# Electron as a Tiny Mirror: Radiation from a Worldline with Asymptotic Inertia

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## Abstract

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## 1. Introduction: Fixed Radiation

## 2. Elements of Electrodynamics: Energy from Moving Electrons

## 3. GO Trajectory for Finite Energy Emission

## 4. Discussions: Mirrors, Electrons, and Black Holes

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Moore, G.T. Quantum theory of the electromagnetic field in a variable-length one-dimensional cavity. J. Math. Phys.
**1970**, 11, 2679–2691. [Google Scholar] [CrossRef] - DeWitt, B.S. Quantum field theory in curved space-time. Phys. Rep.
**1975**, 19, 295–357. [Google Scholar] [CrossRef] - Fulling, S.A.; Davies, P.C.W. Radiation from a moving mirror in two dimensional space-time: Conformal anomaly. Proc. R. Soc. Lond. A
**1976**, 348, 393–414. [Google Scholar] [CrossRef] - Davies, P.C.W.; Fulling, S.A. Radiation from moving mirrors and from black holes. Proc. R. Soc. Lond. A
**1977**, 356, 237–257. [Google Scholar] [CrossRef] - Nikishov, A.; Ritus, V. Emission of scalar photons by an accelerated mirror in (1+1) space and its relation to the radiation from an electrical charge in classical electrodynamics. J. Exp. Theor. Phys.
**1995**, 81, 615–624. Available online: http://jetp.ras.ru/cgi-bin/e/index/e/81/4/p615?a=list (accessed on 30 December 2022). - Ritus, V. The Symmetry, inferable from Bogoliubov transformation, between the processes induced by the mirror in two-dimensional and the charge in four-dimensional space-time. J. Exp. Theor. Phys.
**2003**, 97, 10–23. [Google Scholar] [CrossRef] [Green Version] - Ritus, V. Vacuum-vacuum amplitudes in the theory of quantum radiation by mirrors in 1+1-space and charges in 3+1-space. Int. J. Mod. Phys. A
**2002**, 17, 1033–1040. [Google Scholar] [CrossRef] - Ritus, V. Symmetries and causes of the coincidence of the radiation spectra of mirrors and charges in (1+1) and (3+1) spaces. J. Exp. Theor. Phys.
**1998**, 87, 25–34. [Google Scholar] [CrossRef] - Good, M.R.R.; Singha, C.; Zarikas, V. Extreme electron acceleration with fixed radiation energy. Entropy
**2022**, 24, 1570. [Google Scholar] [CrossRef] - Ritus, V.I. Finite value of the bare charge and the relation of the fine structure constant ratio for physical and bare charges to zero-point oscillations of the electromagnetic field in the vacuum. Phys.-Usp.
**2022**, 65, 468–486. [Google Scholar] [CrossRef] - Good, M.R.R.; Davies, P.C.W. Infrared acceleration radiation. arXiv
**2022**, arXiv:2206.07291. [Google Scholar] [CrossRef] - Good, M.R.R.; Anderson, P.R.; Evans, C.R. Mirror reflections of a black hole. Phys. Rev. D
**2016**, 94, 065010. [Google Scholar] [CrossRef] [Green Version] - Good, M.R.R.; Ong, Y.C. Particle spectrum of the Reissner–Nordström black hole. Eur. Phys. J. C
**2020**, 80, 1169. [Google Scholar] [CrossRef] - Good, M.R.R.; Foo, J.; Linder, E.V. Accelerating boundary analog of a Kerr black hole. Class. Quantum Grav.
**2021**, 38, 085011. [Google Scholar] [CrossRef] - Jackson, J.D. Classical Electrodynamics; John Wiley & Sons, Inc.: Hoboken, NJ, USA, 1999; Available online: https://www.scribd.com/doc/200995304/Classical-Electrodynamics-3rd-Ed-1999-John-David-Jackson (accessed on 30 December 2022).
- Zangwill, A. Modern Electrodynamics; Cambridge University Press: Cambridge, UK, 2012; Available online: https://faculty.kashanu.ac.ir/file/download/page/1604994434-modern-electrodynamics.pdf (accessed on 30 December 2022).
- Griffiths, D.J. Introduction to Electrodynamics; Cambridge University Press: New York, NY, USA, 2017. [Google Scholar] [CrossRef]
- Rindler, W.A. Essential Relativity: Special, General and Cosmological; Springer-Verlag: Berlin/Heidelber, Germany, 1977. [Google Scholar] [CrossRef]
- Good, M.R.R.; Yelshibekov, K.; Ong, Y.C. On horizonless temperature with an accelerating mirror. J. High Energy Phys.
**2017**, 3, 13. [Google Scholar] [CrossRef] [Green Version] - Myrzakul, A.; Xiong, C.; Good, M.R.R. CGHS black hole analog moving mirror and its relativistic quantum information as radiation reaction. Entropy
**2021**, 23, 1664. [Google Scholar] [CrossRef] - Feynman, R.P. Feynman Lectures on Gravitation; CRC Press/Taylor & Francis Group: Boca Raton, FL, USA, 2003. [Google Scholar] [CrossRef]
- Walker, W.R. Particle and energy creation by moving mirrors. Phys. Rev. D
**1985**, 31, 767–774. [Google Scholar] [CrossRef] - Carlitz, R.D.; Willey, R.S. Reflections on moving mirrors. Phys. Rev. D
**1987**, 36, 2327–2335. [Google Scholar] [CrossRef] - Ford, L.H.; Vilenkin, A. Quantum radiation by moving mirrors. Phys. Rev. D
**1982**, 25, 2569–2575. [Google Scholar] [CrossRef] - Good, M.R.; Linder, E.V.; Wilczek, F. Moving mirror model for quasithermal radiation fields. Phys. Rev. D
**2020**, 101, 025012. [Google Scholar] [CrossRef] [Green Version] - Moreno-Ruiz, A.; Bermudez, D. Optical analogue of the Schwarzschild–Planck metric. Class. Quant. Grav.
**2022**, 39, 145001. [Google Scholar] [CrossRef] - Good, M.R.R.; Linder, E.V. Modified Schwarzschild metric from a unitary accelerating mirror analog. New J. Phys.
**2021**, 23, 043007. [Google Scholar] [CrossRef] - Good, M.R.R.; Ong, Y.C. Signatures of energy flux in particle production: A black hole birth cry and death gasp. J. High Energy Phys.
**2015**, 7, 145. [Google Scholar] [CrossRef] - Birrell, N.D.; Davies, P.C.W. Quantum Fields in Curved Space; Cambridge Univercity Press: New York, NY, USA, 1982. [Google Scholar] [CrossRef]
- Good, M.R.R. Extremal Hawking radiation. Phys. Rev. D
**2020**, 101, 104050. [Google Scholar] [CrossRef] - Good, M.R.R. Reflecting at the speed of light. In Memorial Volume for Kerson Huang; Phua, K.K., Low, H.B., Xiong, C., Eds.; World Scientific Co., Ltd.: Singapore, 2017; pp. 113–116. [Google Scholar] [CrossRef] [Green Version]
- Zhakenuly, A.; Temirkhan, M.; Good, M.R.R.; Chen, P. Quantum power distribution of relativistic acceleration radiation: Classical electrodynamic analogies with perfectly reflecting moving mirrors. Symmetry
**2021**, 13, 653. [Google Scholar] [CrossRef] - Myhrvold, N.P. Thermal radiation from accelerated electrons. Ann. Phys.
**1985**, 160, 102–113. [Google Scholar] [CrossRef] - Paithankar, K.; Kolekar, S. Role of the Unruh effect in Bremsstrahlung. Phys. Rev. D
**2020**, 101, 065012. [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] - Bianchi, E.; Smerlak, M. Last gasp of a black hole: Unitary evaporation implies non-monotonic mass loss. Gen. Rel. Grav.
**2014**, 46, 1809. [Google Scholar] [CrossRef] [Green Version] - Bianchi, E.; Smerlak, M. Entanglement entropy and negative energy in two dimensions. Phys. Rev. D
**2014**, 90, 041904. [Google Scholar] [CrossRef] [Green Version] - Abdolrahimi, S.; Page, D.N. Hawking radiation energy and entropy from a Bianchi-Smerlak semiclassical black hole. Phys. Rev. D
**2015**, 92, 083005. [Google Scholar] [CrossRef] [Green Version] - Carter, B. Global structure of the Kerr family of gravitational fields. Phys. Rev.
**1968**, 174, 1559–1571. [Google Scholar] [CrossRef] [Green Version] - Burinskii, A. The Dirac-Kerr electron. Grav. Cosmol.
**2008**, 14, 109–122. [Google Scholar] [CrossRef] [Green Version] - Schmekel, B.S. Quasilocal energy of a charged rotating object described by the Kerr-Newman metric. Phys. Rev. D
**2019**, 100, 124011. [Google Scholar] [CrossRef] [Green Version] - Burinskii, A. The Dirac electron consistent with proper gravitational and electromagnetic field of the Kerr–Newman solution. Galaxies
**2021**, 9, 18. [Google Scholar] [CrossRef] - Letaw, J.R. Vacuum excitation of noninertial detectors on stationary world lines. Phys. Rev. D
**1981**, 23, 1709–1714. [Google Scholar] [CrossRef] - Good, M.; Juárez-Aubry, B.A.; Moustos, D.; Temirkhan, M. Unruh-like effects: Effective temperatures along stationary worldlines. J. High Energy Phys.
**2020**, 06, 059. [Google Scholar] [CrossRef] - Good, M.R.R.; Temirkhan, M.; Oikonomou, T. Stationary worldline power distributions. Int. J. Theor. Phys.
**2019**, 58, 2942–2968. [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] - Good, M.R.R.; Zhakenuly, A.; Linder, E.V. Mirror at the edge of the universe: Reflections on an accelerated boundary correspondence with de Sitter cosmology. Phys. Rev. D
**2020**, 102, 045020. [Google Scholar] [CrossRef] - Castagnino, M.; Ferraro, R. A toy cosmology: Radiation from moving mirrors, the final equilibrium state and the instantaneous model of particle. Ann. Phys.
**1985**, 161, 1–20. [Google Scholar] [CrossRef] - Akhmedov, E.T.; Buividovich, P.V.; Singleton, D.A. De Sitter space and perpetuum mobile. Phys. Atom. Nucl.
**2012**, 75, 525–529. [Google Scholar] [CrossRef] [Green Version] - Polyakov, A.M. Decay of vacuum energy. Nucl. Phys. B
**2010**, 834, 316–329. [Google Scholar] [CrossRef] [Green Version] - Leonhardt, U. The case for a Casimir cosmology. Philos. Trans. R. Soc. A
**2020**, 378, 20190229. [Google Scholar] [CrossRef] [PubMed] - Chen, P. et al. [AnaBHEL Collaboration] AnaBHEL (Analog Black Hole Evaporation via Lasers) experiment: Concept, design, and status. Photonics
**2022**, 9, 1003. [Google Scholar] [CrossRef] - Nico, J.S.; Dewey, M.S.; Gentile, T.R.; Mumm, H.P.; Thompson, A.K.; Fisher, B.M.; Kremsky, I.; Wietfeldt, F.E.; Chupp, T.E.; Cooper, R.L.; et al. Observation of the radioactive decay mode of the free neutron. Nature
**2006**, 444, 1059–1062. [Google Scholar] [CrossRef] [PubMed] - Bales, M.J. et al. [RDK II Collaboration] Precision measurement of the radiative β decay of the free neutron. Phys. Rev. Lett.
**2016**, 116, 242501. [Google Scholar] [CrossRef] [Green Version] - Lynch, M.H.; Good, M.R.R. Experimental observation of a moving mirror. arXiv
**2022**, arXiv:2211.14774. [Google Scholar] [CrossRef] - Chen, P.; Mourou, G. Trajectory of a flying plasma mirror traversing a target with density gradient. Phys. Plasmas
**2020**, 27, 123106. [Google Scholar] [CrossRef] - Chen, P.; Mourou, G. Accelerating plasma mirrors to investigate black hole information loss paradox. Phys. Rev. Lett.
**2017**, 118, 045001. [Google Scholar] [CrossRef] [PubMed]

**Figure 1.**The Larmor power (1) of the Good-Ong (GO) trajectory (6) as a function of time, t, and at final constant speed, $s=0.9$, i.e., Equation (7), with acceleration parameter, $\kappa =1$. This plot illustrates that the Larmor power never emits negative energy flux (NEF) and asymptotically dies off, consistent with a physically finite amount of total radiation energy (2).

**Figure 2.**The Feynman power, $F\xb7v$, associated with the self-force, F (3), of the GO trajectory (6), as a function of time and at final constant speed, $s=0.9$, i.e., Equation (8), with $\kappa =1$. This plot illustrates the Feynman power dies off asymptotically, has a period of negative radiation reaction, and is also consistent with a physically finite amount of total radiation energy (4).

**Figure 3.**A contour plot of the coefficients (13) as a function of in and out modes, ${\omega}^{\prime}$ and $\omega $, where final constant speed, $s=0.444$ and $\kappa =1$. The color gradient darkens for lower values of the count. This plot underscores the symmetry of the modes in the particle per mode squared distribution spectrum of the Bogoliubov $\beta $ coefficients (13).

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**MDPI and ACS Style**

Good, M.R.R.; Ong, Y.C.
Electron as a Tiny Mirror: Radiation from a Worldline with Asymptotic Inertia. *Physics* **2023**, *5*, 131-139.
https://doi.org/10.3390/physics5010010

**AMA Style**

Good MRR, Ong YC.
Electron as a Tiny Mirror: Radiation from a Worldline with Asymptotic Inertia. *Physics*. 2023; 5(1):131-139.
https://doi.org/10.3390/physics5010010

**Chicago/Turabian Style**

Good, Michael R. R., and Yen Chin Ong.
2023. "Electron as a Tiny Mirror: Radiation from a Worldline with Asymptotic Inertia" *Physics* 5, no. 1: 131-139.
https://doi.org/10.3390/physics5010010