Generalized Sparling-Thirring form in the Brans-Dicke theory

TL;DR

This paper extends the Sparling-Thirring form to Brans-Dicke theory, defining a superpotential and pseudo energy-momentum form, and applies it to compute energy in a spherically symmetric vacuum solution.

Contribution

It introduces a generalized Sparling-Thirring form for Brans-Dicke theory and explores its implications for energy definitions in different frames.

Findings

Superpotential and pseudo energy-momentum form are defined in Brans-Dicke theory.

Energy expression in Jordan frame is compared to Einstein frame.

Energy for spherically symmetric vacuum solution is calculated.

Abstract

The definition of the Sparling-Thirring form is extended to the Brans-Dicke theory. By writing the Brans-Dicke field equations in a formally Maxwell-like form, a superpotential and a corresponding pseudo energy-momentum form are defined. The general energy expression provided by the superpotential in the Jordan frame is discussed in relation to the corresponding expression in the Einstein frame. In order to substantiate its formal definition, the generalized Sparling-Thirring form is used to calculate the energy for the spherically symmetric vacuum solution in the Brans-Dicke theory.