BLACK-HOLES IN HIGHER-DERIVATIVE GRAVITY THEORIES

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 a2 of the R2 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 + aR2 theory if D > 4. 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.