A new Geometry for Einstein’s Theory of Relativity & Beyond
A new Geometry for Einstein’s Theory of Relativity & Beyond
Disciplines
Mathematics (70%); Physics, Astronomy (30%)
Keywords
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Synthetic Lorentzian geometry,
Lorentzian length spaces,
Mathematical General Relativity,
Metric Geometry,
Optimal Transport,
Quantum Gravity
General Relativity, Einsteins famous theory of space, time and gravity, has one central message: gravity is the curvature of the universe or spacetime to be precise. The mathematical language in which we usually speak of spacetime curvature is Lorentzian Differential Geometry. It is the somewhat strangely behaved sister-theory of our everyday Euclidean (or Riemannian) Geometry: Lorentzian detours (measured in spacetime distance) are shorter rather than longer as is the case in the Riemannian setting using the usual notion of a distance. One essential drawback of Differential Geometry (Riemannian and Lorentzian alike) is that its central objects must be smooth: One may only speak of the curvature of very nice geometries without corners, edges or spikes. However, physics builds more often than not on rough, non-smooth models, thus providing a strong motivation for a non-smooth geometry. Luckily during the past decades, a powerful formalism that provides a very robust notion of curvature for the non-smooth Riemannian setting has been developed. Based on the mathematical theories of Metric Geometry and Optimal Transport, it has revolutionised Riemannian geometry. In this so-called synthetic setting the prime object is the distance function and curvature is encoded in the convexity properties of an entropy functional. In 2018 our research group formulated the foundations of a synthetic Lorentzian Geometry built on spacetime distance as its central concept. We have thereby built a bridge between the robust curvature framework of Metric Geometry/Optimal Transport and Lorentzian Geometry. Our vision is to cross this bridge and develop a new geometry to tackle some long-standing open problems in fundamental physics like the nature of spacetime singularities in General Relativity, and beyond it by providing a unifying language for approaches to Quantum Gravity that are fundamentally discrete.
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consortium member (01.10.2024 -)
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principal investigator (01.10.2024 -)
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consortium member (01.10.2024 -)
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consortium member (01.10.2024 -)
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consortium member (01.10.2024 -)
- Universität Wien - 100%
- Robert J. Mccann, University of Toronto - Canada
- Jiri Podolsky, Charles University Prague - Czechia
- Melanie Graf, Universität Hamburg - Germany
- Sumati Surya, Raman Research Institute, Bengaluru - India
- Nicola Gigli, SISSA - Italy
- Fabio Cavalletti, Università degli Studi di Milano - Italy
- Annegret Burtscher, University of Nijmegen - Netherlands
- Jan Sbierski, University of Edinburgh - United Kingdom
- Andrea Mondino, University of Oxford - United Kingdom
Research Output
- 1 Citations
- 19 Publications
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2025
Title Hawking's singularity theorem for Lipschitz Lorentzian metrics DOI 10.48550/arxiv.2501.18450 Type Preprint Author Calisti M Link Publication -
2025
Title On the Hauptvermutung of Causal Set Theory DOI 10.48550/arxiv.2503.01719 Type Preprint Author Müller O Link Publication -
2025
Title A synthetic Lorentzian Cartan-Hadamard theorem DOI 10.48550/arxiv.2506.22197 Type Preprint Author Erös D Link Publication -
2025
Title Gromov's reconstruction theorem and measured Gromov-Hausdorff convergence in Lorentzian geometry DOI 10.48550/arxiv.2506.10852 Type Preprint Author Braun M Link Publication -
2025
Title Particle approximation of nonlocal interaction energies DOI 10.48550/arxiv.2506.02905 Type Preprint Author Carazzato D Link Publication -
2025
Title Generalized cones admitting a curvature-dimension condition DOI 10.48550/arxiv.2506.02723 Type Preprint Author Calisti M Link Publication -
2025
Title Spacetime reconstruction by order and number DOI 10.48550/arxiv.2507.01907 Type Preprint Author Braun M Link Publication -
2025
Title Geodesic causality in Kerr spacetimes with |a|?=?M DOI 10.1016/j.geomphys.2025.105589 Type Journal Article Author Sanzeni G Journal Journal of Geometry and Physics Pages 105589 Link Publication -
2025
Title Quantum Geometry of the Light Cone: Fock representation and Spectrum of Radiated Power DOI 10.48550/arxiv.2504.10802 Type Preprint Author Wieland W Link Publication -
2025
Title Optimal transport on the sub-Lorentzian Heisenberg group DOI 10.48550/arxiv.2504.03062 Type Preprint Author Borza S Link Publication -
2025
Title Failure of the measure contraction property via quotients in higher-step sub-Riemannian structures DOI 10.48550/arxiv.2505.09681 Type Preprint Author Borza S Link Publication -
2025
Title Geodesic causality in Kerr spacetimes with $|a|\geq M$ DOI 10.48550/arxiv.2504.17763 Type Preprint Author Mosani K Link Publication -
2024
Title Marginally outer trapped tubes in de Sitter spacetime DOI 10.48550/arxiv.2407.10602 Type Preprint Author Mars M 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 Splitting theorems for weighted Finsler spacetimes via the $p$-d'Alembertian: beyond the Berwald case DOI 10.48550/arxiv.2412.20783 Type Preprint Author Caponio E Link Publication -
2024
Title Ricci curvature bounds and rigidity for non-smooth Riemannian and semi-Riemannian metrics DOI 10.48550/arxiv.2406.06762 Type Preprint Author Kunzinger M Link Publication -
2024
Title Gradient flows of $(K,N)$-convex functions with negative $N$ DOI 10.48550/arxiv.2412.04574 Type Preprint Author Magnabosco M Link Publication -
2024
Title A nonlinear d'Alembert comparison theorem and causal differential calculus on metric measure spacetimes DOI 10.48550/arxiv.2408.15968 Type Preprint Author Beran T Link Publication -
2024
Title An elliptic proof of the splitting theorems from Lorentzian geometry DOI 10.48550/arxiv.2410.12632 Type Preprint Author Braun M Link Publication