# A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics

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## Abstract

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## 1. Introduction

- The zeroth law of black hole mechanics states that the surface gravity κ of a stationary black hole is constant over the horizon, which is essentially the requirement of transitivity of the equilibrium state.
- The first law manifests a relation between variations in the mass M, horizon area A, and angular momentum J if the black hole is perturbed,$$\delta M=\frac{\kappa}{8\pi}\delta A+{\mathrm{\Omega}}_{\mathrm{H}}\delta J,$$
- The second law of black hole mechanics is Hawking’s area theorem, which states that the surface area of the event horizon never decreases with time,$$\delta A\ge 0.$$
- The third law is formulated by stating that it is impossible to achieve $\kappa =0$ in a finite series of physical processes.

#### 1.1. Equilibrium Compatibility

#### 1.2. Zeroth Law Compatibility

#### 1.3. A $\lambda \ne 0$ Parametric Approach

#### 1.4. A Nonparametric Approach

## 2. Results

#### 2.1. Kerr Black Holes

#### 2.2. The Formal Logarithm Approach

#### 2.3. Stability Analysis

## 3. Discussion

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Plots of heat capacities ${C}_{\mathrm{\Omega}}$ (blue solid) and ${C}_{J}$ (red dashed) as functions of h. ${C}_{J}$ diverges at ${h}_{c}=\sqrt{\frac{2}{3}\sqrt{3}-1}$. ${C}_{J}$ is negative for $h<{h}_{c}$ and positive for $h>{h}_{c}$. The heat capacities coincide at the limit values $h=0$ (Schwarzschild holes) and $h=1$ (extreme Kerr holes).

**Figure 2.**Plots of entropy S as a function of the mass–energy parameter M at fixed J, standard (black solid) and formal logarithm (red dashed) approaches. The standard entropy is asymptotically convex (being proportional to ${M}^{2}$ in the large M limit). The new entropy is a straight line because the entropy is proportional to the mass–energy.

**Figure 3.**Plots of temperature T as a function of the mass–energy parameter M at fixed J— standard (black solid) and formal logarithm (red dashed) approaches. The standard temperature has a local maximum. The smaller mass branch corresponds with $h>{h}_{c}$, and the larger mass with $h<{h}_{c}$. The new temperature is a horizontal line because it is an energy-independent constant.

**Figure 4.**Plots of stability curves $\beta \left(M\right)$ at fixed J of standard (black solid) and formal logarithm (red dashed) approaches in the microcanonical treatment. No vertical tangent occurs in either cases. By rotating the figure clockwise with $\frac{\pi}{2}$, the stability curves of the canonical treatment can be obtained; i.e., $-M\left(\beta \right)$ at fixed J. There is a vertical tangent on the stability curve of standard case. The smaller mass branch ($h>{h}_{c}$) is more stable than the larger mass branch ($h<{h}_{c}$). No turning point appears on the stability curve of the formal logarithm approach.

**Figure 5.**Plots of stability curves $-\alpha \left(J\right)$ at fixed M of standard (black solid) and formal logarithm (red dashed) approaches in the microcanonical treatment. No turning point appears on either stability curve.

**Figure 6.**Stability curve $-\alpha \left(J\right)$ at fixed β of the standard case in the canonical treatment. There is a vertical tangent. The more stable positive slope branch corresponds to fast rotation ($h>{h}_{c}$). The less stable negative slope branch corresponds to slow rotation ($h<{h}_{c}$).

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Czinner, V.G.; Iguchi, H.
A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics. *Universe* **2017**, *3*, 14.
https://doi.org/10.3390/universe3010014

**AMA Style**

Czinner VG, Iguchi H.
A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics. *Universe*. 2017; 3(1):14.
https://doi.org/10.3390/universe3010014

**Chicago/Turabian Style**

Czinner, Viktor G., and Hideo Iguchi.
2017. "A Zeroth Law Compatible Model to Kerr Black Hole Thermodynamics" *Universe* 3, no. 1: 14.
https://doi.org/10.3390/universe3010014