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

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arxiv 2408.15968 v2 pith:LJ3CITXD submitted 2024-08-28 math.DG gr-qcmath-phmath.MGmath.MP

Tobias Beran , Mathias Braun , Matteo Calisti , Nicola Gigli , Robert J. McCann , Argam Ohanyan , Felix Rott , Clemens Sämann This is my paper · ORCID
classification math.DGgr-qcmath-phmath.MGmath.MP
keywords measurespacetimescalculuscausalcomparisondifferentiallorentzianmetric
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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.

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