Gravity coupled to a scalar field in extra dimensions

In d + 1 dimensions we solve the equations of motion for the case of gravity minimally or conformally coupled to a scalar field. For the minimally coupled system the equations can either be solved directly or by transforming vacuum solutions, as shown before in 3 + 1 dimensions by Buchdahl. In d + 1 dimensions the solutions have been previously found directly by Xanthopoulos and Zannias. Here we first rederive these earlier results, and then extend Buchdahl’s method of transforming vacuum solutions to d + 1 dimensions. We also review the conformal coupling case, in which d + 1 dimensional solutions can be found by extending Bekenstein’s method of conformal transformation of the minimal coupling solution. Combining the extended versions of Buchdahl transformations and Bekenstein transformations we can in arbitrary dimensions always generate solutions of both the minimal and the conformal equations from known vacuum solutions.