Generalized Sparling–Thirring form in the Brans–Dicke theory

The definition of the Sparling–Thirring form is extended to 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 Brans–Dicke theory.