P. A. Horvathy

Conformal Galilei groups, Veronese curves, and Newton-Hooke spacetimes

Finite-dimensional nonrelativistic conformal Lie algebras spanned by polynomial vector fields of Galilei spacetime arise if the dynamical exponent is z = 2/N with N = 1,2,.... Their underlying group structure and matrix representation are constructed (up to a covering) by means of the Veronese map of degree N. Suitable quotients of the conformal Galilei groups provide us with Newton-Hooke nonrelativistic spacetimes with a quantized reduced negative cosmological constant � = −N.

Carroll versus Newton and Galilei: two dual non-Einsteinian concepts of time

The Carroll group was originally introduced by Levy-Leblond (1965 Ann. Inst. Henri Poincare 3 1) by considering the contraction of the Poincare 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, spacetime structure is proposed. A duality with the Galilean limit c → ∞ is established. Our theory is illustrated by Carrollian electromagnetism.

Kohn's theorem and Galilean symmetry

Abstract The relation between the separability of a system of charged particles in a uniform magnetic field and Galilean symmetry is revisited using Duvalʼs “Bargmann framework”. If the charge-to-mass ratios of the particles are identical, e a / m a = ϵ for all particles, then the Bargmann space of the magnetic system is isometric to that of an anisotropic harmonic oscillator. Assuming that the particles interact through a potential which only depends on their relative distances, the system splits into one representing the center of mass plus a decoupled internal part, and can be mapped further into an isolated system using Niedererʼs transformation. Conversely, the manifest Galilean boost symmetry of the isolated system can be “imported” to the oscillator and to the magnetic systems, respectively, to yield the symmetry used by Gibbons and Pope to prove the separability. For vanishing interaction potential the isolated system is free and our procedure endows all our systems with a hidden Schrodinger symmetry, augmented with independent internal rotations. All these properties follow from the cohomological structure of the Galilei group, as explained by Souriauʼs “decomposition barycentrique” .

Soft gravitons and the memory effect for plane gravitational waves

The “gravitational memory effect” due to an exact plane wave provides us with an elementary description of the diffeomorphisms associated with the analogue of “soft gravitons for this nonasymptotically flat system. We explain how the presence of the latter may be detected by observing the motion of freely falling particles or other forms of gravitational wave detection. Numerical calculations confirm the relevance of the first, second and third time integrals of the Riemann tensor pointed out earlier. Solutions for various profiles are constructed. It is also shown how to extend our treatment to Einstein-Maxwell plane waves and a midisuperspace quantization is given.

Newton-Hooke type symmetry of anisotropic oscillators ∗

Abstract Rotation-less Newton–Hooke-type symmetry, found recently in the Hill problem, and instrumental for explaining the center-of-mass decomposition, is generalized to an arbitrary anisotropic oscillator in the plane. Conversely, the latter system is shown, by the orbit method, to be the most general one with such a symmetry. Full Newton–Hooke symmetry is recovered in the isotropic case. Star escape from a galaxy is studied as an application.

Wigner–Souriau translations and Lorentz symmetry of chiral fermions

Abstract Chiral fermions can be embedded into Souriaus massless spinning particle model by “enslaving” the spin, viewed as a gauge constraint. The latter is not invariant under Lorentz boosts; spin enslavement can be restored, however, by a Wigner–Souriau (WS) translation, analogous to a compensating gauge transformation. The combined transformation is precisely the recently uncovered twisted boost, which we now extend to finite transformations. WS-translations are identified with the stability group of a motion acting on the right on the Poincare group, whereas the natural Poincare action corresponds to action on the left.

The memory effect for plane gravitational waves

Abstract We give an account of the gravitational memory effect in the presence of the exact plane wave solution of Einsteins vacuum equations. This allows an elementary but exact description of the soft gravitons and how their presence may be detected by observing the motion of freely falling particles. The theorem of Bondi and Pirani on caustics (for which we present a new proof) implies that the asymptotic relative velocity is constant but not zero, in contradiction with the permanent displacement claimed by Zeldovich and Polnarev. A non-vanishing asymptotic relative velocity might be used to detect gravitational waves through the “velocity memory effect”, considered by Braginsky, Thorne, Grishchuk, and Polnarev.

Eisenhart lifts and symmetries of time-dependent systems

Certain dissipative systems, such as Caldirola and Kannais damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with

Kohn's theorem and Newton-Hooke symmetry for Hill's equations

n

Killing tensors and canonical geometry

degrees of freedom. In this paper we treat these systems, their projective and conformal symmetries as well as their quantisation from the point of view of the Eisenhart lift to a Bargmann spacetime in