Mass Definitions for Boson Stars in Brans Dicke Gravity

We consider boson stars in Brans Dicke gravity, focusing on general ω weak field solutions and ω = −1 strong field solutions. We discuss three definitions of quasilocal mass adapted to the spherical symmetry of the solutions: the Schwarzschild mass, the Keplerian mass and the Tensor mass, and discuss how the asymptotic values of these masses may be derived from an ADM-like formula. We show that in all solutions there is a region in which the Schwarzschild mass has a negative gradient and that in the weak field limit, the ADM mass of the star is negative for ω < −1. We reject the Schwarzschild and ADM masses as descriptions of the quasilocal and total energies of the star. The Tensor and Keplerian masses are well behaved for all weak field solutions and all ω = −1 strong field solutions. However, the maximum Keplerian mass solution does not correspond to the maximum rest mass solution and so we reject the Keplerian mass as a measure of the star’s energy. The maximum Tensor mass solution does have this property, and we conjecture that this quantity correctly describes the energy of the star. ∗Email: [email protected]