Black Holes in f(R) theories

TL;DR

This paper investigates static, spherically symmetric black hole solutions within $f(R)$ gravity theories, analyzing constant and non-constant curvature cases, and explores their thermodynamic properties in anti-de Sitter space.

Contribution

It provides a perturbative analysis showing Schwarzschild-(anti) de Sitter solutions are predominant in $f(R)$ theories and links black hole thermodynamics to conditions on the $f(R)$ function.

Findings

Solutions are mainly of Schwarzschild-(anti) de Sitter type up to second order perturbations.

Explicit effective cosmological constant expressions are derived in terms of $f(R)$.

Existence of black holes in $f(R)$ theories requires $R_0+f(R_0)<0$, ensuring positive Newton's constant and similar thermodynamics to GR.

Abstract

In the context of $f(R)$ theories of gravity, we address the problem of finding static and spherically symmetric black hole solutions. Several aspects of constant curvature solutions with and without electric charge are discussed. We also study the general case (without imposing constant curvature). Following a perturbative approach around the Einstein-Hilbert action, it is found that only solutions of the Schwarzschild-(anti) de Sitter type are present up to second order in perturbations. Explicit expressions for the effective cosmological constant are obtained in terms of the $f(R)$ function. Finally, we have considered the thermodynamics of black holes in anti-de Sitter space-time and found that this kind of solutions can only exist provided the theory satisfies $R_0+f(R_0)<0$. Interestingly, this expression is related to the condition which guarantees the positivity of the effective Newton's constant in this type of theories. In addition, it also ensures that the thermodynamical properties in $f(R)$ gravities are qualitatively similar to those of standard General Relativity.