Celestial Mechanics, Conformal Structures, and Gravitational Waves

TL;DR

This paper explores the deep geometric and conformal structures underlying Newtonian N-body dynamics, revealing connections to higher-dimensional spacetimes, symmetries, and quantum analogs, with applications to cosmology and gravitational wave theory.

Contribution

It introduces a novel geometric framework linking Newtonian mechanics to Ricci-flat Lorentzian spacetimes and explores conformal symmetries, extending classical results to quantum and cosmological contexts.

Findings

Equations of motion correspond to null geodesics in higher-dimensional Ricci-flat spacetimes.

Conformal symmetries yield insights into classical solutions like Kepler's laws and the virial theorem.

The framework extends to quantum regimes and variable gravitational constants in expanding universes.

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 L\'evy-Leblond). At the quantum level, the $N$-body Schr\"odinger 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.