Trading linearity for ellipticity: a nonsmooth approach to Einstein's theory of gravity and the Lorentzian splitting theorems

TL;DR

This paper explores a nonsmooth, low-regularity approach to Lorentzian geometry and Einstein's gravity, using a negative homogeneity p-d'Alembert operator to establish splitting theorems and connect Lorentzian and Riemannian geometric results.

Contribution

It introduces a novel nonsmooth Lorentzian theory of gravity by sacrificing linearity, enabling elliptic techniques and simplifying proofs of classical splitting theorems.

Findings

Develops a low-regularity splitting theorem using a negative homogeneity p-d'Alembert operator.

Provides a simplified proof of classical Lorentzian splitting results.

Bridges Lorentzian and Riemannian geometry frameworks through nonsmooth analysis.

Abstract

While Einstein's theory of gravity is formulated in a smooth setting, the celebrated singularity theorems of Hawking and Penrose describe many physical situations in which this smoothness must eventually break down. In positive-definite signature, there is a highly successful theory of metric and metric-measure geometry which includes Riemannian manifolds as a special case, but permits the extraction of nonsmooth limits under dimension and curvature bounds analogous to the energy conditions from relativity: here sectional curvature is reformulated through triangle comparison, while Ricci curvature is reformulated using entropic convexity along geodesics of probability measures. This lecture highlights recent progress in the development of an analogous theory in Lorentzian signature, whose ultimate goal is to provide a nonsmooth theory of gravity. In particular, we foreshadow a low-regularity splitting theorem obtained by sacrificing linearity of the d'Alembertian to recover ellipticity. We exploit a negative homogeneity $p$-d'Alembert operator for this purpose. The same technique yields a simplified proof of Eschenberg (1988), Galloway (1989), and Newman's (1990) confirmation of Yau's (1982) conjecture, bringing both Lorentzian splitting results into a framework closer to the Cheeger--Gromoll (1971) splitting theorem from Riemannian geometry.