Gravity coupled to a scalar field in extra dimensions

TL;DR

This paper extends methods for solving gravity coupled to scalar fields in higher dimensions, enabling generation of solutions from vacuum solutions through transformations, applicable to both minimal and conformal couplings.

Contribution

It generalizes Buchdahl's and Bekenstein's solution-generating techniques to arbitrary dimensions, providing a unified approach for minimal and conformal scalar couplings.

Findings

Extended Buchdahl transformations to d+1 dimensions.

Extended Bekenstein conformal transformations to d+1 dimensions.

Unified method for generating solutions from vacuum solutions.

Abstract

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.