Abstract
The article is aimed to explore trajectories of charged particles moving under the effect of Lorentz force of Killing magnetic fields (i.e., closed 2-forms corresponding to Killing vector fields) in 3D pp-waves. First, we obtain magnetic trajectories in trivial magnetic background which are nothing but the geodesics of 3D pp-waves. Next, we characterize Killing vector fields and hence explore associated Killing magnetic trajectories in 3D pp-waves. In particular, we explicitly determine Killing magnetic trajectories in 3D Cahen–Wallach space and provide some pictorial examples.








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References
Barros, M., Romero, A.: Magnetic vortices. Europhys. Lett. 77(3), 34002 (2007)
Barros, M., Cabrerizo, J.L., Fernández, M., Romero, A.: Magnetic vortex filament flows. J. Math. Phys. 48(8), 082904 (2007)
Beem, J.K., Ehrlich, P.E., Easley, K.L.: Global Lorentzian Geometry, 2nd edn. CRC Press, Boca Raton (1996)
Brinkmann, H.W.: Einstein spaces which are mapped conformally on each other. Math. Ann. 94(1), 119–145 (1925)
Cabrerizo, J.L.: Magnetic fields in 2D and 3D sphere. J. Nonlinear Math. Phys. 20(3), 440–450 (2013)
Cahen, M., Wallach, N.: Lorentzian symmetric spaces. Bull. Am. Math. Soc. 76(3), 585–591 (1970)
Druţă-Romaniuc, S.L., Munteanu, M.I.: Magnetic curves corresponding to Killing magnetic fields in \(\mathbb{E} ^3\). J. Math. Phys. 52(11), 113506 (2011)
Druţă-Romaniuc, S.L., Munteanu, M.I.: Killing magnetic curves in a Minkowski 3-space. Nonlinear Anal. Real World Appl. 14(1), 383–396 (2013)
Ehlers, J., Kundt, W.: Exact solutions of the gravitational field equations. In: The Theory of Gravitation, pp. 49–101. Wiley, New York (1962)
Einstein, A.: Näherungsweise integration der feldgleichungen der gravitation, pp. 688–696. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, Berlin (1916)
Einstein, A.: Über gravitationswellen, pp. 154–167. Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften, Berlin (1918)
Feynman, R.P., Leighton, R.B., Sands, M.: The Feynman Lectures on Physics, Vol. II: The New Millennium Edition: Mainly Electromagnetism and Matter. Basic Books, New York (2015)
Gilkey, P., Nikčević, S.: Complete curvature homogeneous pseudo-Riemannian manifolds. Class. Quantum Grav. 21(15), 3755 (2004)
Iqbal, Z.: Differential geometry of magnetic and Killing magnetic trajectories in de Sitter 3-Space. Lobachevskii J. Math. 43(2), 394–409 (2022a)
Iqbal, Z.: Geometrical aspects of motion of charged particles in magnetic and Killing magnetic fields and their corresponding trajectories in anti-de Sitter 3-space. J. Syst. Geometric Theor. 20(2), 191–226 (2022b)
Iqbal, Z., Sengupta, J., Chakraborty, S.: Magnetic trajectories corresponding to Killing magnetic fields in a three-dimensional warped product. Int. J. Geom. Methods Mod. Phys. 17(14), 2050212 (2020)
Kundt, W.: The plane-fronted gravitational waves. Z. Phys. 163(1), 77–86 (1961)
Newman, S.C.: Semi-Riemannian Geometry: The Mathematical Language of General Relativity. Wiley, New York (2019)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, New York (1983)
Sachs, R.: Gravitational waves in general relativity. VI. The outgoing radiation condition. Proc. R. Soc. Lond. A 264(1318), 309–338 (1961)
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, New York (2009)
Sunada, T.: Magnetic flows on a Riemann surface. In: Proceedings of KAIST Mathematics Workshop (Analysis and Geometry), vol. 8, pp. 93–108 (1993)
Talman, R.: Geometric Mechanics: Toward a Unification of Classical Physics, 2nd edn. Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim (2007)
Thidé, B.: Electromagnetic Field Theory. Upsilon Books, Uppsala (2004)
Acknowledgements
I owe a deep sense of gratitude to Dr. Joydeep Sengupta (Aliah University, India) and Prof. Subenoy Chakraborty (Jadavpur University, India) for helping me in various directions. I further extend my deepest gratitude to the honorable editors and anonymous referees for their invaluable time devoted to this article and for their pearls of comments and suggestions which definitely enriched this article.
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Iqbal, Z. Magnetic Trajectories in Killing Magnetic Backgrounds Framed in 3D pp-Waves. J Nonlinear Sci 33, 16 (2023). https://doi.org/10.1007/s00332-022-09871-0
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DOI: https://doi.org/10.1007/s00332-022-09871-0