"Kepler Harmonies" and conformal symmetries
P. -M. Zhang, M. Cariglia, M. Elbistan, G. W. Gibbons, P. A. Horvathy
TL;DR
This work reframes Keplerian motion within the Bargmann (Eisenhart) lift to a 5D spacetime, where Kepler's rescaling becomes a Chrono-Projective conformal symmetry acting on null geodesics. The lifted symmetry yields a conserved scalar charge ${\mathcal Q}=3 t p_t + 2 {\bf x}\cdot{\bm p} + s p_s$; its projection to 3D coordinates connects to the energy $E$ and the Kepler action, enabling Noether-based derivations of the Virial theorem and Kepler's Third Law via $\frac{a^3}{T^2}=\frac{GM_{\odot}}{4\pi^2}$. The formalism extends to an $N$-body Newtonian cosmology, producing a generalized conserved charge ${\mathcal Q} = -3 T E + 2 \sum_a {\bf x}_a \cdot {\bm p}_a + s m$ with $E=\sum_a \frac{{\bm p}_a^2}{2 m_a}+U$ and ${\mathcal L}_N=\sum_a \frac{m_a}{2}({\bf x}'_a)^2 - U$, thereby unifying the geometric origin of these classic relations. Overall, the paper demonstrates that Keplerian laws admit a geometric Noether explanation in a higher-dimensional, conformally related spacetime, offering potential applications to celestial mechanics and Newtonian cosmology.
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.