**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

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.

*© Springer-Verlag Berlin Heidelberg and Società Italiana di Fisica, 2013*