“Kepler Harmonies” and conformal symmetries

Abstract

Kepler's rescaling becomes, when "Eisenhart-Duval lifted" to 5-dimensional "Bargmann" gravitational wave spacetime, an ordinary spacetime symmetry for motion along null geodesics, which are the lifts of Keplerian trajectories. The lifted rescaling generates a well-behaved conserved Noether charge upstairs, which takes an unconventional form when expressed in conventional terms. This conserved quantity seems to have escaped attention so far. Applications include the Virial Theorem and also Kepler's Third Law. The lifted Kepler rescaling is a Chrono-Projective transformation. The results extend to celestial mechanics and Newtonian Cosmology.

Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

“…The projected integral of the motion behaves similarly, as illustrated by the Kepler problem [4,25]. It is worth to notice that the constraint p µ p µ = −m 2 in (II.10b) allows us to express p u as…”

Scaling and conformal symmetries for plane gravitational waves

“…So far we analysed what happens for a possibly time-dependent quadratic-in-X potential profile K(U ). We note that a similar analysis can be carried out for different homogeneous potentials [31].…”

“…However for a Chrono-Projective transformation L Y ξ = ψ ξ cf. (II.23) with ψ = 0 a new, "Chrono-Projective" type conserved quantity is associated [31]). See secs.…”

“…(I. 4) and the conserved charge it generates does not project to a conserved quantity downstairs (but it does "almost" [31]).…”

“…In this paper we return to the original terminology proposed in [32]. The paradigm of a Chrono-Projective transformation is provided by Kepler's third law [31,32]. 2 Symmetries of timelike geodesics were considered recently using non-local conservation laws [38].…”