Thermodynamics of rotating black holes with scalar hair in three dimensions

Introducing a new form of scalar potential $V(\varphi )$, we derive a proper form of the rotating black hole solution in three-dimensional Einstein gravity with nonminimally coupled scalar field and find that the first law of thermodynamics of this new rotating hairy black hole can be protected, where the scalar field parameter $B$ is constrained to relate to the black hole size. We also disclose the Hawking–Page phase transition between this rotating hairy black holes and the pure thermal radiation. Moreover, we study phase transitions between this rotating hairy black hole and rotating Banados-Teitelboim-Zanelli (BTZ) black hole. Considering the matchings for the temperature and angular momentum, we find that the rotating BTZ black hole always has smaller free energy, which is a thermodynamically more preferred phase. Additionally, we evaluate the thermodynamics of the rotating black hole with minimally coupled scalar hair in three dimensions, which exhibits that the thermodynamical behaviors of this rotating hairy black hole are very similar to those of the rotating black hole with nonminimally coupled scalar hair.

Figure 1

$\theta ({\tilde{X}}_{\nu })$ vs ${\tilde{X}}_{\nu }$.

Figure 2

The free energy $F$ of the rotating hairy black hole relates to the temperature $T$ for different $\theta$ when we fix $J=1$ and $l=1$.

Figure 3

The free energies $F$ of ($2+1$)-dimensional rotating hairy black hole (dashed lines) and BTZ black hole (solid lines) vs the temperature $T$ with $l=1$. (a) $J=1$. (b) $\theta =0.2$.

Figure 4

The free energies $F$ of ($2+1$)-dimensional rotating black hole with minimally scalar field (dashed lines) and BTZ black hole (solid line) vs the temperature $T$ when $l=1$.