We study static spherically symmetric solutions of Einstein gravity plus an action polynomial in the Ricci scalar $R$ of arbitrary degree $n$ in an arbitrary dimension $D$. The global properties of all such solutions are derived by studying the phase space of field equations in the equivalent theory of gravity coupled to a scalar field, which is obtained by a field redefinition and conformal transformation. The following uniqueness theorem is obtained: Provided that the coefficient $a_2$ of the $R^2$ term in the Lagrangian polynomial is positive then the only static spherically symmetric asymptotically flat solution with a regular horizon in these models is the Schwarzschild solution. Other branches of solutions with regular horizons, which are asymptotically anti-de Sitter, or de Sitter, are also found. An exact Schwarzschild-de Sitter-type solution is found to exist in the $R+a{R}^2$ theory if $D>\mathrm{}4\mathrm{}$. If terms of cubic or higher order in $R$ are included in the action, then such solutions also exist in four dimensions. The general Schwarzschild-de Sitter-type solution for arbitrary $D$ and $n$ is given. The fact that the Schwarzschild solution in these models does not coincide with the exterior solution of physical bodies such as stars has important physical implications which we discuss. As a byproduct, we classify all static spherically symmetric solutions of $D$-dimensional gravity coupled to a scalar field with a potential consisting of a finite sum of exponential terms.

• Received 6 February 1992

DOI:https://doi.org/10.1103/PhysRevD.46.1475