Celestial Mechanics, Conformal Structures, and Gravitational Waves

TL;DR

This work reframes non-relativistic N-body celestial mechanics within a Bargmann (5D) spacetime, showing Newtonian trajectories correspond to null geodesics and that conformal structures control both classical invariants (e.g., a Lagrange–Laplace–Runge–Lenz-type vector) and time-dependent couplings G(t). The authors develop Vinti–Lynden–Bell conformal mappings, derive generalized virial relations, and connect to Brinkmann–Kaluza–Klein theories, including gravitational dyons and Taub–NUT-like structures, all within a Ricci-flat Bargmann setting. They extend the framework to the quantum regime via a covariant Schrödinger equation, reveal mass prequantization conditions, and show how extended Schrödinger symmetry governs the solutions. The approach yields a unifying geometric perspective on time-dependent gravitation, conformal symmetries, and quantum aspects of non-relativistic gravity with potential applications to expanding-universe dynamics and topologically nontrivial quantum states.

Abstract

The equations of motion for $N$ non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a $(3N+2)$-dimensional Lorentzian, Ricci-flat, spacetime with a covariantly constant null vector. Such a spacetime admits a Bargmann structure and corresponds physically to a generalized pp-wave. Bargmann electromagnetism in five dimensions comprises the two Galilean electro-magnetic theories (Le Bellac and Lévy-Leblond). At the quantum level, the $N$-body Schrödinger equation retains the form of a massless wave equation. We exploit the conformal symmetries of such spacetimes to discuss some properties of the Newtonian $N$-body problem: homographic solutions, the virial theorem, Kepler's third law, the Lagrange-Laplace-Runge-Lenz vector arising from three conformal Killing 2-tensors, and motions under inverse square law forces with a gravitational constant $G(t)$ varying inversely as time (Dirac). The latter problem is reduced to one with time independent forces for a rescaled position vector and a new time variable; this transformation (Vinti and Lynden-Bell) arises from a conformal transformation preserving the Ricci-flatness (Brinkmann). A Ricci-flat metric representing $N$ non-relativistic gravitational dyons is also pointed out. Our results for general time-dependent $G(t)$ are applicable to the motion of point particles in an expanding universe. Finally we extend these results to the quantum regime.