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Clemens Sämann

Publications and source records attributed to Clemens Sämann.

At least 19 recordsLinked to original sources
arXiv2026

Stability of Synthetic Timelike Ricci Bounds under $C^0$-Limits and Applications to Impulsive Gravitational Waves

We investigate the stability of timelike Ricci curvature lower bounds under low-regularity limits of Lorentzian metrics. Specifically, we prove that the synthetic curvature-dimension condition $\mathsf{TCD}^e_p(K,N)$, which provides an optimal transport formulation of the Hawking-Penrose strong energy condition, is stable under locally uniform convergence of smooth Lorentzian metrics, provided a uniform global hyperbolicity assumption holds. As a consequence, smooth locally uniform limits of vacuum spacetimes satisfy the strong energy condition, even though curvature is not controlled a priori. As a main application, we study impulsive gravitational waves - spacetimes with Lipschitz continuous metrics - and show that large classes of such waves satisfy synthetic timelike Ricci curvature lower bounds. In the case of Minkowski background, we further establish synthetic upper Ricci curvature bounds. Our approach relies on constructing suitable smooth approximations with lower bounds on the timelike Ricci, and analyzing the limiting behavior via Lorentzian optimal transport. We also apply the aforementioned stability theorem to weak solutions of the Einstein equations arising from the nonlinear interaction of impulsive gravitational waves. These results yield new geometric insights into low-regularity solutions of the Einstein equations.

gr-qc↗
arXiv2026

Doubling for chronological diamonds in Lorentzian geometry

We define doubling conditions for measured Lorentzian length spaces in terms of chronological diamonds, and prove that such conditions are implied by suitable timelike curvature bounds. In particular, for the first time, we relate doubling to curvature bounds in Lorentzian geometry. As a consequence, we obtain compactness results for Lorentzian generalized cones with respect to the Lorentzian Gromov--Hausdorff convergence introduced by Mondino--Sämann.

math.DG↗
arXiv2026

Curvature bounds, regularity and inextendibility of spacetimes

We provide a completely new relation between curvature bounds and definiteness of the causal character of maximizers by exploiting the robust notion of synthetic curvature. This enables us to relate low-regularity inextendibility of spacetimes to unboundedness of curvature - which is at present unattainable using classical methods - thereby strengthening and complementing the results of Grant-Kunzinger-Saemann (2019) significantly.

gr-qc↗
arXiv2026

Lorentzian Gromov-Hausdorff convergence and pre-compactness

The goal of the paper is to introduce a convergence à la Gromov-Hausdorff for Lorentzian spaces, building on $ε$-nets consisting of causal diamonds and relying only on the time separation function. This yields a geometric notion of convergence, which can be applied to synthetic Lorentzian spaces (Lorentzian pre-length spaces) or smooth spacetimes. Among the main results, we prove a Lorentzian counterpart of the celebrated Gromov's pre-compactness theorem for metric spaces, where controlled covers by balls are replaced by controlled covers by diamonds. This yields a geometric pre-compactness result for classes of globally hyperbolic spacetimes, satisfying a uniform doubling property on Cauchy hypersurfaces and a suitable control on the causality, and a curvature-driven pre-compactness result. The final part of the paper establishes several applications: we show that Chruściel-Grant approximations are an instance of the Lorentzian Gromov-Hausdorff convergence here introduced, we prove that timelike sectional curvature bounds are stable under such a convergence, we introduce timelike blow-up tangents and discuss connections with the main conjecture of causal set theory.

math.DG↗
arXiv2026

An elliptic proof of the splitting theorems from Lorentzian geometry

We provide a new proof of the splitting theorems from Lorentzian geometry, in which simplicity is gained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity (non-uniformly) elliptic $p$-d'Alembert operator for this purpose. This allows us to bring the Eschenburg, Galloway, and Newman Lorentzian splitting theorems into a framework closer to the Cheeger-Gromoll splitting theorem from Riemannian geometry.

math.DG↗
arXiv2026

Generalized cones admitting a curvature-dimension condition

We study (generalized) cones over metric spaces, both in Riemannian and Lorentzian signature. In particular, we establish synthetic lower Ricci curvature bounds à la Lott-Villani-Sturm and Ohta in the metric measure case, and à la Cavalletti-Mondino in Lorentzian signature. Here, a generalized cone is a warped product of a one-dimensional base space, which will be positive or negative definite, over a fiber that is a metric space. We prove that Riemannian or Lorentzian generalized cones over $\mathsf{CD}$-spaces satisfy the (timelike) measure contraction property $\mathsf{(T)MCP}$ - a weaker version of a (timelike) curvature-dimension condition $\mathsf{(T)CD}$. Conversely, if the generalized cone is a $\mathsf{(T)CD}$-space, then the fiber is a $\mathsf{CD}$-space with the appropriate bounds on Ricci curvature and dimension. In proving these results we develop a novel and powerful two-dimensional localization technique, which we expect to be interesting in its own right and useful in other circumstances. We conclude by giving several applications including synthetic singularity and splitting theorems for generalized cones. The final application is that we propose a new definition for lower curvature bounds for metric and metric measure spaces via lower curvature bounds for generalized cones over the given space.

math.DG↗
arXiv2026

Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds

Within the synthetic-geometric framework of Lorentzian (pre-)length spaces developed in Kunzinger and Sämann (Ann. Glob. Anal. Geom. 54(3):399--447, 2018) we introduce a notion of a hyperbolic angle, an angle between timelike curves and related concepts like timelike tangent cone and exponential map. This provides valuable technical tools for the further development of the theory and paves the way for the main result of the article, which is the characterization of timelike curvature bounds (defined via triangle comparison) with an angle monotonicity condition. Further, we improve on a geodesic non-branching result for spaces with timelike curvature bounded below.

math.DG↗
arXiv2025

Conformal transformations of metric spaces and Lorentzian pre-length spaces

We introduce conformal transformations in the synthetic setting of metric spaces and Lorentzian (pre-)length spaces. Our main focus lies on the Lorentzian case, where, motivated by the need to extend classical notions to spaces of low regularity, we provide the first consistent notion of conformal length, and analyse its fundamental properties. We prove that the conformal time separation function $τ_Ω$ (and the causal structure it induces) yields a Lorentzian pre-length structure if the original space is intrinsic and strongly causal. This allows us to construct a notion of conformal transformation between spaces within this class, yielding an equivalence relation. As applications, we show that the conformal length functional agrees with the standard conformal length of (strongly causal) spacetimes. We also prove conformal invariance of angles and causality conditions, give a characterisation of global hyperbolicity via finiteness of $τ_Ω$ for all conformal factors, and establish the behaviour of the Lorentzian Hausdorff measure defined in [MS22a] under conformal changes. Moreover, we apply the same methods to the metric case, which is of interest in its own right. This is exemplified by proving an analog of the Nomizu--Ozeki theorem for metric length spaces, which has the advantage that the resulting complete space is conformally related to the original space.

math.DG↗
arXiv2025

A Lorentzian splitting theorem for continuously differentiable metrics and weights

We prove a splitting theorem for globally hyperbolic, weighted spacetimes with metrics and weights of regularity $C^1$ by combining elliptic techniques for the negative homogeneity $p$-d'Alembert operator from our recent work in the smooth setting with the concept of line-adapted curves introduced here. Our results extend the Lorentzian splitting theorem proved for smooth globally hyperbolic spacetimes by Galloway -- and variants of its weighted counterparts by Case and Woolgar--Wylie -- to this low regularity setting.

math.DG↗
arXiv2025

Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry

We establish Gromov's celebrated reconstruction theorem in Lorentzian geometry. Alongside this result, we introduce and study a natural concept of isomorphy of normalized bounded Lorentzian metric measure spaces. We outline applications to the spacetime reconstruction problem from causal set theory. Lastly, we propose three notions of convergence of (isomorphism classes of) normalized bounded Lorentzian metric measure spaces, for which we prove several fundamental properties.

math.DG↗
arXiv2025

A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes

We introduce a variational first-order Sobolev calculus on metric measure spacetimes. The key object is the maximal weak subslope of an arbitrary causal function, which plays the role of the (Lorentzian) modulus of its differential. It is shown to satisfy certain chain and Leibniz rules, certify a locality property, and be compatible with its smooth analog. In this setup, we propose a quadraticity condition termed infinitesimal Minkowskianity, which singles out genuinely Lorentzian structures among Lorentz-Finsler spacetimes. Moreover, we establish a comparison theorem for a nonlinear yet elliptic $p$-d'Alembertian in a weak form under the timelike measure contraction property. As a particular case, this extends Eschenburg's classical estimate past the timelike cut locus.

math.DG↗
arXiv2024

Cut-and-paste for impulsive gravitational waves with $Λ$: The mathematical analysis

Impulsive gravitational waves are theoretical models of short but violent bursts of gravitational radiation. They are commonly described by two distinct spacetime metrics, one of local Lipschitz regularity, the other one even distributional. These two metrics are thought to be `physically equivalent' since they can be formally related by a `discontinuous coordinate transformation'. In this paper we provide a mathematical analysis of this issue for the entire class of nonexpanding impulsive gravitational waves propagating in a background spacetime of constant curvature. We devise a natural geometric regularisation procedure to show that the notorious change of variables arises as the distributional limit of a family of smooth coordinate transformations. In other words, we establish that both spacetimes arise as distributional limits of a smooth sandwich wave taken in different coordinate systems which are diffeomorphically related.

gr-qc↗
arXiv2021

A Lorentzian analog for Hausdorff dimension and measure

We define a one-parameter family of canonical volume measures on Lorentzian (pre-)length spaces. In the Lorentzian setting, this allows us to define a geometric dimension - akin to the Hausdorff dimension for metric spaces - that distinguishes between e.g. spacelike and null subspaces of Minkowski spacetime. The volume measure corresponding to its geometric dimension gives a natural reference measure on a synthetic or limiting spacetime, and allows us to define what it means for such a spacetime to be collapsed (in analogy with metric measure geometry and the theory of Riemannian Ricci limit spaces). As a crucial tool we introduce a doubling condition for causal diamonds and a notion of causal doubling measures. Moreover, applications to continuous spacetimes and connections to synthetic timelike curvature bounds are given.

math-ph↗
arXiv2021

Generalized cones as Lorentzian length spaces: Causality, curvature, and singularity theorems

We study generalizations of Lorentzian warped products with one-dimensional base of the form $I\times_f X$, where $I$ is an interval, $X$ is a length space and $f$ is a positive continuous function. These generalized cones furnish an important class of Lorentzian length spaces in the sense of [Kunzinger, Sämann; Ann. Glob. Anal. Geom. 54(3):399--447, 2018], displaying optimal causality properties that allow for explicit descriptions of all underlying notions. In addition, synthetic sectional curvature bounds of generalized cones are directly related to metric curvature bounds of the fiber $X$. The interest in such spaces comes both from metric geometry and from General Relativity, where warped products underlie important cosmological models (FLRW spacetimes). Moreover, we prove singularity theorems for these spaces, showing that non-positive lower timelike curvature bounds imply the existence of incomplete timelike geodesics.

math.MG↗
arXiv2020

Lorentz meets Lipschitz

We show that maximal causal curves for a Lipschitz continuous Lorentzian metric admit a $\mathcal{C}^{1,1}$-parametrization and that they solve the geodesic equation in the sense of Filippov in this parametrization. Our proof shows that maximal causal curves are either everywhere lightlike or everywhere timelike. Furthermore, the proof demonstrates that maximal causal curves for an $α$-Hölder continuous Lorentzian metric admit a $\mathcal{C}^{1,\fracα{4}}$-parametrization.

math.DG↗
arXiv2019

Global hyperbolicity for spacetimes with continuous metrics

We show that the definition of global hyperbolicity in terms of the compactness of the causal diamonds and non-total imprisonment can be extended to spacetimes with continuous metrics, while retaining all of the equivalences to other notions of global hyperbolicity. In fact, global hyperbolicity is equivalent to the compactness of the space of causal curves and to the existence of a Cauchy hypersurface. Furthermore, global hyperbolicity implies causal simplicity, stable causality and the existence of maximal curves connecting any two causally related points.

math.DG↗
arXiv2019

Lorentzian length spaces

We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT$(k)$-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

math.DG↗