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    Celestial Mechanics, Conformal Structures, and Gravitational Waves

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    The equations of motion for NN non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a (3N+2)(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 NN-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 NN-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)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 NN non-relativistic gravitational dyons is also pointed out. Our results for general time-dependent G(t)G(t) are applicable to the motion of point particles in an expanding universe. Finally we extend these results to the quantum regime.Comment: 26 pages, LaTe
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