# Analogous Hawking Effect in Dielectric Media and Solitonic Solutions

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## Abstract

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## 1. Introduction

## 2. The Covariant Generalization of the Hopfield Model

#### 2.1. Linearized Quantum Theory

#### 2.2. Exact Solitonic Solutions

## 3. The $\phi \psi $-Model

#### 3.1. Current Conservation and Inner Product

#### 3.2. Quantization of the Fields

#### 3.3. Amplitudes and Thermality

- A WKB analysis is produced, which holds in the asymptotic region and also in part of the so called linear region, except for a neighborhood of the horizon $x=0$. WKB wave functions are obtained [28].
- A near horizon approximation is developed [28]. Saddle point techniques and contour integrals around branch cut(s) are common tools in this calculation.

#### 3.4. More on the Linearized Equation

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Black Hole Metric for the Nondispersive Case

**Figure A1.**Example of the dielectric perturbation geometry. ${x}_{+}$ and ${x}_{-}$ indicate the black hole and white hole horizon positions, respectively.

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1. | Notice that we cannot expect for the model to be relativistically invariant, since Poincaré invariance is broken by the field $\mathbf{P}$, which selects a preferential class of frames: the ones where the dielectric is at rest. |

2. | For the Minkowski metric ${\eta}_{\mu \nu}$ the standard signature $(+,-,-,-)$ used for quantum field theory is chosen. |

3. | Henceforth, as in [19], we will use the accent $\phantom{\rule{4pt}{0ex}}\stackrel{\u02d8}{}\phantom{\rule{4pt}{0ex}}$ to denote a spacetime dependence on the given parameter. |

4. | By negative Kerr effect, we indicate a decrease in the refractive index of the medium in response to the passage of an electromagnetic pulse. |

**Figure 1.**This picture is the same as in [19]. The thick black lines represent the dispersion relations as seen in the lab frame, shown for positive frequencies and wave-numbers. The grey lines represent the axes of a frame boosted with velocity $\mathit{v}$. There are two positive branches for the transverse dispersion relation (curved thick lines): $0\le \omega <{\omega}_{0}$ and $\overline{\omega}\le \omega <\infty $. From the expression of the group velocity we see that for any given value of ${\nu}_{g}$, there are always two corresponding positive values ${\omega}_{1}$ and ${\omega}_{2}$, one for each positive branch. These points determine the superluminal and subluminal regions, w.r.t. the given group velocity.

**Figure 2.**Asymptotic dispersion relation in the external region for the Cauchy case in the comoving frame. The monotone branch is ${G}_{+}$, the non-monotone one is ${G}_{-}$. The line at $\omega =$ const is also drawn, and relevant states introduced in the text are explicitly indicated.

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**MDPI and ACS Style**

Belgiorno, F.; Cacciatori, S.L.
Analogous Hawking Effect in Dielectric Media and Solitonic Solutions. *Universe* **2020**, *6*, 127.
https://doi.org/10.3390/universe6080127

**AMA Style**

Belgiorno F, Cacciatori SL.
Analogous Hawking Effect in Dielectric Media and Solitonic Solutions. *Universe*. 2020; 6(8):127.
https://doi.org/10.3390/universe6080127

**Chicago/Turabian Style**

Belgiorno, Francesco, and Sergio L. Cacciatori.
2020. "Analogous Hawking Effect in Dielectric Media and Solitonic Solutions" *Universe* 6, no. 8: 127.
https://doi.org/10.3390/universe6080127