Revisiting the static spherically symmetric solutions of gravity with a conformally coupled scalar field

We revisit the static spherically symmetric solutions of Einstein’s general relativity with a conformally coupled scalar field in arbitrary dimensions. Using a four rank tensor introduced earlier we recast the field equations in a manifestly symmetric form to elucidate a somewhat less-known feature of dual mapping between solutions. We also show that there is a two-parameter subfamily of solutions which enjoy a duality symmetry and in four dimensions both the BBMB black hole and the Barcelo–Visser wormhole belong to this subfamily. Along the way, we rederive the full three-parameter family of solutions by direct integration of the field equations and a natural choice of ansatz which arguably has several advantages over other previously known methods.