Eisenhart lifts and symmetries of time-dependent systems

Abstract

Certain dissipative systems, such as Caldirola and Kannai's damped simple harmonic oscillator, may be modelled by time-dependent Lagrangian and hence time dependent Hamiltonian systems with n 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 n + 2 dimensions, equipped with its covariantly constant null Killing vector field. Reparametrization of the time variable corresponds to conformal rescalings of the Bargmann metric. We show how the Arnold map lifts to Bargmann spacetime. We contrast the greater generality of the Caldirola-Kannai approach with that of Arnold and Bateman. At the level of quantum mechanics, we are able to show how the relevant Schrödinger equation emerges naturally using the techniques of quantum field theory in curved spacetimes, since a covariantly constant null Killing vector field gives rise to well defined one particle Hilbert space.Time-dependent Lagrangians arise naturally also in cosmology and give rise to the phenomenon of Hubble friction. We provide an account of this for Friedmann-Lemaître and Bianchi cosmologies and how it fits in with our previous discussion in the non-relativistic limit.

Conformal symmetries and integrals of the motion in pp waves with external electromagnetic fields

“…pp-wave space-times play a role for the Eisenhart-Duval lift [12][13][14]16] of 2-dimensional classical dynamics. Analysing the meaning of the charges in this context, we have shown that after expressing them in terms of transversal coordinates, the term with m reduces to a constant.…”

Geometry of the isotropic oscillator driven by the conformal mode

“…(19) also implies that the spacetime (12) admits a geodesic null congruence with vanishing expansion, shear and vorticity and thus belongs to the Kundt class. The distinguished vector field ξ µ can be used to construct the trace-free energy-momentum tensor 4 T 4 The factor d 2π , d being the dimension of the x-subspace, is chosen for further convenience.…”

“…where ǫ = 0 for the null geodesics and ǫ = −1 for the time-like geodesics. Given the general solution in (4), the dynamics of the extra variable s is unambiguously fixed by the first order ordinary differential equation (34). Thus the original dynamical system (1) is recovered if one implements the null reduction along s.…”

“…In general, geometrization of a Lagrangian conservative system, whose kinetic term involves a positive definite metric, amounts to reformulating its equations of motion as the geodesic equations in a properly chosen curved spacetime or embedding them into the geodesics of a larger theory such that the time evolution of the extra degrees of freedom is unambiguously fixed, provided the dynamics of the original model is known. The Jacobi metric [1] and the Eisenhart lift [2,3] represent the conventional tools of that kind (for a recent application to time-dependent systems see [4]). While the former method yields a Riemannian metric, the latter gives a Lorentzian metric thus paving the way for applications in the general relativistic context.…”

General theory of Galilean gravity

“…Our initial motivation for this research was to have a deeper understanding of the torsion condition. In nonrelativistic holography the bulk metrics are a pp-wave generalization of AdS spaces [38], clearly related to so called Eisenhart-Duval lift metrics [23,26,[39][40][41][42][43]. However, Eisenhart-Duval lift metrics have always been studied in the context of zero torsion.…”