Uniqueness of non-trivial spherically symmetric black hole solution in special classes of F(R) gravitational theory

Abstract

We show, in detail, that the only non-trivial black hole (BH) solutions for a neutral as well as a charged spherically symmetric space-times, using the class F(R)=R±F1(R), must-have metric potentials in the form h(r)=12−2Mr and h(r)=12−2Mr+q2r2. These BHs have a non-trivial form of Ricci scalar, i.e., R=1r2 and the form of F1(R)=∓R3M. We repeat the same procedure for (Anti-)de Sitter, (A)dS, space-time and got the metric potentials of neutral as well as charged in the form h(r)=12−2Mr−2Λr23 and h(r)=12−2Mr+q2r2−2Λr23, respectively. The Ricci scalar of the (A)dS space-times has the form R=1+8r2Λr2 and the form of F1(R)=∓2R−8Λ3M. We calculate the thermodynamical quantities, Hawking temperature, entropy, quasi-local energy, and Gibbs-free energy for all the derived BHs, that behaves asymptotically as flat and (A)dS, and show that they give acceptable physical thermodynamical quantities consistent with the literature. Finally, we prove the validity of the first law of thermodynamics for those BHs.