# Geometric Model of Black Hole Quantum N-portrait, Extradimensions and Thermodynamics

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## Abstract

**:**

## 1. Introduction

- (i)
- the metric admits an extremal configuration for a black hole mass ${M}_{\mathrm{e}}={M}_{\mathrm{P}}$ and radius ${r}_{\mathrm{e}}={L}_{\mathrm{P}}$;
- (ii)
- the extremal configuration represents the fundamental qubit of the system; any larger black holes are governed by a unique universal parameter N, which is set via a holographic relation similar to that between voxels and pixels [44];
- (iii)
- the metric coincides with the Schwarzschild metric away from the Planck scale, i.e., in the $N\gg 1$ limit.

## 2. Holographic Metric in $(3+1)$-dimensions

#### Derivation in Terms of Non-Local Gravity Actions

## 3. Higher Dimensional Holographic Metric

## 4. AdS Background and Hawking-Page Phase Transition

- for $b>{b}_{\mathrm{c}}$, the temperature admits both a minimum value ${T}_{\mathrm{min}}$ and a maximum value ${T}_{\mathrm{max}}$;
- for $b<{b}_{\mathrm{c}}$, the temperature is a monotonically increasing function of ${r}_{+}$;
- for $b={b}_{\mathrm{c}}$, the minimum and the maximum temperature coalesce, ${T}_{\mathrm{min}}={T}_{\mathrm{max}}={T}_{\mathrm{c}}$, in a point where the curve exhibit an inflection, $\mathrm{d}T/\mathrm{d}{r}_{+}={\mathrm{d}}^{2}T/\mathrm{d}{r}_{+}^{2}=0$.

- (i)
- for $T<{T}_{\mathrm{min}}$, there exist just one horizon radius ${r}_{1}$. The heat capacity of the black hole $C=\mathrm{d}M/\mathrm{d}T$ is positive defined, the systems is locally stable but the free energy is positive. This means that the black hole is a meta-stable state. The AdS-thermal background results the favorable state.
- (ii)
- For ${T}_{\mathrm{min}}<T<{T}_{\mathrm{max}}$, there are three horizon radii, ${r}_{1}<{r}_{2}<{r}_{3}$. While at ${r}_{1}$ and ${r}_{3}$ the heat capacity is positive, this is not the case at ${r}_{2}$ that is locally unstable. This corresponds the case of a mixed phase. The configuration in ${r}_{2}$ can decay to ${r}_{1}$ to ${r}_{3}$ or in thermal AdS. From Figure 7, one can see that ${r}_{1}$ is always globally unstable since $F({r}_{1})>0$, while ${r}_{3}$ may have either positive or negative free energy.
- (iii)
- From Figure 8 we see that there is a value ${T}_{\mathrm{tr}}$ at which the curve intersects itself by forming a cross at the vertex of the swallow tail. The value ${T}_{\mathrm{tr}}$ is the temperature at which the phase transition between small and big black holes occurs [135]. For ${T}_{\mathrm{min}}<T<{T}_{\mathrm{tr}}$ the pure AdS background is still favorable state, even if small black holes, ${r}_{1}$, and big black holes, ${r}_{3}$, are possible. The smaller black hole ${r}_{1}$ is the meta-stable state since it sits in local minimum of the free energy in Figure 7.
- (iv)
- For ${T}_{\mathrm{tr}}<T<{T}_{\mathrm{HP}}$ the AdS thermal state is still favorable state and the bigger black holes are in the meta-stable state since $F({r}_{1})>F({r}_{3})>0$. Here ${T}_{\mathrm{HP}}$ is the Hawking-Page temperature, i.e., $F({T}_{\mathrm{HP}})=0$. For ${T}_{\mathrm{HP}}<T<{T}_{\mathrm{max}}$, bigger black holes are the favorable state while AdS radiation and ${r}_{1}$ are meta-stable states. This can be seen by the fact that $F({r}_{3})<0$. For $T>{T}_{\mathrm{max}}$, there exist only one horizon radius and the black hole is the favorable state while the AdS radiation is the meta-stable state. From Figure 6, we see that ${T}_{\mathrm{max}}\simeq 0.03{L}_{\mathrm{P}}$.
- (v)
- As in the standard Schwarzschild-AdS case there exists a temperature ${T}_{\mathrm{coll}}$, above which the AdS thermal radiation cannot longer sustain itself. The value of such a temperature is of the order ${T}_{\mathrm{coll}}\sim {({L}_{\mathrm{P}}/b)}^{1/2}{M}_{\mathrm{P}}$ [130]. Accordingly there exist another regime: for $T>{T}_{\mathrm{coll}}$, the AdS radiation will inevitably collapse in a black hole. For $b<{b}_{\mathrm{coll}}$, the collapse of the AdS radiation occurs above the local maximum temperature, i.e., ${T}_{\mathrm{coll}}>{T}_{\mathrm{max}}$. A rough estimate of parameters suggests that ${b}_{\mathrm{coll}}\sim {10}^{3}{L}_{\mathrm{P}}$.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Size vs. Energy relation in Planck units. (

**a**) the size of particles, estimated by the Compton wavelength ${\lambda}_{\mathrm{C}}\sim {M}^{-1}$, compared to the size of black holes, determined by Schwarzschild radius ${r}_{+}\sim M$. The shaded area is inaccessible. At sub-Planckian energy scales, a length scale ambiguity arises; (

**b**) the self-complete gravity paradigm proposes a solution.

**Figure 2.**(

**a**) Temperature of the holographic metric in $(3+1)$-dimensions; (

**b**) heat capacity $C\equiv \mathrm{d}M/\mathrm{d}T$. The shades area is inaccessible $r<{r}_{\mathrm{e}}={L}_{\mathrm{P}}$. The asymptote in the heat capacity occurs at the maximum temperature. The positive heat capacity phase corresponds to the cool down phase also known as “SCRAM phase”.

**Figure 3.**The function $-{g}_{00}$ versus r for $n=2$. The three curves corresponds to three different values of the mass parameter M.

**Figure 4.**(

**a**) Temperatures of the higher dimensional holographic metric (n is the number of spatial dimensions); (

**b**) ration between the holographic metric temperature and that of the Schwarzschild-Tangherlini temperature black hole, ${T}_{\mathrm{Schwarz}}=(1+n)/(4\pi {r}_{+})$.

**Figure 5.**Plot of the Hawking temperature Equation (42) as function of the horizon radius ${r}_{+}$ with ${L}_{\mathrm{P}}=1$. The temperature goes to zero for ${r}_{+}={L}_{\mathrm{P}}$. The dashed line shows the behaviour of the standard Schwarzschild-AdS temperature. The $b\to \infty $ case corresponds to the temperature of the asymptotically flat holographic metric.

**Table 1.**Maximum radii ${r}_{\mathrm{max}}$ and maximum temperatures ${T}_{\mathrm{max}}\equiv T({r}_{\mathrm{max}})$ of the holographic black hole in $(3+n)$ spatial dimensions.

n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
---|---|---|---|---|---|---|---|---|

${r}_{\mathrm{max}}/{L}_{*}$ | 2.06 | 1.60 | 1.48 | 1.41 | 1.36 | 1.33 | 1.30 | 1.28 |

${T}_{\mathrm{max}}/{M}_{*}$ | 0.024 | 0.07 | 0.12 | 0.18 | 0.25 | 0.31 | 0.38 | 0.44 |

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**MDPI and ACS Style**

Frassino, A.M.; Köppel, S.; Nicolini, P.
Geometric Model of Black Hole Quantum *N*-portrait, Extradimensions and Thermodynamics. *Entropy* **2016**, *18*, 181.
https://doi.org/10.3390/e18050181

**AMA Style**

Frassino AM, Köppel S, Nicolini P.
Geometric Model of Black Hole Quantum *N*-portrait, Extradimensions and Thermodynamics. *Entropy*. 2016; 18(5):181.
https://doi.org/10.3390/e18050181

**Chicago/Turabian Style**

Frassino, Antonia M., Sven Köppel, and Piero Nicolini.
2016. "Geometric Model of Black Hole Quantum *N*-portrait, Extradimensions and Thermodynamics" *Entropy* 18, no. 5: 181.
https://doi.org/10.3390/e18050181