The role of elliptic integrals in calculating the gravitational lensing of a charged Weyl black hole surrounded by plasma

M. Fathi   ; J. R. Villanueva

Vestnik KRAUNC. Fiziko-matematičeskie nauki, Tome 36 (2021) no. 3, pp. 165-188

Résumé

In this paper, we mainly aim at highlighting the importance of (hyper-)elliptic integrals in the study of gravitational effects caused by strongly gravitating systems. For this, we study the application of elliptic integrals in calculating the light deflection as it passes a plasmic medium, surrounding a charged Weyl black hole. To proceed with this, we consider two specific algebraic ansatzes for the plasmic refractive index, and we characterize the photon sphere for each of the cases. This will be used further to calculate the angular diameter of the corresponding black hole shadow. We show that the complexity of the refractive index expressions, can result in substantially different types of dependencies of the light behavior on the spacetime parameters.

Détail

Keywords: elliptic integrals, light deflection, black holes.

Bibliographie

[1] Eddington A. S., Space, Time, and Gravitation: An Outline of the General Relativity Theory, Cambridge University Press, 1920

[2] Hagihara Y., “Theory of the Relativistic Trajeetories in a Gravitational Field of Schwarzschild”, Japanese Journal of Astronomy and Geophysics, 8, 67

[3] Darwin C. G., “The gravity field of a particle”, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 249:257 (1959), 180–194 | Zbl

[4] Darwin C. G., “The gravity field of a particle. II”, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 263:1312 (1961), 39–50 | Zbl

[5] Rauch K. P. , Blandford R. D., “Optical Caustics in a Kerr Spacetime and the Origin of Rapid X-Ray Variability in Active Galactic Nuclei”, The Astrophysical Journal, 421, 46 | DOI

[6] Beckwith K., Done C., “Extreme gravitational lensing near rotating black holes”, Monthly Notices of the Royal Astronomical Society, 359:4 (2005), 1217–1228 | DOI

[7] Hackmann E., Kagramanova V., Kunz J., Lammerzahl C., “Analytic solutions of the geodesic equation in higher dimensional static spherically symmetric spacetimes”, Phys. Rev. D, 78 (2008), 124018 | DOI

[8] Hackmann E., Lammerzahl C., “Complete analytic solution of the geodesic equation in schwarzschild– (anti-)de sitter spacetimes”, Phys. Rev. Lett., 100 (2008), 171101 | DOI | Zbl

[9] Hackmann E., Lammerzahl C., “Geodesic equation in schwarzschild-(anti-) de sitter space-times: Analytical solutions and applications”, Phys. Rev. D, 78 (2008), 024035 | DOI

[10] Hackmann E., Lammerzahl C., “Hyperelliptic functions and geodesic equations”, PAMM, 8:1 (2008), 10723–10724 | DOI | Zbl

[11] Bisnovatyi-Kogan G. S., Tsupko O. Y., “Strong gravitational lensing by schwarzschild black holes”, Astrophysics, 51 (2008), 99–111 | DOI

[12] Hackmann E., Kagramanova V., Kunz J., Lammerzahl C., “Analytic solutions of the geodesic equation in axially symmetric space-times”, EPL (Europhysics Letters), 88 (2009), 30008 | DOI

[13] Hackmann E., Lammerzahl C., Kagramanova V., Kunz J., “Analytical solution of the geodesic equation in kerr-(anti-) de sitter space-times”, Phys. Rev. D, 81 (2010), 044020 | DOI

[14] Hackmann E., Hartmann B., Lammerzahl C., Sirimachan P., “Complete set of solutions of the geodesic equation in the space-time of a schwarzschild black hole pierced by a cosmic string”, Phys. Rev. D, 81 (2010), 064016 | DOI

[15] Grunau S., Kagramanova V., “Geodesics of electrically and magnetically charged test particles in the reissner-nordstrom space-time: Analytical solutions”, Phys. Rev. D, 83 (2011), 044009 | DOI

[16] Hackmann E., Lammerzahl C., “Observables for bound orbital motion in axially symmetric space-times”, Phys. Rev. D, 85 (2012), 044049 | DOI

[17] Gibbons G. W., Vyska M., “The application of weierstrass elliptic functions to Schwarzschild null geodesics”, Classical and Quantum Gravity, 29 (2012), 065016 | DOI | Zbl

[18] Hackmann E., Lammerzahl C., Mac’{\i}as A., Maceda M., “Analytical solution methods for geodesic motion”, AIP Conference Proceedings, 1577:1 (2014), 78–88 | DOI

[19] Munoz G., “Orbits of massless particles in the schwarzschild metric: Exact solutions”, American Journal of Physics, 82:6 (2014), 564–573 | DOI

[20] Lammerzahl C., Hackmann E., “Analytical Solutions for Geodesic Equation in Black Hole Spacetimes”, Springer Proc. Phys., 170 (2016), 43–51 | DOI | Zbl

[21] De Falco V., Falanga M., and Stella L., “Approximate analytical calculations of photon geodesics in the schwarzschild metric”, Astronomy Astrophysics, 595 (2016), A38 | DOI

[22] Barlow N. S., Weinstein S. J., Faber J. A., “An asymptotically consistent approximant for the equatorial bending angle of light due to kerr black holes”, Classical and Quantum Gravity, 34 (2017), 135017 | DOI | Zbl

[23] Jusufi K., Sarkar N., Rahaman F., Banerjee A., Hansraj S., “Deflection of light by black holes and massless wormholes in massive gravity”, The European Physical Journal C, 78 (2018), 349 | DOI

[24] Ghaffarnejad H., Amirmojahedi M., Niad H., “Gravitational lensing of charged ayon-beatogarcia black holes and nonlinear effects of maxwell fields”, Advances in High Energy Physics, 2018 (2018), 3067272 | DOI | Zbl

[25] Villanueva J. R., Tapia F., Molina M., Olivares M., “Null paths on a toroidal topological black hole in conformal Weyl gravity”, Eur. Phys. J., C78 (2018), 10

[26] Hsiao Y.-W., Lee D.-S., Lin C.-Y., “Equatorial light bending around kerr-newman black holes”, Phys. Rev. D, 101 (2020), 064070 | DOI

[27] Vankov K., “Particle Orbits in General Relativity: from Planetary Solar System to Black Hole”, Environment., 2017 | Zbl

[28] Bisnovatyi-Kogan G. S., Tsupko O. Yu., “Gravitational Lensing in Presence of Plasma: Strong Lens Systems, Black Hole Lensing and Shadow”, Universe, 3:3 (2017), 57 | DOI

[29] Payandeh F., Fathi M., “Spherical Solutions due to the Exterior Geometry of a Charged Weyl Black Hole”, Int. J. Theor. Phys., 51 (2012), 2227–2236 | DOI | Zbl

[30] Fathi M., Olivares M., Villanueva J. R., “Classical tests on a charged weyl black hole: bending of light, shapiro delay and sagnac effect”, The European Physical Journal C, 80 (2020), 51 | DOI

[31] Fathi M., Kariminezhaddahka M., Olivares M., Villanueva J. R., “Motion of massive particles around a charged Weyl black hole and the geodetic precession of orbiting gyroscopes”, European Physical Journal C, 80 (2020), 377 | DOI

[32] Fathi M., Olivares M., Villanueva J. R., “Gravitational Rutherford scattering of electrically charged particles from a charged Weyl black hole”, The European Physical Journal Plus, 136 (2021), 420 | DOI

[33] Fathi M., Olivares M., Villanueva J. R., “Ergosphere, photon region structure, and the shadow of a rotating charged weyl black hole”, Galaxies, 9:2 (2021) | DOI

[34] Navarro J. F., Frenk C. S., White S. D. M., “Simulations of X-ray clusters”, Monthly Notices of the Royal Astronomical Society, 275 (1995), 720–740 | DOI

[35] Navarro J. F., Frenk C. S., White S. D. M., “The structure of cold dark matter halos”, The Astrophysical Journal, 462 (1996), 563 | DOI

[36] Gordon W., “Zur lichtfortpflanzung nach der relativitatstheorie”, Annalen der Physik, 377:22 (1923), 421–456 | DOI

[37] Plebanski J., “Electromagnetic waves in gravitational fields”, Phys. Rev., 118 (1960), 1396–1408 | DOI | Zbl

[38] de Felice F., “On the gravitational field acting as an optical medium”, General Relativity and Gravitation, 2 (1971), 347–357 | DOI

[39] J. L. Synge, Relativity: The general theory. Series in Physics, North-Holland Publication Co, Amsterdam, 1960

[40] Ehlers J., “Transition from the wave optics to geometrical optics in general relativity theory”, Z. Naturforsch., 22a (1968), 1328–32

[41] Chen B., Kantowski R., “Including absorption in gordon’s optical metric”, Phys. Rev. D, 79 (2009), 104007 | DOI

[42] Chen B., Kantowski R., “Distance redshift from an optical metric that includes absorption”, Phys. Rev. D, 80 (2009), 044019 | DOI

[43] Thompson R. T., “Covariant electrodynamics in linear media: Optical metric”, Phys. Rev. D, 97 (2018), 065001 | DOI

[44] Mannheim P. D., Kazanas D., “Exact vacuum solution to conformal weyl gravity and galactic rotation curves”, Astrophysical Journal, 342 (1989), 635–638 | DOI

[45] Bach R., “Zur Weylschen Relativitatstheorie und derWeylschen Erweiterung des Krummungstensorbegriffs”, Mathematische Zeitschrift, 9 (1921), 110–135 | DOI | Zbl

[46] Szekeres P., “Conformal Tensors”, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences, 304:1476 (1968), 113–122 | Zbl

[47] Perlick V., Tsupko O. Yu., Bisnovatyi-Kogan G. S., “Influence of a plasma on the shadow of a spherically symmetric black hole”, Phys. Rev., D92:10 (2015), 104031

[48] P. Byrd and M. Friedman, Handbook of elliptic integrals for engineers and scientists, Grundlehren der mathematischen Wissenschaften, Springer-Verlag, 1971 | Zbl

[49] Chandrasekhar S., The mathematical theory of black holes, Oxford classic texts in the physical sciences, Oxford Univ. Press, Oxford, 2002

[50] Tsukamoto N., “Black hole shadow in an asymptotically flat, stationary, and axisymmetric spacetime: The kerr-newman and rotating regular black holes”, Phys. Rev. D, 97 (2018), 064021 | DOI

[51] Perlick V., Tsupko O. Y., “Light propagation in a plasma on kerr spacetime: Separation of the Hamilton-Jacobi equation and calculation of the shadow”, Phys. Rev. D, 95 (2017), 104003 | DOI

[52] Singh B. P., Ghosh S. G., “Shadow of Schwarzschild–Tangherlini black holes”, Annals Phys., 395 (2018), 127–137 | DOI | Zbl