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

Abstract

We construct two new classes of analytical solutions in three-dimensional spacetime and in the framework of $f(R)$ gravity. The first class 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 momentums $J$ and $J_1$, and an effective cosmological constant ${\Lambda }_{eff}$. Moreover, these solutions do not restore the $3$-dimensional Bañados-Teitelboim-Zanelli (BTZ) solutions of general relativity (GR) which implies the novelty of the obtained BHs in $f(R)$ gravity. Depending on the range of the parameters, these solutions admit rotating/non-rotating asymptotically AdS/dS BH interpretation in spite that the field equation of $f(R)$ has no cosmological constant. Interestingly enough, we observe that in contrast to BTZ solution which has only causal singularity and scalar invariants are constant everywhere, the scalar invariants of these solutions indicate strong singularity for the spacetime. Furthermore, we construct the forms of the $f(R)$ function showing that they behave as polynomial functions. Finally, we show that the obtained solutions are stable from the viewpoint that heat capacity has a positive value, and also from the condition of Ostrogradski which state that the second derivative of $f(R)$ should have a positive value.

20 pages Latex

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