# Thermodynamics of Noncommutative Quantum Kerr Black Holes

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## Abstract

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## 1. Introduction

## 2. Equations of State

## 3. Results and Discussion

#### 3.1. Response Functions

#### 3.2. Thermodynamic Stability and Phase Transition

## 4. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Abbreviations

AdS | Anti de Sitter |

BH | Bekenstein–Hawking |

WDW | Wheeler–DeWitt |

ADM | Arnowitt–Deser–Misner |

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**Figure 1.**Comparison between Bekenstein–Hawking (solid line) and quantum corrected (dash-dot line) entropies. (

**a**) Entropy as a function of internal energy for $J=1$, $S=S(U,1)$; (

**b**) entropy as a function of angular momentum for $U=1$, $S=S(1,J)$.

**Figure 2.**Plots of Bekenstein–Hawking and quantum corrected temperatures for $\Gamma =1$. (

**a**) $T(U,1)$ (solid) vs. ${T}^{\u2605}(U,1)$ (dash-dot) as a function of internal energy considering $J=1$; (

**b**) the same plots of temperature for variations in angular momentum at $U=1$.

**Figure 3.**Plots of angular velocity for both Bekenstein–Hawking and quantum corrected entropies. (

**a**) Ω as a function of internal energy for $J=1$; (

**b**) angular velocity as a function of angular momentum considering $U=10$.

**Figure 4.**Specific heat capacity at constant J for a Kerr black hole considering different values of Γ exhibiting a discontinuity; the following values of noncommutativity parameter are considered: $\Gamma =1$ (solid), $\Gamma =0.99$ (dashed-dot), $\Gamma =0.98$ (dashed) and $\Gamma =0.7$ (dotted). (

**a**) ${C}_{J}$ is presented as a function of energy at $J=1$; (

**b**) plots of specific heat as a function of angular momentum for $U=1$.

**Figure 5.**Thermodynamic planes considering different values of Γ showing the location of divergence found in specific heat; $\pm {C}_{J}$ indicates the sign of this response function in each region; in these thermodynamic planes, the following values of Γ are presented: $\Gamma =1$ (solid), $\Gamma =0.99$ (dashed-dot), $\Gamma =0.98$ (dashed) and $\Gamma =0.7$ (dotted). (

**a**) Plane Ω–T divided by a line representing discontinuity given in Equation (23); (

**b**) plane S–U shows a parabola dividing regions where ${C}_{J}$ is positive or negative; (

**c**) plane T–U depicts another straight line separating both regions.

**Figure 6.**Noncommutative quantum corrected isothermal rotational susceptibility. (

**a**) ${\chi}_{T}^{\u2605}$ as a function of energy for $J=1$, exhibiting a monotonically-growing function; given that noncommutativity does not greatly affect ${\chi}^{\u2605}$, the following values for this parameter are chosen: $\Gamma =1$ (solid), $\Gamma =0.99$ (dashed-dot), $\Gamma =0.98$ (dashed) and $\Gamma =0.2$ (dotted); (

**b**) curves for isothermal susceptibility as a function of J considering $U=1$.

**Figure 7.**Specific heat capacity at constant angular velocity for different values of Γ. (

**a**) ${C}_{\Omega}$ is plotted as a function of internal energy for $J=1$; for these response functions, the following values of Γ were considered: $\Gamma =1$ (solid), $\Gamma =0.99$ (dashed-dot), $\Gamma =0.98$ (dashed) and $\Gamma =0.7$ (dotted); (

**b**) curves of ${C}_{\Omega}$ for $U=1$ as a function of angular momentum.

**Figure 8.**Coefficient of thermally-induced rotation for noncommutative Bekenstein–Hawking entropy. (

**a**) ${\alpha}_{\Omega}$ as a function of internal energy, for $J=1$; where the following values of the noncommutativity parameter are plotted: $\Gamma =1$ (solid), $\Gamma =0.99$ (dashed-dot), $\Gamma =0.98$ (dashed) and $\Gamma =0.7$ (dotted); (

**b**) plots of the same coefficient varying angular momentum, for internal energy at $U=1$.

**Figure 9.**Isotherms in plane Ω–J for a Kerr black hole. Different temperatures were tested, the exterior isotherm corresponds to the lower temperatures. The van der Waals loop does not appear in any of these isotherms. (

**a**) In this figure the corresponding isotherms for low temperatures are presented as a function of angular momentum; (

**b**) this figure presents temperature isotherms for higher temperatures.

**Table 1.**Comparison between thermodynamic properties of noncommutative Bekenstein–Hawking and noncommutative quantum corrected entropies.

Response Functions | Equations of State | Fundamental Relation |
---|---|---|

${C}_{J}>{C}_{J}^{\u2605}$ | $T<{T}^{\u2605}$ | $S>{S}^{\u2605}$ |

${\chi}_{T}<{\chi}_{T}^{\u2605}$ | $\Omega ={\Omega}^{\u2605}$ | - |

${\chi}_{S}={\chi}_{S}^{\u2605}$ | - | - |

${C}_{\Omega}<{C}_{\Omega}^{\u2605}$ | - | - |

${\alpha}_{\Omega}>{\alpha}_{\Omega}^{\u2605}$ | - | - |

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**MDPI and ACS Style**

Escamilla-Herrera, L.F.; Mena-Barboza, E.A.; Torres-Arenas, J.
Thermodynamics of Noncommutative Quantum Kerr Black Holes. *Entropy* **2016**, *18*, 406.
https://doi.org/10.3390/e18110406

**AMA Style**

Escamilla-Herrera LF, Mena-Barboza EA, Torres-Arenas J.
Thermodynamics of Noncommutative Quantum Kerr Black Holes. *Entropy*. 2016; 18(11):406.
https://doi.org/10.3390/e18110406

**Chicago/Turabian Style**

Escamilla-Herrera, Lenin F., Eri A. Mena-Barboza, and José Torres-Arenas.
2016. "Thermodynamics of Noncommutative Quantum Kerr Black Holes" *Entropy* 18, no. 11: 406.
https://doi.org/10.3390/e18110406