# Classical Limit for Dirac Fermions with Modified Action in the Presence of a Black Hole

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## Abstract

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## 1. Introduction

^{3}He superfluid in [7]. It allows to calculate in a demonstrative way Hawking radiation [8] (see also, for example, [9,10,11,12] and references therein).

## 2. Dirac Fermions in the Black Hole in the Painleve–Gullstrand Reference Frame

## 3. Covariant Formulation of the Theory and Its Classical Limit

## 4. The Stress–Energy Tensor of the Non-Interacting Classical Particles

#### 4.1. General Expression for the Stress–Energy Tensor

#### 4.2. The Stress–Energy Tensor in the Limit $\mu \to \infty $

#### 4.3. Expression for the Stress–Energy Tensor for Finite $\mu ,$ in the Case When the Substance Is Co-Moving with the Space Flow

## 5. Description of the Gravitational Collapse in the Generalized Painlevé-Gullstrand Coordinates

## 6. Classical Dynamics of Particles

## 7. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Velocity of “vacuum” (vertical axis) as a function of r (horizontal axis) for $Q=0.4$ and $M=0.5\text{}{m}_{P}$.

**Figure 2.**The typical Fermi surface form in the plane $({p}_{r},{p}_{\perp})$ within the black hole at the values of parameters $M=0.5\text{}{m}_{P}$, $m=0.01\text{}{m}_{P}$, $\mu ={m}_{P}$ and $Q=0.4$. Here the vertical axis corresponds to ${p}_{r}$ while the horizontal axis corresponds to ${p}_{\perp}$. For these values the external horizon is placed at ${r}_{+}=0.64/{m}_{P}$. We represent the Fermi surface at $r=0.5/{m}_{P}$.

**Figure 3.**The radial trajectory of the particle that falls down to the black hole (red solid line): The dependence of radial coordinate r in the units of $1/{m}_{P}$ on time (in the same units); radial momentum ${p}_{r}$ in the units of ${m}_{P}$ as a function of time (dashed blue line). The vertical axis corresponds to r and ${p}_{r}$, while the horizontal axis corresponds to time. Values of parameters: $M=0.5\text{}{m}_{P}$, $m=0.1\text{}{m}_{P}$ and $\mu ={m}_{P}$, while $Q=0.4$. The external horizon is represented by the dotted green line. The particle starts falling at $r\left(0\right)=1.2/\mu $ and $p\left(0\right)=0$. One can see that this particle falls together with “vacuum”. It reaches the center of the BH and stays there.

**Figure 4.**The radial trajectory of the particle that falls down to the black hole and escapes from it (red solid line): The dependence of radial coordinate r in the units of $1/{m}_{P}$ on time (in the same units); radial momentum ${p}_{r}$ in the units of ${m}_{P}$ as a function of time (dashed blue line). The vertical axis corresponds to r and ${p}_{r}$, while the horizontal axis corresponds to time. Values of parameters: $M=0.5\text{}{m}_{P}$, $m=0.1\text{}{m}_{P}$, $\mu ={m}_{P}$ and $Q=0.4$. The external horizon is represented by the dotted green line. The particle starts falling at $r\left(0\right)=1.2/\mu $ and $p\left(0\right)=0.1\mu $. This particle falls more slow than “vacuum”. It reaches the vicinity of the center of the BH. There the repulsion force pushes it away, it escapes from the BH, then falls again, etc.

**Figure 5.**The radial trajectory of the particle that traverses the black hole (red solid line): The dependence of radial coordinate r in the units of $1/{m}_{P}$ on time (in the same units); radial momentum ${p}_{r}$ in the units of ${m}_{P}$ as a function of time (dashed blue line). The vertical axis corresponds to r and ${p}_{r}$, while the horizontal axis corresponds to time. Values of parameters: $M=0.5\text{}{m}_{P}$, $m=0.1\text{}{m}_{P}$, $\mu ={m}_{P}$ and $Q=0.4$. The external horizon is represented by the dotted green line. The particle starts falling at $r\left(0\right)=1.2/\mu $ and $p\left(0\right)=-0.1\mu $. This particle falls faster than “vacuum”. It reaches the center of the BH and crosses it. The repulsion force accelerates it, and the particle escapes at the diametrically opposite point.

**Figure 6.**The radial trajectory of the particle that falls down to the black hole and escapes from it: The dependence of radial coordinate r in the units of $1/{m}_{P}$ on time. The vertical axis corresponds to r, while the horizontal axis corresponds to time. Values of parameters: $M=6\text{}{m}_{P}$, $m=0.01\text{}{m}_{P}$, $\mu ={m}_{P}$ and $Q=0.1$. The external horizon h is not represented here, but the motion starts at $r\left(0\right)=1.2\text{}h$ and $p\left(0\right)=0.01\text{}\mu $. It is supposed that this figure represents qualitatively the typical trajectory for $M\gg \mu \sim {m}_{P}\gg m$.

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Lewkowicz, M.; Zubkov, M.
Classical Limit for Dirac Fermions with Modified Action in the Presence of a Black Hole. *Symmetry* **2019**, *11*, 1294.
https://doi.org/10.3390/sym11101294

**AMA Style**

Lewkowicz M, Zubkov M.
Classical Limit for Dirac Fermions with Modified Action in the Presence of a Black Hole. *Symmetry*. 2019; 11(10):1294.
https://doi.org/10.3390/sym11101294

**Chicago/Turabian Style**

Lewkowicz, Meir, and Mikhail Zubkov.
2019. "Classical Limit for Dirac Fermions with Modified Action in the Presence of a Black Hole" *Symmetry* 11, no. 10: 1294.
https://doi.org/10.3390/sym11101294