Black Holes in Higher Derivative Gravity Theories

TL;DR

This paper investigates static spherically symmetric solutions in higher derivative gravity theories, establishing conditions for Schwarzschild solutions and discovering new solutions with de Sitter or anti-de Sitter asymptotics across various dimensions.

Contribution

It provides a uniqueness theorem for Schwarzschild solutions in polynomial Ricci scalar gravity and finds new Schwarzschild-de Sitter solutions in higher derivative models.

Findings

Schwarzschild solution is unique under certain conditions in these models.

Existence of Schwarzschild-de Sitter solutions in higher derivative gravity.

Classification of scalar field solutions with exponential potentials.

Abstract

We study static spherically symmetric solutions of Einstein gravity plus an action polynomial in the Ricci scalar, $R$, of arbitrary degree, $n$, in 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 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 $R+aR^2$ 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.