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

Robert J. McCann

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.

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

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 -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.
Paper Structure (6 sections, 4 theorems, 25 equations)

This paper contains 6 sections, 4 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. General relativity: Einstein's gravity and field equation
  3. Special relativity: elliptic vs hyperbolic geometry
  4. A crash course in differential geometry
  5. Energy conditions, causality, and singularity theorems
  6. Smooth Lorentzian splitting theorems

Key Result

Theorem 3

Let $(M^n,g_{ij})$ be a connected smooth spacetime satisfying the strong energy condition (SEC) and containing a timelike line. If $M$ is (a) timelike geodesically complete, then $M$ is a geometric product of ${\mathbf R}$ with a (Ricci nonnegative, complete) Riemannian submanifold $\Sigma^{n-1}$.

Theorems & Definitions (6)

  • Example 1: When do convex functions split?
  • Example 2: When do smooth Riemannian manifolds split? CheegerGromoll71
  • Theorem 3: Lorentzian splitting Newman90 conjectured in Yau82
  • Theorem 4: Nonsmooth $p$-d'Alembert comparison BeranOctet24+ BraunGigliMcCannOhanyanSaemann24+
  • Theorem 5: Equi-Lipschitz estimate Eschenburg88 GallowayHorta96
  • Lemma 6: Equi-semiconcavity BraunGigliMcCannOhanyanSaemann24+