ar X iv : h ep - t h / 05 12 18 8 v 1 1 5 D ec 2 00 5 Celestial Mechanics , Conformal Structures , and Gravitational Waves

Newton’s equations for the motion of N non-relativistic 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 electro-magnetic theories pointed out by Le Bellac and Lévy-Leblond. At the quantum level, the N-body Schrödinger 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 LagrangeLaplace-Runge-Lenz vector arising from three conformal Killing 2-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 non-relativistic gravitational dyons. Our results for 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. ‡mailto: [email protected] §UMR 6207 du CNRS associée aux Universités d’Aix-Marseille I et II et Université du Sud Toulon-Var; Laboratoire affilié à la FRUMAM-FR2291 ¶mailto: [email protected] ‖mailto: [email protected]