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Covariant Cherenkov radiation and its friction force

Will Price1,*, Martin S. Formanek2,†, and Johann Rafelski1,‡

  • *Contact author: wprice@arizona.edu
  • †Contact author: martin.formanek@eli-beams.eu
  • ‡Contact author: johannr@arizona.edu
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Phys. Rev. D 113, 116025 – Published 15 June, 2026

DOI: https://doi.org/10.1103/x81q-1b3v

Abstract

We derive the covariant generalization of the Frank-Tamm formula describing the Cherenkov radiation by a charged particle moving uniformly with a speed faster than the local speed of light within a homogeneous dielectric medium. We use our result to derive the covariant Cherenkov radiation reaction force and obtain a 4-force explicitly orthogonal to particle 4-velocity consistent with a relativistic friction force. We present the photon emission spectrum that is dependent primarily on the dielectric properties of the medium. We hint at a possible use of this work to interpret an excess of soft photons seen in relativistic hadron collisions.

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References (45)

  1. H. M. Fried and B. Muller, Vacuum Structure in Intense Fields (Springer Science & Business Media, New York, 2012), Vol. 255.
  2. J. Larmor, On the theory of the magnetic influence on spectra; and on the radiation from moving ions, London Edinburgh Dublin Philos. Mag. J. Sci. 44, 503 (1897).
  3. I. M. Frank and I. E. Tamm, Coherent visible radiation of fast electrons passing through matter, C. R. Acad. Sci. URSS 14, 109 (1937).
  4. I. E. Tamm, in Selected Papers I.E Tamm: Part I: Electrodynamics, edited by B. M. Bolotovsky and V. Y. Frenkel (Springer-Verlag, Berlin, 1991).
  5. D. Melrose, Quantum Plasmadynamics: Unmagnetized Plasmas (Springer, New York, 2008).
  6. J. Dunkel and P. Hänggi, Relativistic Brownian motion, Phys. Rep. 471, 1 (2009).
  7. M. Formanek, A. Steinmetz, and J. Rafelski, Radiation reaction friction: Resistive material medium, Phys. Rev. D 102, 056015 (2020).
  8. A. O. Barut, Electrodynamics and Classical Theory of Fields & Particles (Dover, New York, N.Y., 1980).
  9. G. A. Schott, Electromagnetic Radiation and the Mechanical Reactions Arising From It (Cambridge University Press, Cambridge, England, 1912).
  10. H. A. Lorentz, The Theory of Electrons (Dover Publications, New York, 1952).
  11. E. Poisson, An introduction to the Lorentz-Dirac equation, arXiv:gr-qc/9912045.
  12. H. Spohn, Dynamics of Charged Particles and Their Radiation Field (Cambridge University Press, Cambridge, England, 2004).
  13. F. Rohrlich, Classical Charged Particles (World Scientific, Singapore, 2007).
  14. P. Caldirola, A new model of classical electron, Nuovo Cimento (1955–1965) 3, 297 (1956).
  15. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Course of Theoretical Physics Vol. 2 (Pergamon Press, Oxford, 1975).
  16. G. W. Ford and R. F. O’Connell, Relativistic form of radiation reaction, Phys. Lett. A 174, 182 (1993).
  17. S. E. Gralla, A. I. Harte, and R. M. Wald, Rigorous derivation of electromagnetic self-force, Phys. Rev. D 80, 024031 (2009).
  18. W. Price, M. Formanek, and J. Rafelski, Radiation reaction and limiting acceleration, Phys. Rev. D 105, 016024 (2022).
  19. A. Di Piazza, C. Muller, K. Z. Hatsagortsyan, and C. H. Keitel, Extremely high-intensity laser interactions with fundamental quantum systems, Rev. Mod. Phys. 84, 1177 (2012).
  20. D. A. Burton and A. Noble, Aspects of electromagnetic radiation reaction in strong fields, Contemp. Phys. 55, 110 (2014).
  21. T. G. Blackburn, Radiation reaction in electron-beam interactions with high-intensity lasers, Plasma Phys. 4, 5 (2020).
  22. A. Gonoskov, T. G. Blackburn, M. Marklund, and S. S. Bulanov, Charged particle motion and radiation in strong electromagnetic fields, Rev. Mod. Phys. 94, 045001 (2022).
  23. A. Fedotov, A. Ilderton, F. Karbstein, B. King, D. Seipt, H. Taya, and G. Torgrimsson, Advances in QED with intense background fields, Phys. Rep. 1010, 1 (2023).
  24. R. Durrer, The cosmic microwave background: The history of its experimental investigation and its significance for cosmology, Classical Quantum Gravity 32, 124007 (2015).
  25. G. F. Ellis, R. Maartens, and M. A. H. MacCallum, Relativistic Cosmology (Cambridge University Press, Cambridge, England, 2012).
  26. G. M. Hale and M. R. Querry, Optical constants of water in the 200-nm to 200-μm wavelength region, Appl. Opt. 12, 555 (1973).
  27. M. H. Lynch, E. Cohen, Y. Hadad, and I. Kaminer, Accelerated-Cherenkov radiation and signatures of radiation reaction, New J. Phys. 21, 083038 (2019).
  28. T. H. O’Dell, The electrodynamics of magneto-electric media, Philos. Mag. 7, 1653 (1962).
  29. E. J. Post, Formal Structure of Electromagnetics (Dover Publications, New York, 1997).
  30. W. Price, Covariant theory of electromagnetism and radiation in vacuum and media, Ph.D. thesis, The University of Arizona, 2025.
  31. S. Schuster and M. Visser, Effective metrics and a fully covariant description of constitutive tensors in electrodynamics, Phys. Rev. D 96, 124019 (2017).
  32. L. Bel, Radiation states and the problem of energy in general relativity, Gen. Relativ. Gravit. 32, 2047 (2000).
  33. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1998).
  34. M. Formanek, C. Grayson, J. Rafelski, and B. Müller, Current-conserving relativistic linear response for collisional plasmas, Ann. Phys. (Amsterdam) 434, 168605 (2021).
  35. J. S. Schwinger, W.-Y. Tsai, and T. Erber, Classical and quantum theory of synergic synchrotron—Čerenkov radiation, Ann. Phys. (N.Y.) 96, 303 (1976).
  36. J. Rafelski, Relativity Matters. From Einstein’s EMC2 to Laser Particle Acceleration and Quark-Gluon Plasma (Springer, New York, 2017).
  37. A. Schild, On the radiation emitted by an accelerated point charge, J. Math. Anal. Appl. 1, 127 (1960).
  38. F. C. Jones, Lorentz-invariant formulation of Cherenkov radiation by tachyons, Phys. Rev. D 6, 2727 (1972).
  39. V. F. Perepelitsa, Lienard-Wiechert potentials for charged tachyons and several remarks on the tachyon Cherenkov radiation, arXiv:1502.06551.
  40. R. Lehnert and R. Potting, Vacuum Čerenkov radiation, Phys. Rev. Lett. 93, 110402 (2004).
  41. A. Belogianni, W. Beusch, T. Brodbeck, F. Dzheparov, B. French, P. Ganoti, J. Kinson, A. Kirk, V. Lenti, I. Minashvili et al., Observation of a soft photon signal in excess of QED expectations in pp interactions, Phys. Lett. B 548, 129 (2002).
  42. A. J. Macleod, A. Noble, and D. A. Jaroszynski, Cherenkov radiation from the quantum vacuum, Phys. Rev. Lett. 122, 161601 (2019).
  43. J. Schwinger, L. L. DeRaad, Jr., K. A. Milton, and W.-Y. Tsai, Classical Electrodynamics (Perseus Books, Reading, MA, 1998).
  44. M. Abramhowitz and A. Stegun, Pocketbook of Mathemmatical Functions: Abridged edition of Handbook of Mathematical Functions (Verlag Harri Deutsch, Frankfurt, 1984).
  45. G. N. Afanasiev, S. M. Eliseev, and Y. P. Stepanovsky, Transition of the light velocity in the Vavilov–Cerenkov effect, Proc. R. Soc. A 454, 1049 (1998).

Outline

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Phys. Rev. D 113, 116025– Published 15 June, 2026
  • Received 3 March 2026
  • Accepted 1 May 2026
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