Generalized (2+1) dimensional black hole by Noether symmetry

Abstract

We use the Noether symmetry approach to find $f(R)$ theory of $(2+1)$ dimensional gravity and $(2+1)$ dimensional black hole solution consistent with this $f(R)$ gravity and the associated symmetry. We obtain $f({R})=D_1 R({n}/{n+1})({R}/{K})^{1/n}+D_2 R+D_3$, where the constant term $D_3$ plays no dynamical role. Then, we find general spherically symmetric solution for this $f(R)$ gravity which is potentially capable of being as a black hole. Moreover, in the special case $D_1=0, D_2={1}$, namely $f(R)=R+D_3$, we obtain a generalized BTZ black hole which, other than common conserved charges $m$ and $J$, contains a new conserved charge Q. It is shown that this conserved charge corresponds to the freedom in the choice of the constant term $D_3$ and represents symmetry of the action under the transformation $R \rightarrow R'=R+D_3$ along the killing vector $\partial_{R}$. The ordinary BTZ black hole is obtained as the special case where $D_3$ is {\it fixed} to be proportional to the infinitesimal cosmological constant and consequently the symmetry is broken via Q=0. We study the thermodynamics of the generalized BTZ black hole and show that its entropy can be described by the Cardy-Verlinde formula.