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Non-smooth spacetime geometry

Non-smooth spacetime geometry

Roland Steinbauer (ORCID: 0000-0001-8972-7502)
  • Grant DOI 10.55776/P33594
  • Funding program Principal Investigator Projects
  • Status ended
  • Start February 1, 2021
  • End January 31, 2025
  • Funding amount € 389,308
  • E-mail

Disciplines

Mathematics (70%); Physics, Astronomy (30%)

Keywords

    Mathematical General relativity, Low regularity, Singularity Theorems, Lorentzian causality theory, Metric Geometry, Lorentzian length spaces

Abstract

General Relativity (GR), Einsteins theory of space, time and matter describes gravity via spacetime curvature: Every form of mass or energy curves the surrounding spacetime and this curvature in turn governs the motion of particles through spacetime. GR continues to be a spectacular success also in the 21st century. Indeed, around the time of its centenary in 2015, one of its mayor predictions, the existence of gravitational waves, has been directly confirmed. On the other hand, GR, over the decades, has also exerted a strong influence on various fields of mathematics with many cross-fertilizations leading to milestone results as e.g. the singularity theorems of Penrose and Hawking. These results predict, under physically reasonable conditions, the occurrence of spacetime singularities such as black holes. One of the problems on this interface between physics and mathematics concerns the very language in which GR is formulated: Lorentzian geometry (LG) is the mathematical theory that describes curved geometries, which are the stage for GR. Its central object is the metric, which describes how lengths and angles are measured at different points in spacetime. Traditionally, LG has been formulated for smooth metrics only, that is, for metrics that vary continuously when one travels from point to point in spacetime. On the other hand, physically realistic models of e.g. stars, demand that the mass density jumps at their surface. This however, via the fundamental principles of GR, forces the metric to vary abruptly as well and hence to be non-smooth. This fundamental issue has of course been well-known for a long time, but has only rarely been addressed in the literature. This has dramatically changed roughly a decade ago when researchers in Mathematical Relativity started to systematically investigate non-smooth metrics. Since then a growing number of woks has appeared and turned the study of non-smooth metrics into a thriving line of research, with the project team playing a key role. In a previous project we have, on the one hand, proved the classical singularity theorems of Penrose and Hawking for (certain) non-smooth metrics. On the other hand, we have put forward a new approach to LG which is based on methods of synthetic geometry, where e.g. curvature is measured by how much the sum of angles in triangles deviates from the flat case (180 degrees). In this new project we aim at enhancing and generalizing the methods developed previously proving advanced singularity theorems, especially where the energy might become negative locally. The latter provides an interface to the (yet unfinished) quantum theory of gravity where energies are expected to fluctuate and to become negative. Also we want to turn the new synthetic approach to LG, the so-called Lorentzian length spaces into a mature theory. In short, we will develop a synthesis of analytic and synthetic methods to significantly push the understanding of non-smooth spacetime geometries.

Research institution(s)
  • Universität Wien - 100%
Project participants
  • Clemens Sämann, Universität Wien , national collaboration partner
  • Michael Kunzinger, Universität Wien , national collaboration partner
International project participants
  • Robert J. Mccann, University of Toronto - Canada
  • Annegret Burtscher, University of Nijmegen - Netherlands
  • Melanie Graf, University of Washington - USA

Research Output

  • 132 Citations
  • 21 Publications
Publications
  • 2025
    Title A Toponogov globalisation result for Lorentzian length spaces
    DOI 10.1007/s00208-025-03167-w
    Type Journal Article
    Author Beran T
    Journal Mathematische Annalen
    Pages 1-32
    Link Publication
  • 2024
    Title Cut-and-paste for impulsive gravitational waves with ?: the mathematical analysis
    DOI 10.1007/s11005-024-01804-0
    Type Journal Article
    Author Sämann C
    Journal Letters in Mathematical Physics
    Pages 58
    Link Publication
  • 2024
    Title On curvature bounds in Lorentzian length spaces
    DOI 10.1112/jlms.12971
    Type Journal Article
    Author Beran T
    Journal Journal of the London Mathematical Society
    Link Publication
  • 2024
    Title Examples of cosmological spacetimes without CMC Cauchy surfaces
    DOI 10.1007/s11005-024-01843-7
    Type Journal Article
    Author Ling E
    Journal Letters in Mathematical Physics
    Pages 96
    Link Publication
  • 2023
    Title Hyperbolic angles in Lorentzian length spaces and timelike curvature bounds
    DOI 10.1112/jlms.12726
    Type Journal Article
    Author Beran T
    Journal Journal of the London Mathematical Society
    Pages 1823-1880
    Link Publication
  • 2022
    Title Null Distance and Convergence of Lorentzian Length Spaces
    DOI 10.1007/s00023-022-01198-6
    Type Journal Article
    Author Kunzinger M
    Journal Annales Henri Poincaré
    Pages 4319-4342
    Link Publication
  • 2022
    Title Mapping method of group classification
    DOI 10.1016/j.jmaa.2022.126209
    Type Journal Article
    Author Opanasenko S
    Journal Journal of Mathematical Analysis and Applications
    Pages 126209
    Link Publication
  • 2022
    Title Strong Traces to Degenerate Parabolic Equations
    DOI 10.1137/21m1425530
    Type Journal Article
    Author Erceg M
    Journal SIAM Journal on Mathematical Analysis
    Pages 1775-1796
    Link Publication
  • 2023
    Title Gluing of Lorentzian length spaces and the causal ladder
    DOI 10.1088/1361-6382/ace585
    Type Journal Article
    Author Rott F
    Journal Classical and Quantum Gravity
    Pages 175002
    Link Publication
  • 2023
    Title Gluing constructions for Lorentzian length spaces
    DOI 10.1007/s00229-023-01469-4
    Type Journal Article
    Author Beran T
    Journal manuscripta mathematica
    Pages 667-710
    Link Publication
  • 2023
    Title The splitting theorem for globally hyperbolic Lorentzian length spaces with non-negative timelike curvature
    DOI 10.1007/s11005-023-01668-w
    Type Journal Article
    Author Beran T
    Journal Letters in Mathematical Physics
    Pages 48
    Link Publication
  • 2023
    Title Timelike Ricci bounds for low regularity spacetimes by optimal transport
    DOI 10.1142/s0219199723500499
    Type Journal Article
    Author Braun M
    Journal Communications in Contemporary Mathematics
    Pages 2350049
    Link Publication
  • 2024
    Title Marginally outer trapped tubes in de Sitter spacetime
    DOI 10.1007/s11005-024-01884-y
    Type Journal Article
    Author Mars M
    Journal Letters in Mathematical Physics
    Pages 141
    Link Publication
  • 2024
    Title Hawking-Type Singularity Theorems for Worldvolume Energy Inequalities
    DOI 10.1007/s00023-024-01502-6
    Type Journal Article
    Author Graf M
    Journal Annales Henri Poincaré
    Pages 1-36
    Link Publication
  • 2022
    Title The Singularity Theorems of General Relativity and Their Low Regularity Extensions
    DOI 10.1365/s13291-022-00263-7
    Type Journal Article
    Author Steinbauer R
    Journal Jahresbericht der Deutschen Mathematiker-Vereinigung
    Pages 73-119
    Link Publication
  • 2022
    Title Penrose junction conditions with ?: geometric insights into low-regularity metrics for impulsive gravitational waves
    DOI 10.1007/s10714-022-02977-6
    Type Journal Article
    Author Podolský J
    Journal General Relativity and Gravitation
    Pages 96
    Link Publication
  • 2024
    Title The equivalence of smooth and synthetic notions of timelike sectional curvature bounds
    DOI 10.1090/proc/17022
    Type Journal Article
    Author Beran T
    Journal Proceedings of the American Mathematical Society
    Pages 783-797
    Link Publication
  • 2022
    Title Velocity averaging for diffusive transport equations with discontinuous flux
    DOI 10.1112/jlms.12694
    Type Journal Article
    Author Erceg M
    Journal Journal of the London Mathematical Society
    Pages 658-703
    Link Publication
  • 2022
    Title The Hawking–Penrose Singularity Theorem for C1-Lorentzian Metrics
    DOI 10.1007/s00220-022-04335-8
    Type Journal Article
    Author Kunzinger M
    Journal Communications in Mathematical Physics
    Pages 1143-1179
    Link Publication
  • 2021
    Title A note on the Gannon–Lee theorem
    DOI 10.1007/s11005-021-01481-3
    Type Journal Article
    Author Schinnerl B
    Journal Letters in Mathematical Physics
    Pages 142
    Link Publication
  • 2021
    Title Causal simplicity and (maximal) null pseudoconvexity
    DOI 10.1088/1361-6382/ac2be1
    Type Journal Article
    Author Hedicke J
    Journal Classical and Quantum Gravity
    Pages 227002
    Link Publication

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