TY - JOUR

T1 - Cosmological singularities in Bakry-Émery spacetimes

AU - Galloway, Gregory J.

AU - Woolgar, Eric

N1 - Funding Information:
The work of EW was supported by a Discovery Grant from the Natural Sciences and Engineering Research Council (NSERC) of Canada (Grant No. RGPIN 203614 ). The work of GJG was supported by NSF grant DMS-1313724 and by a grant from the Simons Foundation (Grant No. 63943 ). The work of GJG was also supported by NSF grant 0932078 000 , while in residence at MSRI, Berkeley, during Fall, 2013. Both authors wish to express their gratitude to the Park City Math Institute 2013 Summer Research Program, where this work was begun.

PY - 2014/12/1

Y1 - 2014/12/1

N2 - We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmological type, both for zero and for positive cosmological constant. That is, we find conditions under which every timelike geodesic is incomplete. These conditions are given by "open" inequalities, so we examine the borderline (equality) cases and show that certain singularities are avoided in these cases only if the geometry is rigid; i.e., if it splits as a Lorentzian product or, for a positive cosmological constant, a warped product, and the weight function is constant along the time direction. Then the product case is future timelike geodesically complete while, in the warped product case, worldlines of certain conformally static observers are complete. Our results answer a question posed by J Case. We then apply our results to the cosmology of scalar-tensor gravitation theories. We focus on the Brans-Dicke family of theories in 4 spacetime dimensions, where we obtain "Jordan frame" singularity theorems for big bang singularities.

AB - We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a Bakry-Émery-Ricci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the Bakry-Émery-Ricci tensor and obtain singularity theorems of a cosmological type, both for zero and for positive cosmological constant. That is, we find conditions under which every timelike geodesic is incomplete. These conditions are given by "open" inequalities, so we examine the borderline (equality) cases and show that certain singularities are avoided in these cases only if the geometry is rigid; i.e., if it splits as a Lorentzian product or, for a positive cosmological constant, a warped product, and the weight function is constant along the time direction. Then the product case is future timelike geodesically complete while, in the warped product case, worldlines of certain conformally static observers are complete. Our results answer a question posed by J Case. We then apply our results to the cosmology of scalar-tensor gravitation theories. We focus on the Brans-Dicke family of theories in 4 spacetime dimensions, where we obtain "Jordan frame" singularity theorems for big bang singularities.

KW - Bakry-Émery

KW - Brans-Dicke theory

KW - Cosmological singularity

KW - Lorentzian

KW - Scalar-tensor theory

KW - Singularity theorem

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U2 - 10.1016/j.geomphys.2014.08.016

DO - 10.1016/j.geomphys.2014.08.016

M3 - Article

AN - SCOPUS:84908010348

VL - 86

SP - 359

EP - 369

JO - Journal of Geometry and Physics

JF - Journal of Geometry and Physics

SN - 0393-0440

ER -