On the Energy-Momentum Tensor of the Scalar Field in Scalar--Tensor Theories of Gravity

TL;DR

This paper clarifies the proper definition of the scalar field energy-momentum tensor in scalar-tensor gravity theories, highlighting its distinction from the gravitational contribution and establishing conditions for physical viability.

Contribution

It introduces a new connection to explicitly separate the scalar field energy-momentum tensor from gravitational dynamics in scalar-tensor theories.

Findings

The naive scalar energy-momentum tensor is inadequate due to gravitational contributions.

A new connection allows explicit identification of the scalar energy-momentum tensor.

Conditions are derived to keep the scalar energy density positive.

Abstract

We study the dynamical description of gravity, the appropriate definition of the scalar field energy-momentum tensor, and the interrelation between them in scalar-tensor theories of gravity. We show that the quantity which one would naively identify as the energy-momentum tensor of the scalar field is not appropriate because it is spoiled by a part of the dynamical description of gravity. A new connection can be defined in terms of which the full dynamical description of gravity is explicit, and the correct scalar field energy-momentum tensor can be immediately identified. Certain inequalities must be imposed on the two free functions (the coupling function and the potential) that define a particular scalar-tensor theory, to ensure that the scalar field energy density never becomes negative. The correct dynamical description leads naturally to the Einstein frame formulation of scalar-tensor gravity which is also studied in detail.