New black hole solutions in three-dimensional f(R) gravity

Abstract

We construct two new classes of analytical solutions in three-dimensional spacetime in the framework of f(R) gravity whose field equations did not have cosmological constant. These solutions cannot coincide with Einstein’s general relativity due to the non-allowable vanishing of the dimensional parameter that characterizes the higher-order curvature. The first class of this solutions represents a non-rotating black hole (BH) while the second class corresponds to a rotating BH solution. The Ricci scalar of these BH solutions have non-trivial values and are described by the gravitational mass M, two angular momenta J and J1, and an effective cosmological constant Λeff that mainly comes from the contribution of the higher-order curvature terms. Interestingly enough, we observe that in contrast to Bañados–Teitelboim–Zanelli (BTZ) solution, which has only causal singularities and its scalar invariants are constant everywhere, the scalar invariants of our solutions indicate strong singularities for spacetime. This implies that the higher order derivative terms that come from f(R) gravity changes the casual structure of the spacetime. Indeed, the scalar invariants of the solutions behave as O1rn where n>0. The behavior of the invariants as r→0 become strong as compared with the behavior of black holes that reproduce from GR. Furthermore, we construct the forms, rotating/non-rotating, of the f(R) function showing that they behave as polynomial functions. Finally, we show that the obtained solutions are stable because of the positivity of their heat capacity, and also from the condition of Ostrogradski which states that the second derivative of f(R) should have a positive value.