Niederer’s transformation, time-dependent oscillators and polarized gravitational waves

Abstract

It is noted that the Niederer transformation can be used to find the explicit relation between time-dependent linear oscillators, including the most interesting case when one of them is harmonic. A geometric interpretation of this correspondence is provided by certain subclasses of pp-waves; in particular the ones strictly related to the proper conformal transformations. This observation allows us to show that the pulses of plane gravitational wave exhibiting the maximal conformal symmetry are analytically solvable. Particularly interesting is the circularly polarized family for which some aspects (such as the classical cross section, velocity memory effect and impulsive limit) are discussed in more detail. The role of the additional integrals of motion, associated with the conformal generators, is clarified by means of Ermakov-Lewis invariants. Possible applications to the description of interaction of electromagnetic beams with matter are also indicated.

From polarized gravitational waves to analytically solvable electromagnetic beams

“…(6.10) with χ = 0. This fact perfectly agrees with the gravitational picture where K (1) is a conformal generator for the gravitational plane wave with the profile H (1) , see [21]. In consequence eqs.…”

“…(4.11) corresponding the purely gravitational case, i.e. A = 0 and g = g (1,2) , have been discussed in [17,21] (see Sec. 2), while for the Minkowski spacetime endowed with the electromagnetic field given by (3.15) and (3.16) in Sec.…”

Scaling and conformal symmetries for plane gravitational waves

“…The memory problem requires finding the geodesics in the gravitational wave spacetime, and the integrability of the geodesic equations is related to conformal symmetry [13,[20][21][22].…”

“…1) where X ∈ R 2 and U, V are the transversal resp. light-cone coordinates on the gravitational wave space-time play also an important role, namely for the integrability of the geodesic equations [20][21][22]. In fact, (I.1) carries lightlike geodesics to lightlike geodesics, as illustrated on fig.2.…”

Energy-momentum and angular-momentum of a gyratonic pp -waves spacetime

“…The ppwaves space-time were investigated in the 1950's and 1960's specially by Peres, Pirani and Bondi [2,3], and most recently in the Refs. [4][5][6][7][8][9][10][11].…”