Einstein gravity coupled to a massless conformal scalar field in arbitrary space-time dimensions

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Abstract
For space‐times of arbitrary dimensionality two transformations are obtained that applied to any solution of the Einstein equations coupled to a minimally coupled scalar field construct a solution of the Einstein equations coupled to a conformal scalar field. The transformations are conformal for the metrics and algebraic for the reparametrization of the scalar fields. With these generating techniques static, spherically symmetric, asymptotically flat solutions of the Einstein‐conformal scalar field equations in arbitrary space‐time dimensions are obtained. Black hole solutions are systematically searched for and it is found that the only (nonvacuum) one exists in four space‐time dimensions. This is the Bekenstein black hole for gravity coupled to a conformal scalar field.