Three dimensional charged black holes in Gauss-Bonnet gravity

TL;DR

The paper addresses the construction of regular, charged black holes in three-dimensional Gauss-Bonnet gravity using a zero-point length regulator, yielding finite gravitational and electromagnetic potentials but nonzero curvature invariants that indicate a central singularity at $r=0$. It derives static BTZ-like solutions within a scalar-tensor GB framework and analyzes the thermodynamics in an extended phase space, showing that the entropy is reduced by the stringy length $l_0$ and that the field equations at the horizon can be written as a first-law identity, with the Gauss-Bonnet coupling $\alpha$ dropping out of the horizon thermodynamics. The entropy is given by $S=\frac{\pi r_+}{2}\left(1-\frac{l_0^2}{r_+^2}\right)$, and the mass/temperature relations are modified accordingly, with $P=\frac{1}{8\pi l^2}$ and $V=\pi r_+^2$ in the extended phase framework. The work highlights how stringy quantum gravity effects modify low-dimensional black hole thermodynamics, including a rotating solution discussed in the appendix, thereby enriching the interplay between gravity, thermodynamics, and quantum corrections.

Abstract

By using the zero-point length effect, we construct a new class of charged black hole solutions in the framework of three dimensional Gauss-Bonnet (GB) gravity with Maxwell electrodynamics. The gravitational and electromagnetic potentials are finite and regular everywhere, however, the computation of scalar curvature invariants suggest the presence of a singularity at the origin. We also explore thermodynamics of the obtained solutions and reveal that the entropy of the black hole decreases due to the stringy effects. The thermodynamic and conserved quantities are computed and also the validity of the first law of thermodynamics on the black hole horizon is verified. Finally, the spinning black hole solution is also reported.

Figures (2)

• Figure 1: Left panel: Plot of $f_{GB}(r)$ as a function of $r$. We have set $M=1$, $Q=0.1$, $l=10$, $l_0=0.1$ and $\alpha=0.001$. This particular case shows the presence of the horizon, hence the solution describes a black holes. Right panel: Plot of the energy density $\rho$ (given by Eq. (3)) as a function of $r$. We have set $M=1$ and $l_0=0.1$.
• Figure 2: Plot of Ricci scalar and Kretschmann scalar. We have set $M=1$, $Q=0.1$, $l=10$, $l_0=0.1$ and $\alpha=0.001$. This particular case shows the presence singularities.