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

Tobias Beran, Mathias Braun, Matteo Calisti, Nicola Gigli, Robert J. McCann, Argam Ohanyan, Felix Rott, Clemens Sämann

TL;DR

This work extends Lorentzian geometry to nonsmooth metric measure spacetimes by developing a variational first-order Sobolev calculus driven by the maximal weak subslope |d f|, which serves as the Lorentzian modulus of df. A nonlinear Lagrangian–Hamiltonian duality and the notion of infinitesimal Minkowskianity provide a robust framework for cotangent/tangent duality, horizontal and vertical derivatives, and weak p-d'Alembertian notions. The authors establish a weak p-d'Alembert comparison under timelike curvature-dimension bounds TMCP, generalizing Eschenburg-style estimates beyond the cut locus, and develop a Lorentzian lifting theory that connects curves of measures to dynamical plans, enabling Brenier–McCann-type characterizations in this nonsmooth setting. They further integrate synthetic timelike curvature bounds with a detailed non-branching theory and a rich calculus of perturbations, chain and Leibniz rules, offering a foundation for nonsmooth Lorentzian analysis with potential implications for spacetime splitting results and quantum gravity formalisms.

Abstract

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.

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

TL;DR

This work extends Lorentzian geometry to nonsmooth metric measure spacetimes by developing a variational first-order Sobolev calculus driven by the maximal weak subslope |d f|, which serves as the Lorentzian modulus of df. A nonlinear Lagrangian–Hamiltonian duality and the notion of infinitesimal Minkowskianity provide a robust framework for cotangent/tangent duality, horizontal and vertical derivatives, and weak p-d'Alembertian notions. The authors establish a weak p-d'Alembert comparison under timelike curvature-dimension bounds TMCP, generalizing Eschenburg-style estimates beyond the cut locus, and develop a Lorentzian lifting theory that connects curves of measures to dynamical plans, enabling Brenier–McCann-type characterizations in this nonsmooth setting. They further integrate synthetic timelike curvature bounds with a detailed non-branching theory and a rich calculus of perturbations, chain and Leibniz rules, offering a foundation for nonsmooth Lorentzian analysis with potential implications for spacetime splitting results and quantum gravity formalisms.

Abstract

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 -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.
Paper Structure (38 sections, 89 theorems, 378 equations)

This paper contains 38 sections, 89 theorems, 378 equations.

Table of Contents

  1. Introduction
  2. Order-completeness; tangent notions for worldlines, curves of measures, and measures over worldlines
  3. Smooth paradigm: Lagrangian-Hamiltonian duality
  4. The Sobolev cone of cotangent fields; infinitesimal Minkowskianity
  5. Nonsmooth Lagrangian--Hamiltonian duality
  6. Timelike Ricci curvature bounds and applications
  7. Two-sided perturbations, hyperbolic norms, and compatibility with smooth notions
  8. Curves in the nonsmooth Lorentzian setting
  9. Infinity conventions
  10. Notions of spacetimes
  11. The spacetime of Borel probability measures
  12. Causal speed
  13. Left-continuous causal paths and their topology
  14. Action and geodesics
  15. Lifting curves of probability measures to measures on curves
  16. ...and 23 more sections

Key Result

Lemma 2.4

Let $({\rm M},\uptau,\ell)$ be a Polish metric spacetime. Then emeralds are closed. If moreover $\ell_+$ is real valued and upper semicontinuous, then it is bounded above on each emerald.

Theorems & Definitions (234)

  • Definition 1.1: Lorentzian $\mathsf{TCD}$ spaces
  • Remark 1.2: Beyond global hyperbolicity and $q<0$
  • Definition 1.3: Weak subslope
  • Definition 1.4: Infinitesimal Minkowskianity
  • Definition 2.1: Notions of spacetimes
  • Remark 2.2: The choice of the topology
  • Remark 2.3: Push-up property
  • Lemma 2.4: Upper semicontinuous time separations attain maxima on emeralds
  • proof
  • Lemma 2.5: Rough timelike geodesy yields lower semicontinuous time to a point
  • ...and 224 more