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

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)=D1R(n/n+1)(R/K)1/n+D2R+D3, where the constant term D3 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 D1=0,D2=1, namely f(R)=R+D3, 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 D3 and represents symmetry of the action under the transformation R→R′=R+D3 along the killing vector ∂R. The ordinary BTZ black hole is obtained as the special case where D3 is 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.