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Exact solution of the Dirac equation in spherical Rindler coordinates

Andre G. Campos*, Karen Z. Hatsagortsyan, and Christoph H. Keitel

  • *Contact author: agontijo@mpi-hd.mpg.de
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Phys. Rev. D 112, 085005 – Published 8 October, 2025

DOI: https://doi.org/10.1103/vvby-8vrj

Abstract

The exact solution to the Dirac equation corresponding to a bound state in the spherical Rindler metric is derived and its relation to the near-horizon Schwarzschild metric is investigated. It is shown how to represent these solutions with a specially chirped free-electron wave packet. The spherical Rindler frame stands out as being invariant under Lorentz transformations, in contrast with the 1+1D Rindler coordinates, and has been used in the description of the so-called expanding bubble spacetime. With heuristic modification of the wave function of the chirped free-electron wave packet quantitative predictions for the quantum properties of the electron crossing the black hole horizon are obtained.

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Laboratory cosmology with free spinorial wave packets

Andre G. Campos, Karen Z. Hatsagortsyan, and Christoph H. Keitel
Phys. Rev. D 112, L081701 (2025)

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References (28)

  1. W. Greiner, B. Müller, and J. Rafelski, Quantum Electrodynamics of Strong Fields (Springer, New York, 1985).
  2. U. H. Gerlach, Thermal ambience of expanding event horizon in Minkowski space-time, Phys. Rev. D 28, 761 (1983).
  3. U. H. Gerlach, Paired accelerated frames, Int. J. Mod. Phys. A 11, 3667 (1996).
  4. E. Witten, Instability of the Kaluza-Klein vacuum, Nucl. Phys. B195, 481 (1982).
  5. D. R. Brill and M. D. Matlin, Geodesic motion in a Kaluza-Klein bubble spacetime, Phys. Rev. D 39, 3151 (1989).
  6. D.-i. Hwang, B.-H. Lee, W. Lee, and D.-h. Yeom, Bubble collision in curved spacetime, Nucl. Phys. B, Proc. Suppl. 246, 196 (2014).
  7. B. Bhawal and C. Vishveshwara, Scalar waves in the Witten bubble spacetime, Phys. Rev. D 42, 1996 (1990).
  8. A. Davidson, Tachyonic compactification, Phys. Rev. D 35, 1811 (1987).
  9. A. G. Campos, K. Z. Hatsagortsyan, C. H. Keitel, companion Letter, Laboratory cosmology with free spinorial wave packets, Phys. Rev. D 112, 081701 (2025).
  10. A. Lasenby, C. Doran, J. Pritchard, A. Caceres, and S. Dolan, Bound states and decay times of fermions in a Schwarzschild black hole background, Phys. Rev. D 72, 105014 (2005).
  11. S. Dolan, C. Doran, and A. Lasenby, Fermion scattering by a schwarzschild black hole, Phys. Rev. D 74, 064005 (2006).
  12. G. H. Gossel, J. C. Berengut, and V. V. Flambaum, Energy levels of a scalar particle in a static gravitational field close to the black hole limit, Gen. Relativ. Gravit. 43, 2673 (2011).
  13. J. G. Rosa and S. R. Dolan, Massive vector fields on the schwarzschild spacetime: Quasinormal modes and bound states, Phys. Rev. D 85, 044043 (2012).
  14. S. Kar, Curious variant of the Bronnikov-Ellis spacetime, Phys. Rev. D 105, 024013 (2022).
  15. B. Czech, J. L. Karczmarek, F. Nogueira, and M. Van Raamsdonk, Rindler quantum gravity, Classical Quantum Gravity 29, 235025 (2012).
  16. K. Izumi and T. Shiromizu, DGP braneworld with a bubble of nothing, Phys. Rev. D 90, 046005 (2014).
  17. A. G. Campos and L. Fabbri, Relativistic dynamical inversion in manifestly covariant form, Phys. Rev. Res. 4, 023140 (2022).
  18. D. Hestenes, Proper dynamics of a rigid point particle, J. Math. Phys. (N.Y.) 15, 1778 (1974).
  19. D. Hestenes, Observables, operators, and complex numbers in the Dirac theory, J. Math. Phys. (N.Y.) 16, 556 (1975).
  20. D. Hestenes, Real spinor fields, J. Math. Phys. (N.Y.) 8, 798 (1967).
  21. W. E. Baylis, Classical eigenspinors and the Dirac equation, Phys. Rev. A 45, 4293 (1992).
  22. T. Takabayasi, Relativistic hydrodynamics of the Dirac matter. Part I. General theory, Prog. Theor. Phys. Suppl. 4, 1 (1957).
  23. J. Yvon, Équations de Dirac-Madelung, J. Phys. Radium 1, 18 (1940).
  24. V. B. Berestetskii, E. M. Lifshitz, and L. P. Pitevskii, Quantum Electrodynamics (Pergamon, Oxford, 1982).
  25. H. Weyl, The Theory of Groups and Quantum Mechanics (Courier Corporation, New York, 1950).
  26. C. Møller, The Theoy of Relativity (Clarendon Press, Oxford, 1952).
  27. V. S. Popov, Imaginary-time method in quantum mechanics and field theory, Phys. At. Nucl. 68, 686 (2005).
  28. A. I. Akhiezer and V. B. Berestetskii, Kvantovaya Elektrodynamika (in Russian) (Nauka, Moscow, 1969).

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Phys. Rev. D 112, 085005– Published 8 October, 2025
  • Received 21 January 2025
  • Accepted 4 August 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

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