Cosmological singularities in BakryÉmery spacetimes
Abstract
We consider spacetimes consisting of a manifold with Lorentzian metric and a weight function or scalar field. These spacetimes admit a BakryÉmeryRicci tensor which is a natural generalization of the Ricci tensor. We impose an energy condition on the BakryÉmeryRicci 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 scalartensor gravitation theories. We focus on the BransDicke family of theories in 4 spacetime dimensions, where we obtain 'Jordan frame' singularity theorems for big bang singularities.
 Publication:

Journal of Geometry and Physics
 Pub Date:
 December 2014
 DOI:
 10.1016/j.geomphys.2014.08.016
 arXiv:
 arXiv:1312.3410
 Bibcode:
 2014JGP....86..359G
 Keywords:

 BakryÉmery;
 Lorentzian;
 Singularity theorem;
 Cosmological singularity;
 BransDicke theory;
 Scalartensor theory;
 Mathematics  Differential Geometry;
 General Relativity and Quantum Cosmology
 EPrint:
 15 pages