No abstract

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

The Carroll group was originally introduced by Lévy-Leblond [1] by considering the contraction of the Poincaré group as c → 0. In this paper an alternative definition, based on the geometric properties of a non-Minkowskian, non-Galilean but nevertheless boost-invariant, space-time structure is proposed. A "duality" with the Galilean limit c → ∞ is established. Our theory is illustrated by Carrollian electromagnetism.

Taking advantage of the two-parameter central extension of the planar Galilei group, we construct a non relativistic particle model in the plane. Owing to the extra structure, the coordinates do not commute. Our model can be viewed as the non-relativistic counterpart of the relativistic anyon considered before by Jackiw and Nair. For a particle moving in a magnetic field perpendicular to the plane, the two parameters combine with the magnetic field to provide an effective mass. For vanishing effective mass the phase space admits a two-dimensional reduction, which represents the condensation to collective "Hall" motions, and justifies the rule called "Peierls substitution". Quantization yields the wave functions proposed by Laughlin to describe the Fractional Quantum Hall Effect.