Celestial mechanics, conformal structures, and gravitational waves

Abstract

Newton's equations for the motion of $N$ nonrelativistic point particles attracting according to the inverse square law may be cast in the form of equations for null geodesics in a ($3N+2$)-dimensional Lorentzian spacetime which is Ricci flat and admits a covariantly constant null vector. Such a spacetime admits a Bargmann structure and corresponds physically to a plane-fronted gravitational wave (generalized pp wave). Bargmann electromagnetism in five dimensions actually comprises the two distinct Galilean electromagnetic theories pointed out by Le Bellac and L\'evy-Leblond. At the quantum level, the $N$-body Schr\"odinger equation may be cast into 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, in particular, (i) homographic solutions, (ii) the virial theorem, (iii) Kepler's third law, (iv) the Lagrange-Laplace-Runge-Lenz vector arising from three conformal Killing two-tensors, and (v) the motion under time-dependent inverse-square-law forces whose strength varies inversely as time in a manner originally envisaged by Dirac in his theory of a time-dependent gravitational constant $G(t)$. It is found that the problem can be reduced to one with time-independent inverse-square-law forces for a rescaled position vector and a new time variable. This transformation (Vinti and Lynden-Bell) is shown to arise from a particular conformal transformation of spacetime which preserves the Ricci-flat condition originally pointed out by Brinkmann. We also point out (vi) a Ricci-flat metric representing a system of $N$ nonrelativistic gravitational dyons. Our results for a general time-dependent $G(t)$ are also applicable by suitable reinterpretation to the motion of point particles in an expanding universe. Finally we extend these results to the quantum regime.