Behavior of Quasilocal Mass Under Conformal Transformations

TL;DR

This paper demonstrates that the referenced quasilocal mass in scalar-tensor gravity theories remains invariant under conformal transformations when the conformal factor depends monotonically on the scalar field, extending previous formalisms.

Contribution

It extends the Brown-York formalism to scalar-tensor theories and generalizes the Hawking-Horowitz reference term to prove conformal invariance of quasilocal mass.

Findings

Referenced quasilocal mass is conformally invariant under specific conditions.

Derived explicit expression for quasilocal mass in scalar-tensor theories.

Confirmed invariance through applications to black hole solutions.

Abstract

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. 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.