Title: Cosmological singularity theorems and splitting theorems for N-Bakry-Émery spacetimes

We study Lorentzian manifolds with a weight function such that the N-Bakry-Émery 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-Émery” N = ∞ 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 ∈ (n, ∞) and N ∈ (−∞, 1]. In the N ∈ (n, ∞) case, no bound on f is required, while for N ∈ (−∞, 1] and N = ∞, 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 a prior work on the N-Bakry-Émerymore » curvature in Riemannian signature when N = 1 and appears to be a general feature.« less

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, Alberta T6G 2G1 (Canada)

215 Carnegie Building, Department of Mathematics, Syracuse University, Syracuse, New York 13244 (United States)

Publication Date:

OSTI Identifier:

22479632

Resource Type:

Journal Article

Resource Relation:

Journal Name: Journal of Mathematical Physics; Journal Volume: 57; Journal Issue: 2; Other Information: (c) 2016 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

Country of Publication:

United States

Language:

English

Subject:

71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; BRANES; KALUZA-KLEIN THEORY; SINGULARITY; SPACE-TIME; STRING MODELS; STRING THEORY