# Glimpses on the Micro Black Hole Planck Phase

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## Abstract

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## 1. Introduction

## 2. Gups

## 3. Translation and Rotation Invariance of the Gups

## 4. From the Uncertainty Principle to the Mass–Temperature Relation

## 5. Minimum Masses, Maximum Temperatures

## 6. Emission Rate Equation

## 7. Micro Black Hole Lifetime

## 8. Emission Rate Equation at the End Point

## 9. Entropy and Heat Capacity

#### 9.1. Entropy

- for $n>0$$$\begin{array}{c}\hfill S=\frac{2\pi {k}_{B}}{{\omega}_{n}}\left(\frac{{y}^{n+2}}{n+2}-\frac{\beta}{4n}{y}^{n}+O({\beta}^{1+\frac{n}{2}})\right)\end{array}$$
- for $n=0$$$\begin{array}{ccc}\hfill S& =& 2\pi {k}_{B}{\int}_{{y}_{MIN}}^{y}\left(y-\frac{\beta}{4}{y}^{-1}+O({\beta}^{2})\phantom{\rule{0.166667em}{0ex}}\alpha \phantom{\rule{0.166667em}{0ex}}{y}^{-3}\right)dy\hfill \\ & =& 2\pi {k}_{B}{\left[\frac{{y}^{2}}{2}-\frac{\beta}{4}logy+O({\beta}^{2})\frac{\alpha}{2{y}^{2}}\right]}_{\sqrt{\beta}}^{y}\hfill \\ & =& 2\pi {k}_{B}\left[\frac{{y}^{2}}{2}-\frac{\beta}{4}\left(logy+2+K\right)+\frac{\beta}{8}log\beta +O({\beta}^{2})\right]\hfill \end{array}$$

- for $n>0$$$\begin{array}{ccc}& S& =\frac{2\pi {k}_{B}}{{\omega}_{n}}\times \hfill \\ & \times & \left(\frac{{y}^{n+2}}{n+2}-\frac{\beta}{2}{\left(\frac{{\omega}_{n}}{2}\right)}^{\frac{1}{n+1}}\left(\frac{n+1}{n(n+2)}\right){y}^{\frac{n(n+2)}{n+1}}+\dots \right)\hfill \end{array}$$
- for $n=0$

#### 9.2. Heat Capacity

- If $\beta =0$, then $C<0$ for any $\mathsf{\Theta}$. Black holes are bodies with negative specific heat.
- If $\beta =0$, then C approaches 0 ($C\to {0}^{-}$) only when $\overline{\mathsf{\Theta}}\to +\infty $.
- If $\beta >0$, then we have $C=0$ for $\overline{\mathsf{\Theta}}={\overline{\mathsf{\Theta}}}_{MAX}=\frac{1}{\sqrt{\beta}}$, i.e. $\mathsf{\Theta}=(n+1)/(2\pi \sqrt{\beta})$.

## 10. Conclusions and Outlooks

- Both principles predict remnants of finite rest mass as the end product of Hawking evaporation of black holes.
- For deformation parameter $\beta \simeq 1.5$ or greater the mass thresholds predicted by MBH GUP are remarkably lower than those of ST GUP, meaning that the production of a micro black hole is largely enhanced by the MBH GUP. In particular, micro black holes should have been detectable in any number of extra dimensions, at the designed energy for LHC-like machines, or larger.
- The micro black hole lifetimes predicted by MBH GUP are in general always longer that those predicted by the ST GUP (and the difference is particularly noticeable for $N=5,6,\dots $ spatial dimensions). However the lifetimes predicted by both GUPs are, roughly, one order of magnitude shorter than those predicted by the standard Hawking evaporation (based on the standard Heisenberg principle) (see Figure 7 and Figure 8).
- The GUP-corrected entropy, for both GUPs, is lower than the standard Hawking entropy. In fact, as we see in Equations (118), (119) and (124), the leading corrective terms in $\beta $ have negative signs. The reason for this diminished entropy, in respect to the standard entropy (which is computed on the basis of the Heisenberg principle), can be traced back to the squeezing of the volume of the elementary cells in phase space, due to the new measure introduced by the GUPs (on this, see also Appendix C).
- The heat capacity predicted by both GUPs drops to zero in a finite time, at a finite temperature, meaning that the evaporation stops, and the black hole in its final stage cannot exchange energy with the surrounding space. It ceases then to interact thermodynamically with the environment. In the present framework, this is a strong indication for the existence of a final evaporation product of finite rest mass, denoted as remnant. We can expect therefore that for this remnant, which interacts with the environment only gravitationally, through its mass, the very concept of temperature becomes meaningless, being it an object with zero heat capacity, comparable for example with an elementary particle.

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

**Figure A1.**Functions $m(\tau )$ with $m(0)=15$ and ${m}_{min}=1$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A2.**Functions $m(\tau )$ with $m(0)=10$ and ${m}_{min}=1$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A3.**Functions $m(\tau )$ with $m(0)=5$ and ${m}_{min}=1$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A4.**Functions $m(\tau )$ with $m(0)=2$ and ${m}_{min}=1$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A5.**Functions $m(\tau )$ with $m(0)=15$ and ${m}_{min}=4$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A6.**Functions $m(\tau )$ with $m(0)=10$ and ${m}_{min}=4$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

**Figure A7.**Functions $m(\tau )$ with $m(0)=6$ and ${m}_{min}=4$. Green: standard Hawking emission; Blue: GUP corrected; Red: GUP + Squeezing corrected.

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**Figure 1.**Mass–temperature relation for the ST GUP, $\beta =2$, for $N=3,4,5$ from the bottom line to the top.

**Figure 2.**Minimal masses for stringy (ST) (blue line) and micro black hole (MBH) (green line) generalized uncertainty principles (GUPs), for $\beta =2$, and ${M}_{Pl}\simeq 1$ TeV, for $N\ge 4$. Red line is the LHC energy limit.

**Figure 3.**Minimal masses for ST and MBH GUPs, for $\beta =2$, with ${M}_{Pl}$ variable as in ADD model (red line).

**Figure 5.**Diagrams of ${\tau}_{bh}$ for Stringy and MBH GUPs, for N = 3 and $\beta =2$. Mass and time in Planck units.

**Figure 6.**Diagrams of ${\tau}_{bh}$ vs. m. Green: emission of photons only. Red: emission of gravitons only. Upper diagram obtained with MBH GUP, lower diagram with stringy GUP. $N=4$ and $\beta =2$. Mass and time in Planck units (in $N=4$).

**Figure 7.**Diagrams of ${\tau}_{bh}$ vs. m. Green: emission of photons only. Red: emission of gravitons only. Upper diagram obtained with MBH GUP, lower diagram with stringy GUP. $N=5$ and $\beta =2$. Mass and time in Planck units (in $N=5$).

**Figure 8.**Diagrams of ${\tau}_{bh}$ vs. m in the Hawking limit $\beta \to 0$. Green: emission of photons only. Red: emission of gravitons only. $N=5$. Mass and time in Planck units (in $N=5$).

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Scardigli, F.
Glimpses on the Micro Black Hole Planck Phase. *Symmetry* **2020**, *12*, 1519.
https://doi.org/10.3390/sym12091519

**AMA Style**

Scardigli F.
Glimpses on the Micro Black Hole Planck Phase. *Symmetry*. 2020; 12(9):1519.
https://doi.org/10.3390/sym12091519

**Chicago/Turabian Style**

Scardigli, Fabio.
2020. "Glimpses on the Micro Black Hole Planck Phase" *Symmetry* 12, no. 9: 1519.
https://doi.org/10.3390/sym12091519