Abstract
Using the correspondence between solutions of gravitational and gauge theories (the so-called classical double copy conjecture) some electromagnetic fields with vortices are constructed, for which the Lorentz force equations are analytically solvable. The starting point is a certain class of plane gravitational waves exhibiting the conformal symmetry. The notion of the Niederer transformation, crucial for the solvability, is analyzed in the case of the Lorentz force equation on the curved spacetimes as well as its derivation by means of integrals of motion (associated with conformal generators preserving these vortices) is presented. Furthermore, some models discussed recently in the context of the intense laser beams are constructed from their gravitational counterparts, with the special emphasis put on the focusing property, and new solvable examples are presented.
Highlights
Gravitational waves have been intensively studied since the invention of general relativity
The pair ðg; A Þ is the one for which the transverse part of the Lorentz force equations is given by Eq (4.11)
In the present work, using the idea of the classical double copy, we construct null electromagnetic fields which are explicitly solvable and directly generalize the electromagnetic vortices considered, in the context of singular optics, in Ref. [33]. Since these results are strictly related to the notion of Niederer’s transformation, we analyze the geometric extension of the latter in the case of the plane gravitational spacetimes endowed with the crossed electromagnetic fields (3.2) and (3.3); in this approach the transverse part of the Lorentz force equations transformed by the Niederer map can be obtained by means of a new metric and a new, crossed, electromagnetic field
Summary
Gravitational waves have been intensively studied since the invention of general relativity. Classes consists of a metric family describing linearly or circularly polarized gravitational pulses This exceptional family admits the proper conformal transformations; it can be used to model impulsive gravitational waves with the Dirac delta profile. Some gravitational phenomena (such as singularities, focusing, classical cross section, and the velocity memory effect) can be analyzed analytically, in the Dirac delta limit [17,21] Such a situation is strictly related to the existence of the so-called Niederer transformation [21,22,23] (see Sec. II for some details). [39,40]; they capture some essential features of the transverse magnetic beam of laser light near the beam axis We apply this procedure to the linearly polarized gravitational metrics and obtain an explicitly solvable model described in Ref. VII we give a summary with an outlook for further studies
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