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

TL;DR

This paper explores how the Bakry-Emery-Ricci tensor influences singularity and splitting theorems in scalar-tensor gravitation theories, extending classical results to the Jordan frame and examining specific models like Brans-Dicke theory.

Contribution

It demonstrates that key singularity and splitting theorems hold in scalar-tensor gravitation theories using the Bakry-Emery tensor, including applications to dilaton and Brans-Dicke theories.

Findings

Hawking-Penrose singularity theorem applies in the Jordan frame.

Splitting theorem holds for scalar-tensor theories with the Bakry-Emery tensor.

Theorems may fail for certain Brans-Dicke parameter ranges, notably omega<-1.

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.