Scalar-tensor gravitation and the Bakry-Emery-Ricci tensor

Abstract: The Bakry-Emery generalized Ricci tensor arises in scalar-tensor gravitation theories in the conformal gauge known as the Jordan frame. Recent results from the mathematics literature show that standard singularity and splitting theorems that hold when an energy condition is applied in general relativity also hold when that energy condition is applied to the Bakry-Emery tensor. We show here that a direct consequence is that the Hawking-Penrose singularity theorem and the timelike splitting theorem hold for scalar-tensor theory in the Jordan frame. As examples, we consider dilaton gravity (including totally anti-symmetric torsion) and the Brans-Dicke family of scalar-tensor theories. For Brans-Dicke theory the theorems do not extend to cover the entire space of values of the Brans-Dicke family parameter omega, and so may fail to hold for omega<-1. Observations show that this range of values does not describe our Universe, but the result is in accord with examples in the literature of Brans-Dicke spacetimes that have no singularity in the Jordan frame and do not split as a Riemannian product.