New perspectives on the d'Alembertian from general relativity. An invitation

TL;DR

This paper reviews a nonlinear, elliptic-like generalization of the d'Alembertian from mathematical relativity, discussing recent estimates, geometric representations, applications, and open problems in the context of Lorentzian geometry.

Contribution

It introduces and analyzes a nonlinear $p$-d'Alembertian operator, providing new comparison estimates, geometric formulas, and applications in spacetime analysis.

Findings

Control of the timelike cut locus via optimal transport

Exact convex geometric representation formulas

New estimates for the $p$-d'Alembertian of Lorentz distance functions

Abstract

This survey has multiple objectives. First, we motivate and review a new distributional notion of the d'Alembertian from mathematical relativity, more precisely, a nonlinear $p$-version thereof, where $p$ is a nonzero number less than one. This operator comes from natural Lagrangian actions introduced relatively recently. Unlike its classical linear yet hyperbolic counterpart, it is nonlinear yet has elliptic characteristics. Second, we describe recent comparison estimates for the $p$-d'Alembertian of Lorentz distance functions (notably a point or a spacelike hypersurface). Their new contribution implied by prior works on optimal transport through spacetime is a control of the timelike cut locus. Third, we illustrate exact representation formulas for these $p$-d'Alembertians employing methods from convex geometry. Fourth, several applications and open problems are presented.