Eur. Phys. J. C (2013) 73: 2657
https://doi.org/10.1140/epjc/s10052-013-2657-6

Regular Article - Theoretical Physics

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

¹ Center for Excellence in Astronomy and Astrophysics of IRAN (CEAAI-RIAAM), Maragha, 55134-441, Iran
² Department of Physics, Azarbaijan Shahid Madani University, Tabriz, 53741-161, Iran

^* e-mail: f.darabi@azaruniv.edu

Received: 15 August 2013
Published online: 26 November 2013

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→R′=R+D ₃ along the killing vector ∂ _R . The ordinary BTZ black hole is obtained as the special case where D ₃ 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.