Behavior of quasilocal mass under conformal transformations

We show that in a generic scalar-tensor theory of gravity, the “referenced” quasilocal mass of a spatially bounded region in a classical solution is invariant under conformal transformations of the spacetime metric. We first extend the Brown-York quasilocal formalism to such theories to obtain the “unreferenced” quasilocal mass and prove it to be conformally invariant. However, this quantity is typically divergent. It is, therefore, essential to subtract from it a properly defined reference term to obtain a finite and physically meaningful quantity, namely, the referenced quasilocal mass. The appropriate reference term in this case is defined by generalizing the Hawking-Horowitz prescription, which was originally proposed for general relativity. For such a choice of reference term, the referenced quasilocal mass for a general spacetime solution is obtained. This expression is shown to be a conformal invariant provided the conformal factor is a monotonic function of the scalar field. We apply this expression to the case of static spherically symmetric solutions with arbitrary asymptotics to obtain the referenced quasilocal mass of such solutions. Finally, we demonstrate the conformal invariance of our quasilocal mass formula by applying it to specific cases of four-dimensional charged black hole spacetimes, of both the asymptotically flat and non-flat kinds, in conformally related theories.