A new class of analytic charged spherically symmetric black hole solutions, which behave asymptotically as flat or (anti-)de Sitter spacetimes, is derived for specific classes of $f(R)$ gravity, i.e., $f(R)=R-2\alpha \sqrt{R}$ and $f(R)=R-2\alpha \sqrt{R-8\mathrm{\Lambda }}$, where $\mathrm{\Lambda }$ is the cosmological constant. These black holes are characterized by the dimensional parameter $\alpha$ that makes solutions deviate from the standard solutions of general relativity. The Kretschmann scalar and squared Ricci tensor are shown to depend on the parameter $\alpha$, which is not allowed to be zero. Thermodynamical quantities, like entropy, Hawking temperature, quasilocal energy, and the Gibbs free energy are calculated. From these calculations, it is possible to put a constraint on the dimensional parameter $\alpha$ to have $0<\alpha <0.5$ so that all thermodynamical quantities have a physical meaning. The interesting result of these calculations is the possibility of a negative black hole entropy. Furthermore, present calculations show that for negative energy particles inside a black hole,behave as if they have a negative entropy. This fact gives rise to instability for $f_{RR}<0$. Finally, we study the linear metric perturbations of the derived black hole solution. We show that for the odd-type modes our black hole is always stable and has a radial speed with fixed value equal to 1. We also use the geodesic deviation to derive further stability conditions.

• Received 23 February 2019
• Revised 1 April 2019

DOI:https://doi.org/10.1103/PhysRevD.99.104018