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Gravitational light deflection and shadows in a warm plasma

Barbora Bezděková*

Volker Perlick†

  • Department of Physics, Faculty of Natural Sciences, University of Haifa, Haifa 3498838, Israel and Haifa Research Center for Theoretical Physics and Astrophysics, University of Haifa, Haifa 3498838, Israel

  • *Contact author: bbezdeko@campus.haifa.ac.il
  • †Contact author: volker.perlick@uni-bremen.de
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Phys. Rev. D 114, 084043 – Published 14 October, 2026

DOI: https://doi.org/10.1103/l4hp-smpp

Abstract

Analytical studies of light bending in a dispersive medium near compact objects, e.g., black holes or neutron stars, are most challenged by a suitable definition of the medium. The most realistic model would be a hot magnetized plasma. In such a medium, however, an analytical description of light rays is very difficult. Therefore, usually an isotropic dispersive medium is assumed in analytical calculations. While it is possible to formulate equations for a general refractive index, which some studies do, most attention in the literature is given to the particular case of a cold, nonmagnetized electron-ion plasma. Whereas this model covers many astrophysically relevant situations, there are indications that in some cases the plasma temperature is so high that the approximation of a cold plasma is no longer valid. For this reason, we consider in this paper a warm, nonmagnetized electron-ion plasma, where the temperature is not set equal to zero but assumed to be small enough, such that relevant equations can be linearized with respect to it. After discussing the general equations for light rays in such a medium on a general-relativistic spacetime, we specify to the axially symmetric and stationary case which includes the spherically symmetric and static case. In particular, we calculate the influence of a warm plasma on the bending angle. In the spherically symmetric and static case, we also calculate the shadow in a warm plasma. We illustrate the general results with a static (respectively corotating) and an infalling warm plasma on Schwarzschild and Kerr spacetimes.

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References (34)

  1. J. A. Bittencourt, Waves in hot magnetized plasmas, in Fundamentals of Plasma Physics (Springer, New York, 2004), pp. 515–559.
  2. J. L. Synge, Relativity: The General Theory (North-Holland Publishing Company, Amsterdam, 1960).
  3. D. O. Muhleman and I. D. Johnston, Radio propagation in the solar gravitational field, Phys. Rev. Lett. 17, 455 (1966).
  4. D. O. Muhleman, R. D. Ekers, and E. B. Fomalont, Radio interferometric test of the general relativistic light bending near the sun, Phys. Rev. Lett. 24, 1377 (1970).
  5. V. Perlick, Ray Optics, Fermat’s Principle, and Applications to General Relativity (Springer, Berlin-Heidelberg-New York, 2000).
  6. G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Gravitational radiospectrometer, Gravitation Cosmol. 15, 20 (2009).
  7. G. S. Bisnovatyi-Kogan and O. Y. Tsupko, Gravitational lensing in a non-uniform plasma, Mon. Not. R. Astron. Soc. 404, 1790 (2010).
  8. O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Gravitational lensing in plasma: Relativistic images at homogeneous plasma, Phys. Rev. D 87, 124009 (2013).
  9. X. Er and S. Mao, Effects of plasma on gravitational lensing, Mon. Not. R. Astron. Soc. 437, 2180 (2014).
  10. A. Rogers, Frequency-dependent effects of gravitational lensing within plasma, Mon. Not. R. Astron. Soc. 451, 4536 (2015).
  11. V. Perlick, O. Y. Tsupko, and G. S. Bisnovatyi-Kogan, Influence of a plasma on the shadow of a spherically symmetric black hole, Phys. Rev. D 92, 104031 (2015).
  12. V. Perlick and O. Y. Tsupko, Light propagation in a plasma on Kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow, Phys. Rev. D 95, 104003 (2017).
  13. V. Perlick and O. Y. Tsupko, Light propagation in a plasma on Kerr spacetime. II. Plasma imprint on photon orbits, Phys. Rev. D 109, 064063 (2024).
  14. J. W. Freeman and R. E. Lopez, The cold solar wind, J. Geophys. Res. 90, 9885 (1985).
  15. F. Bagenal, Giant planet magnetospheres, Annu. Rev. Earth Planet Sci. 20, 289 (1992).
  16. A. Czechowski and S. Grzedzielski, A cold plasma layer at the heliopause, Adv. Space Res. 16, 321 (1995).
  17. R. Genzel, F. Eisenhauer, and S. Gillessen, The galactic center massive black hole and nuclear star cluster, Rev. Mod. Phys. 82, 3121 (2010).
  18. G. Lanzuisi et al., NuSTAR measurement of coronal temperature in two luminous, high-redshift quasars, Astrophys. J. 875, L20 (2019).
  19. K. Akiyama et al., First M87 Event Horizon Telescope results. VIII. Magnetic field structure near the event horizon, Astrophys. J. 910, L13 (2021).
  20. O. Y. Tsupko, Deflection of light rays by a spherically symmetric black hole in a dispersive medium, Phys. Rev. D 103, 104019 (2021).
  21. B. Bezděková and J. Bičák, Light deflection in plasma in the Hartle-Thorne metric and in other axisymmetric spacetimes with a quadrupole moment, Phys. Rev. D 108, 084043 (2023).
  22. B. Bezděková, O. Y. Tsupko, and C. Pfeifer, Deflection of light rays in a moving medium around a spherically symmetric gravitating object, Phys. Rev. D 109, 124024 (2024).
  23. C. Pfeifer, B. Bezděková, and O. Y. Tsupko, Light deflection in axially symmetric stationary spacetimes filled with a moving medium, Phys. Rev. D 112, 104064 (2025).
  24. P. C. Clemmow and A. J. Willson, The dispersion equation in plasma oscillations, Proc. R. Soc. A. 237, 117 (1956).
  25. K. Imre, Oscillations in a relativistic plasma, Phys. Fluids 5, 459 (1962).
  26. F. C. Michel, Accretion of matter by condensed objects, Astrophys. Space Sci. 15, 153 (1972).
  27. V. Perlick and O. Y. Tsupko, Calculating black hole shadows: Review of analytical studies, Phys. Rep. 947, 1 (2022).
  28. S. Issaoun et al., The size, shape, and scattering of Sagittarius A* at 86 GHz: First VLBI with ALMA, Astrophys J. 871, 30 (2019).
  29. J. Jiang, A. C. Fabian, T. Dauser, L. Gallo, J. A. García, E. Kara, M. L. Parker, J. A. Tomsick, D. J. Walton, and C. S. Reynolds, High density reflection spectroscopy—II. The density of the inner black hole accretion disc in AGN, Mon. Not. R. Astron. Soc. 489, 3436 (2019).
  30. L. Mallick et al., High-density disc reflection spectroscopy of low-mass active galactic nuclei, Mon. Not. R. Astron. Soc. 513, 4361 (2022).
  31. O. Y. Tsupko and G. S. Bisnovatyi-Kogan, Hills and holes in the microlensing light curve due to plasma environment around gravitational lens, Mon. Not. R. Astron. Soc. 491, 5636 (2020).
  32. B. Bezděková, V. Perlick, and J. Bičák, Light propagation in a plasma on an axially symmetric and stationary spacetime: Separability of the Hamilton-Jacobi equation and shadow, J. Math. Phys. (N. Y.) 63, 092501 (2022).
  33. K. Schulze-Koops, V. Perlick, and D. J. Schwarz, Sachs equations for light bundles in a cold plasma, Classical Quantum Gravity 34, 215006 (2017).
  34. M. Sárený and V. Balek, Effect of black hole-plasma system on light beams, Gen. Relativ. Gravit. 51, 141 (2019).

Outline

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Phys. Rev. D 114, 084043– Published 14 October, 2026
  • Received 17 December 2025
  • Accepted 17 September 2026
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