussing on the question of the stability of the Cauchy horizon of the naked singularity.
Filipe Nazaré
Elastic Reaction to Gravitational Waves
We study the interaction between gravitational waves and elastic bodies within the framework of relativistic elasticity. Starting from the Lagrangian formulation of relativistic elasticity, we derive the linearized equations governing the response of a homogeneous and isotropic solid to a weak gravitational wave by expanding the action to quadratic order in the elastic field derivatives and to linear order in the metric perturbation. In this way, Dyson’s interaction term from the effective potential approach emerges naturally from the relativistic theory. We then apply the formalism to a thin rectangular elastic plate aligned with the direction of propagation and polarization of a plus-polarized gravitational wave. For a material with vanishing Poisson ratio, the equations decouple and admit explicit solutions. We obtain closed-form expressions for the induced displacements and for the energy deposited on the plate by both short gravitational wave bursts and continuous monochromatic waves. Finally, we compute the gravitational wave emission generated by the oscillating plate itself under continuous excitation. These results provide a fully relativistic derivation of the elastic response to gravitational waves together with explicit solvable examples relevant to resonant gravitational wave detectors.
Francisco Pais
Boundedness and Decay of Solutions of the Massless Dirac Equation on a Schwarzschild Background
Black hole stability problems have been given considerable attention in the past decades. Central to this program is the well-know Teukolsky equation, a linearized version of the Einstein equations around the Kerr family whose analysis constitutes a stepping stone in the proof of more complicated nonlinear stability results. A remarkable observation of Teukolsky is that many physically-relevant equations, such as Klein-Gordon (|s| = 0), Dirac (|s| = 1/2), Maxwell (|s| = 1), and Rarita-Schwinger (|s| = 3/2), can be seen as special cases of a unified equation of motion -- the so-called Teukolsky master equation -- which parametrizes each of these equations by the spin-weight s of the respective field. The past few years have thus been populated by several results regarding the Teukolsky equation, namely in what concerns boundedness and decay of solutions with integer spin weight. In this talk, we will show how one can obtain similar results for the massless Dirac equation, equivalent to the spin |s| = 1/2 Teukolsky equations, on a Schwarzschild background, and how to overcome some of the main obstructions that have impeded the analysis for some time now, such as the lack of a weak energy condition. Time-permitting, we will make some comments about the other half-integer spin equation (Rarita-Schwinger), as well as (obstructions to) the generalization of the results presented to the Kerr family.
Davide Manini
On the geometry of synthetic null hypersurfaces and the Null Energy Condition
In this talk, I will present a joint work with Fabio Cavalletti (Milan) and Andrea Mondino (Oxford), where we develop a synthetic framework for the geometric and analytic study of null (lightlike) hypersurfaces in non-smooth spacetimes. Drawing from optimal transport and recent advances in Lorentzian geometry and causality theory, we define a synthetic null hypersurface as a triple (H, G, m): H is a closed achronal set in a topological causal space, G is a gauge function encoding affine parametrizations along null generators, and m is a Radon measure serving as a synthetic analog of the rigged measure. This generalizes classical differential geometric structures to potentially singular spacetimes. A central object is the synthetic null energy condition (NC^e(N )), defined via the concavity of an entropy power functional along optimal transport, with parameterization given by the gauge G. This condition is invariant under changes of gauge and measure within natural equivalence classes. It agrees with the classical Null Energy Condition in the smooth setting and it applies to low-regularity spacetimes. A key property of NC^e(N ) is the stability under convergence of synthetic null hypersurfaces, inspired by measured Gromov--Hausdorff convergence. As a first application, we obtain a synthetic version of Hawking’s area theorem. Moreover, we extend the celebrated Penrose singularity theorem to continuous spacetimes and we prove the existence of trapped regions in the general setting of topological causal spaces satisfying the synthetic null energy condition.