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

TL;DR

This paper develops a variational Sobolev calculus on metric measure spacetimes, introducing a Lorentzian differential modulus, and proves a comparison theorem for a nonlinear d'Alembertian under certain geometric conditions.

Contribution

It introduces a novel first-order Sobolev calculus for causal functions and establishes a comparison theorem for nonlinear d'Alembertian operators in Lorentzian geometry.

Findings

Defined maximal weak subslope as a Lorentzian differential modulus.

Proved chain and Leibniz rules for the subslope.

Established a comparison theorem for nonlinear d'Alembertian under measure contraction.

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.