# Small Black/White Hole Stability and Dark Matter

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## Abstract

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## 1. Remnants

## 2. White Holes

## 3. Quantum Processes and Time Scales

## 4. Timescales

## 5. Stability

## 6. Black and White Hole Processes

- Black hole volume increase and white hole volume decrease$$\begin{array}{ccc}\hfill |B,m,v\rangle & \to & |B,m,v+\delta v\rangle ,\hfill \end{array}$$$$\begin{array}{ccc}\hfill |W,m,v\rangle & \to & |W,m,v-\delta v\rangle .\hfill \end{array}$$This is simply determined by the Einstein’s equations if nothing else happens. The variation is computed in [26] to be governed by$$\frac{dv}{d\mathrm{t}}=\pm 3\sqrt{3}\pi {m}_{o}^{2}.$$
- White to black instability$$|W,m,v\rangle \to |B,m,v\rangle .$$This process is allowed by classical general relativity in the absence of any perturbation when there is a second asymptotic region, as it is apparent from the Left panel of Figure 3; but it can also be triggered by an external perturbation [25]. Notice that the volume does not change: this is due to the fact that this is a local process in the horizon region, which does not modify the interior. The lifetime of a white hole under decay to a black hole has been estimated to be proportional to its Schwarzschild radius [25]:$${\tau}_{\phantom{\rule{0.166667em}{0ex}}W\to B}\sim m.$$This is equivalent to a transition probability per unit of time$$p\sim {m}^{-1}.$$
- Hawking evaporation$$|B,m,v\rangle \to |B,m-\delta m,v\rangle .$$This is a process that decreases the mass of a black hole, produced by negative energy entering the hole when a Hawking quantum is radiated. It is a phenomenon described by the classical backreaction on the geometry of the dynamics of a quantum field. Hawking radiation theory gives$$\frac{dm}{d\mathrm{t}}=\frac{\hslash}{{m}^{2}}.$$Giving the lifetime for a massive black hole$${\tau}_{B}\sim \frac{{m}^{3}}{\hslash}.$$
- Black to white tunnelling$$|B,m,v\rangle \to |W,m,v\rangle .$$This is a genuine quantum gravitational process [11,16,30]. Its probability per unit of time is still unclear. We take here the conservative estimate derived in [15] using covariant Loop Quantum Gravity [31], which agrees with the semiclassical expectation for tunnelling phenomena, namely that this probability is suppressed by the semiclassical standard tunnelling factor$${e}^{-\frac{S}{\hslash}}\sim {e}^{-\frac{{m}^{2}}{\hslash}}$$$$p\sim {e}^{-\frac{{m}^{2}}{\hslash}}/m$$

## 7. Dynamical Evolution

## 8. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**

**Left:**in the extended Schwarzschild spacetime, which is stationary, the (light grey) region outside $r=2m+\u03f5$ (dotted line) is equally the outside of a black and a white hole.

**Center:**A collapsing star (dark grey) replaces the white hole region ( WH) in the non-stationary collapse metric.

**Right:**The time revered process. The difference between the last two can only be detected looking at the past, or the future.

**Figure 3.**

**Left:**Cauchy surfaces in extended Schwarzschild spacetime below and above the central sphere.

**Center:**Internal portion of a Cauchy surface describing a black hole formed by a collapsed star.

**Right:**Its time reversal, or white hole.

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**MDPI and ACS Style**

Rovelli, C.; Vidotto, F.
Small Black/White Hole Stability and Dark Matter. *Universe* **2018**, *4*, 127.
https://doi.org/10.3390/universe4110127

**AMA Style**

Rovelli C, Vidotto F.
Small Black/White Hole Stability and Dark Matter. *Universe*. 2018; 4(11):127.
https://doi.org/10.3390/universe4110127

**Chicago/Turabian Style**

Rovelli, Carlo, and Francesca Vidotto.
2018. "Small Black/White Hole Stability and Dark Matter" *Universe* 4, no. 11: 127.
https://doi.org/10.3390/universe4110127