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Celestial Mechanics, Conformal Structures, and Gravitational Waves
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abstract
The equations of motion for $N$ non-relativistic particles attracting according to Newton's law are shown to correspond to the equations for null geodesics in a $(3N+2)$-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 $N$-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 $N$-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 $G(t)$ 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 $N$ non-relativistic gravitational dyons is also pointed out. Our results for general time-dependent $G(t)$ are applicable to the motion of point particles in an expanding universe. Finally we extend these results to the quantum regime.
Forward citations
Cited by 15 Pith papers
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Planar Carrollian statistical physics is well-defined thanks to central extensions and rotation, yielding logarithmic entropy scaling with disc area and two-dimensional ideal-gas pressure.
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Gauge vs (hidden) physical symmetries of FLRW cosmologies
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p-brane Newton--Cartan Geometry
A p-brane Newton-Cartan structure admits a torsion-free compatible affine connection exactly when it is Augustinian, and all such connections are classified by longitudinal-frame field strengths.
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Affine Extension of the Free-Particle--Oscillator--Inverted-Oscillator Triangle
The linear-potential (Airy) system is an affine–Heisenberg extension of the free-particle–oscillator–inverted-oscillator conformal triangle, with Airy states obtained from accelerated-frame, oscillator-condensation, a...
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The authors derive exact analytic solutions for gravitational wave displacement memory when the flyby profile is approximated by the hyperbolic Scarf potential, with magic amplitude values |g|=(2n+1) sqrt(n(n+1)).
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Peierls substitution and Hall motion in exotic Carroll dynamics
Exotic Carroll particles with two-parameter central extension move by an anomalous Hall law, and their critical reduction reproduces the Dunne-Jackiw-Trugenberger model underlying the Peierls substitution.
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Penrose limits and TsT for fibered $I$-branes
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