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

Eric Woolgar, William Wylie

Research output: Research - peer-reviewArticle

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Abstract

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-émery curvature in Riemannian signature when N = 1 and appears to be a general feature.

LanguageEnglish (US)
Article number022504
JournalJournal of Mathematical Physics
Volume57
Issue number2
DOIs
StatePublished - Feb 1 2016

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Space-time
Singularity
Theorem
theorems
Tensor
tensors
Lorentzian Manifolds
Warped Product
Dimensional Reduction
Gravitation
String Theory
Weight Function
Rigidity
Geodesic
Boundedness
Signature
Curvature
Physics
Scalar
Approximation

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

Cosmological singularity theorems and splitting theorems for N-Bakry-émery spacetimes. / Woolgar, Eric; Wylie, William.

In: Journal of Mathematical Physics, Vol. 57, No. 2, 022504, 01.02.2016.

Research output: Research - peer-reviewArticle

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