# Unruh Effect and Information Entropy Approach

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## Abstract

**:**

## 1. Introduction

## 2. Probability and Entropy

## 3. Unruh Effect

## 4. Unruh Entropy

- In order to obey, the energy conservation law N should be finite;
- In the case of (2 + 1) or (3 + 1)-dimensional space-time, the Unruh horizon should be considered as a radiation source of finite size.

## 5. Asymptotics of Unruh Entropy

## 6. Generalization to Intrinsic Degrees of Freedom

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

- Unruh, W.G. Notes on black-hole evaporation. Phys. Rev. D
**1976**, 14, 870–892. [Google Scholar] [CrossRef] [Green Version] - Hawking, S.W. Particle creation by black holes. Comm. Math. Phys.
**1975**, 43, 199–220. [Google Scholar] [CrossRef] - Bekenstein, J.D. Black holes and entropy. Phys. Rev. D
**1973**, 7, 2333–2346. [Google Scholar] [CrossRef] - Bekenstein, J.D. Statistical black-hole thermodynamics. Phys. Rev. D
**1975**, 12, 3077–3085. [Google Scholar] [CrossRef] - Crispino, L.C.B.; Higuchi, A.; Matsas, G.E.A. The Unruh effect and its applications. Rev. Mod. Phys.
**2008**, 80, 787–838. [Google Scholar] [CrossRef] [Green Version] - Letaw, J.R. Stationary world lines and the vacuum excitation of noninertial detectors. Phys. Rev. D
**1981**, 23, 1709–1714. [Google Scholar] [CrossRef] - Obadia, N.; Milgrom, M. On the Unruh effect for general trajectories. Phys. Rev. D
**2007**, 75, 065006. [Google Scholar] [CrossRef] [Green Version] - Letaw, J.R.; Pfautsch, J.D. Quantized scalar field in rotating coordinates. Phys. Rev. D
**1980**, 22, 1345–1351. [Google Scholar] [CrossRef] - Korsbakken, J.I.; Leinaas, J.M. The Fulling-Unruh effect in general stationary accelerated frames. Phys. Rev. D
**2004**, 70, 084016. [Google Scholar] [CrossRef] [Green Version] - Takagi, S. Vacuum noise and stress induced by uniform acceleration: Hawking-Unruh effect in Rindler manifold of arbitrary dimension. Prog. Theor. Phys. Suppl.
**1986**, 88, 1–142. [Google Scholar] [CrossRef] - Bisognano, J.J.; Wichmann, E.H. On the duality connection for a Hermitian scalar field. J. Math. Phys.
**1975**, 16, 985–1007. [Google Scholar] [CrossRef] [Green Version] - Bisognano, J.J.; Wichmann, E.H. On the duality connection for quantum fields. J. Math. Phys.
**1976**, 17, 303–321. [Google Scholar] [CrossRef] [Green Version] - Unruh, W.G.; Weiss, N. Acceleration radiation in interacting field theories. Phys. Rev. D
**1984**, 29, 1656–1662. [Google Scholar] [CrossRef] - Becattini, F. Thermodynamic equilibrium with acceleration and the Unruh effect. Phys. Rev. D
**2018**, 97, 085013. [Google Scholar] [CrossRef] [Green Version] - Becattini, F.; Rindori, D. Extensivity, entropy current, area law and Unruh effect. Phys. Rev. D
**2019**, 99, 125011. [Google Scholar] [CrossRef] [Green Version] - Prokhorov, G.Y.; Teryaev, O.V.; Zakharov, V.I. Unruh effect for fermions from the Zubarev density operator. Phys. Rev. D
**2019**, 99, 071901. [Google Scholar] [CrossRef] [Green Version] - Zubarev, D.N.; Prozorkevich, A.V.; Smolyanskii, S.A. Derivation of nonlinear generalized equations of quantum relativistic hydrodynamics. Theor. Math. Phys.
**1979**, 40, 821–831. [Google Scholar] [CrossRef] - Becattini, F.; Buzzegoli, M.; Grossi, F. Reworking the Zubarev’s approach to non-equilibrium quantum statistical mechanics. Particles
**2019**, 2, 197–207. [Google Scholar] [CrossRef] [Green Version] - Page, D.N. Information in black hole radiation. Phys. Rev. Lett.
**1993**, 71, 3473. [Google Scholar] [CrossRef] [Green Version] - Page, D.N. Hawking radiation and black hole thermodynamics. New J. Phys.
**2005**, 7, 203. [Google Scholar] [CrossRef] - Stoica, O.C. Revisiting the Black Hole Entropy and the Information Paradox. Adv. High Energy Phys.
**2018**, 2018, 4130417. [Google Scholar] [CrossRef] - Svaiter, B.F.; Svaiter, N.F. Inertial and noninertial particle detectors and vacuum fluctuations. Phys. Rev. D
**1992**, 46, 5267–5277. [Google Scholar] [CrossRef] - Sriramkumar, L.; Padmanabhan, T. Response of finite time particle detectors in noninertial frames and curved space-time. Class. Quant. Grav.
**1996**, 13, 2061–2079. [Google Scholar] [CrossRef] [Green Version] - Higuchi, A.; Matsas, G.E.A.; Peres, C.B. Uniformly accelerated finite-time detectors. Phys. Rev. D
**1993**, 48, 3731–3734. [Google Scholar] [CrossRef] [Green Version] - Kothawala, D.; Padmanabhan, T. Response of Unruh-DeWitt detector with time-dependent acceleration. Phys. Lett. B
**2010**, 690, 201–206. [Google Scholar] [CrossRef] [Green Version] - Chen, P.; Tajima, T. Testing Unruh radiation with ultraintense lasers. Phys. Rev. Lett.
**1999**, 83, 256–259. [Google Scholar] [CrossRef] [Green Version] - Louko, J.; Satz, A. Transition rate of the Unruh-DeWitt detector in curved spacetime. Class. Quant. Grav.
**2008**, 25, 055012. [Google Scholar] [CrossRef] - Akhmedova, V.; Pilling, T.; de Gill, A.; Singleton, D. Comments on anomaly versus WKB/tunneling methods for calculating Unruh radiation. Phys. Lett. B
**2009**, 673, 227–231. [Google Scholar] [CrossRef] - Bradler, K. Eavesdropping of quantum communication from a non-inertial frame. Phys. Rev. A
**2007**, 75, 022311. [Google Scholar] [CrossRef] [Green Version] - Han, M.; Olson, S.J.; Dowling, J.P. Generating entangled photons from the vacuum by accelerated measurements: Quantum information theory meets the Unruh-Davies effect. Phys. Rev. A
**2008**, 78, 022302. [Google Scholar] [CrossRef] [Green Version] - Parentani, R.; Massar, S. The Schwinger mechanism, the Unruh effect and the production of accelerated black holes. Phys. Rev. D
**1997**, 55, 3603–3613. [Google Scholar] [CrossRef] [Green Version] - Bell, J.S.; Leinaas, J.M. Electrons as accelerated thermometers. Nucl. Phys. B
**1983**, 212, 131–150. [Google Scholar] [CrossRef] [Green Version] - Bell, J.S.; Leinaas, J.M. The Unruh effect and quantum fluctuations of electrons in storage rings. Nucl. Phys. B
**1987**, 284, 488–508. [Google Scholar] [CrossRef] [Green Version] - Akhmedov, E.T.; Singleton, D. On the relation between Unruh and Sokolov-Ternov effects. Int. J. Modern Phys. A
**2007**, 22, 4797–4823. [Google Scholar] [CrossRef] [Green Version] - Banerjee, R.; Majhi, B.R. Hawking black body spectrum from tunneling mechanism. Phys. Lett. B
**2009**, 675, 243–245. [Google Scholar] [CrossRef] [Green Version] - Roy, D. The Unruh thermal spectrum through scalar and fermion tunneling. Phys. Lett. B
**2009**, 681, 185–189. [Google Scholar] [CrossRef] [Green Version] - Barshay, S.; Troost, W. A possible origin for temperature in strong interactions. Phys. Lett.
**1978**, 73B, 437–439. [Google Scholar] [CrossRef] - Grillo, A.F.; Srivastava, Y. Intrinsic temperature of confined systems. Phys. Lett. B
**1979**, 85, 377–380. [Google Scholar] [CrossRef] - Hosoya, A. Moving mirror effects in hadronic reactions. Progr. Theor. Phys.
**1979**, 61, 280–293. [Google Scholar] [CrossRef] [Green Version] - Barshay, S.; Braun, H.; Gerber, J.P.; Maurer, G. Possible evidence for fluctuations in the hadronic temperature. Phys. Rev. D
**1980**, 21, 1849–1853. [Google Scholar] [CrossRef] - Kharzeev, D.; Tuchin, K. From color glass condensate to quark gluon plasma through the event horizon. Nucl. Phys. A
**2005**, 753, 316–334. [Google Scholar] [CrossRef] [Green Version] - Castorina, P.; Kharzeev, D.; Satz, H. Thermal hadronization and Hawking-Unruh radiation in QCD. Eur. Phys. J. C
**2007**, 52, 187–201. [Google Scholar] [CrossRef] - Castorina, P.; Satz, H. Hawking-Unruh hadronization and strangeness production in high energy collisions. Adv. High Energy Phys.
**2014**, 2014, 376982. [Google Scholar] [CrossRef] - Biro, T.S.; Gyulassy, M.; Schram, Z. Unruh gamma radiation at RHIC? Phys. Lett. B
**2012**, 708, 276–279. [Google Scholar] [CrossRef] [Green Version] - Liu, F.; Wang, E.; Wang, X.-N.; Xu, N.; Zhang, B.-W. Proceedings, 28th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2019): Wuhan, China, November 3–9, 2019. Nucl. Phys. A
**2021**, 1005, 122104. [Google Scholar] - Antinori, F.; Dainese, A.; Giubellino, P.; Greco, V.; Lombardi, M.P.; Scomparin, E. Proceedings, 27th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 2018): Venice, Italy, May 14–19, 2018. Nucl. Phys. A
**2019**, 982, 1–1066. [Google Scholar] - Marolf, D.; Minic, D.; Ross, S.F. Notes on spacetime thermodynamics and the observer dependence of entropy. Phys. Rev. D
**2004**, 69, 064006. [Google Scholar] [CrossRef] [Green Version] - Jaynes, E.T. Information theory and statistical mechanics. In Lectures in Theoretical Physics; Ford, K.W., Ed.; Benjamin: New York, NY, USA, 1963; Volume 3, pp. 181–218. [Google Scholar]
- Jaynes, E.T. Prior probabilities. IEEE Trans. Syst. Sci. Cybern.
**1968**, 4, 227–241. [Google Scholar] [CrossRef] - Pathak, A. Elements of Quantum Computation and Quantum Communication, 1st ed.; Taylor & Francis: London, UK, 2013; pp. 1–340. [Google Scholar]
- Kharzeev, D.E. Quantum information approach to high energy interactions. Phil. Trans. R. Soc. A
**2021**, 380, 20210063. [Google Scholar] [CrossRef] - Schwinger, J. On Gauge Invariance and Vacuum Polarization. Phys. Rev.
**1951**, 82, 664–679. [Google Scholar] [CrossRef]

**Figure 1.**(Color online) The entropy $H(n,E|N,T)$ of Unruh radiation given by Equation (17) for fermions $(N=2)$ as function of $m/T$ and $M/T$.

**Figure 2.**(Color online) The same as Figure 1 but for bosons. The spectrum of bosons contains (

**a**) $N=100$ and (

**b**) $N=1000$ particles.

**Figure 3.**(Color online) Asymptotic behavior of entropy $H(n,E|N,T)$ given by Equation (23) at $T\to 0$ as function of $m/T$.

**Figure 4.**(Color online) High-temperature asymptotics of the entropy $H(n,E|N,T)$ of Unruh radiation given by Equation (28) for fermions $(N=2)$ as a function of m and M.

**Figure 5.**(Color online) The same as Figure 4 but for bosons with (

**a**) $N=100$ and (

**b**) $N=1000$ particles in the spectrum.

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**MDPI and ACS Style**

Teslyk, M.; Teslyk, O.; Zadorozhna, L.; Bravina, L.; Zabrodin, E.
Unruh Effect and Information Entropy Approach. *Particles* **2022**, *5*, 157-170.
https://doi.org/10.3390/particles5020014

**AMA Style**

Teslyk M, Teslyk O, Zadorozhna L, Bravina L, Zabrodin E.
Unruh Effect and Information Entropy Approach. *Particles*. 2022; 5(2):157-170.
https://doi.org/10.3390/particles5020014

**Chicago/Turabian Style**

Teslyk, Maksym, Olena Teslyk, Lidiia Zadorozhna, Larisa Bravina, and Evgeny Zabrodin.
2022. "Unruh Effect and Information Entropy Approach" *Particles* 5, no. 2: 157-170.
https://doi.org/10.3390/particles5020014