1 research outputs found
Celestial Mechanics, Conformal Structures, and Gravitational Waves
The equations of motion for non-relativistic particles attracting
according to Newton's law are shown to correspond to the equations for null
geodesics in a -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 -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
-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 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 non-relativistic
gravitational dyons is also pointed out. Our results for general time-dependent
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