Cosmological singularity theorems and splitting theorems for N-Bakry-Emery spacetimes

TL;DR

This paper extends singularity and splitting theorems for Lorentzian manifolds with N-Bakry-Emery curvature bounds, relevant in scalar-tensor gravitation theories, covering new N ranges and weaker conditions on the weight function.

Contribution

It generalizes existing theorems to finite N values and weaker conditions on the weight function, broadening the applicability in physics and geometry.

Findings

Extended singularity theorems to N in (n,∞) and (-∞,1]

Extended splitting theorems with warped product structures

Weaker conditions on the weight function for certain N ranges

Abstract

We study Lorentzian manifolds with a weight function such that the $N$-Bakry-\'Emery tensor is bounded below. Such spacetimes arise in the physics of scalar-tensor gravitation theories, including Brans-Dicke theory, theories with Kaluza-Klein dimensional reduction, and low-energy approximations to string theory. In the "pure Bakry-\'Emery" $N= \infty$ case with $f$ uniformly bounded above and initial data suitably bounded, cosmological-type singularity theorems are known, as are splitting theorems which determine the geometry of timelike geodesically complete spacetimes for which the bound on the initial data is borderline violated. We extend these results in a number of ways. We are able to extend the singularity theorems to finite $N$-values $N\in (n,\infty)$ and $N\in (-\infty,1]$. In the $N\in (n,\infty)$ case, no bound on $f$ is required, while for $N\in (-\infty,1]$ and $N= \infty$, we are able to replace the boundedness of $f$ by a weaker condition on the integral of $f$ along future-inextendible timelike geodesics. The splitting theorems extend similarly, but when $N=1$ the splitting is only that of a warped product for all cases considered. A similar limited loss of rigidity has been observed in prior work on the $N$-Bakry-\'Emery curvature in Riemannian signature when $N=1$, and appears to be a general feature.