Ricci-flat spacetimes admitting higher rank Killing tensors

Cariglia, Marco; Galajinsky, Anton

doi:10.1016/j.physletb.2015.04.001

Cited by 16 publications

(15 citation statements)

References 25 publications

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“…One question that comes to mind is if one can give a projective interpretation of the phenomenon of R-separation of the Schrödinger equation [25,26,33]. We also notice how the results discussed here imply that recent efforts in obtaining new non-trivial Einstein spaces from dynamical systems [8,19] can be extended to the more general case of Einstein-Weyl spaces, see [6]. A considerable amount of research has been produced in the area, as a non-exhaustive example see [4,28,29].…”

Section: Discussionmentioning

confidence: 75%

Null lifts and projective dynamics

Cariglia

2015

Annals of Physics

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We describe natural Hamiltonian systems using projective geometry. The null lift procedure endows the tangent bundle with a projective structure where the null Hamiltonian is identified with a projective conic and induces a Weyl geometry. Projective transformations generate a set of known and new dualities between Hamiltonian systems, as for example the phenomenon of coupling-constant metamorphosis. We conclude outlining how this construction can be extended to the quantum case for Eisenhart-Duval lifts.Comment: 11 pages, no figures. Some minor changes, typos amende

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Section: Discussionmentioning

confidence: 75%

Null lifts and projective dynamics

Cariglia

2015

Annals of Physics

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“…, x n ), (13) does not solve the vacuum Einstein equations. This is to be contrasted with (1) which is Ricci-flat provided the potential is a harmonic function [18], while for spacetimes of the ultrahyperbolic signature the potential should be an additive function [11]. In view of the ultrahyperbolic signature, it proves problematic to unambiguously link the Hamiltonian of the original mechanics to the time translation generator in spacetime.…”

Section: Eisenhart Lift For Higher Derivative Models Via Ostrogradskymentioning

confidence: 99%

“…The geodesic equations associated with (32) do reproduce (29), while the evolution of s is fixed provided the general solution of (29) is known. For earlier application of the Eisenhart lift to mechanics in pseudo-Euclidean space see [11,12]. Turning to the odd values of n = 2m + 1, the condition that the function x (n) V + U depends on x and its derivatives of even order only implies…”

Section: Canonical Transformation Of Ostrogradsky's Hamiltonian and Ementioning

confidence: 99%

“…In a series of recent works [5]- [10] various Lorentzian spacetimes admitting irreducible Killing tensors of rank greater than two have been constructed by applying the Eisenhart lift to specific integrable models. In [11,12] Ricci-flat spacetimes of the ultrahyperbolic signature which support higher rank Killing tensors or possess maximally superintegrable geodesic flows have been built along similar lines. Hidden symmetries of the Eisenhart lift metrics and the Dirac equation with flux have been studied in [13].…”

Section: Introductionmentioning

confidence: 99%

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Eisenhart lift for higher derivative systems

Galajinsky

Masterov

2017

Physics Letters B

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The Eisenhart lift provides an elegant geometric description of a dynamical system of second order in terms of null geodesics of the Brinkmann-type metric. In this work, we attempt to generalize the Eisenhart method so as to encompass higher derivative models. The analysis relies upon Ostrogradsky's Hamiltonian. A consistent geometric description seems feasible only for a particular class of potentials. The scheme is exemplified by the Pais-Uhlenbeck oscillator.Comment: V2: 12 pages, minor improvements, references added; the version to appear in PL

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“…Ricci-flat metrics of ultrahyperbolic signature, which admit higher rank Killing tensors, were constructed in[6].…”

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confidence: 99%

Eisenhart lift of 2-dimensional mechanics

Fordy

Galajinsky

2019

Eur. Phys. J. C

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The Eisenhart lift is a variant of geometrization of classical mechanics with d degrees of freedom in which the equations of motion are embedded into the geodesic equations of a Brinkmann-type metric defined on (d+2)-dimensional spacetime of Lorentzian signature. In this work, the Eisenhart lift of 2-dimensional mechanics on curved background is studied. The corresponding 4-dimensional metric is governed by two scalar functions which are just the conformal factor and the potential of the original dynamical system. We derive a conformal symmetry and a corresponding quadratic integral, associated with the Eisenhart lift. The energy-momentum tensor is constructed which, along with the metric, provides a solution to the Einstein equations. Uplifts of 2-dimensional superintegrable models are discussed with a particular emphasis on the issue of hidden symmetries. It is shown that for the 2-dimensional Darboux-Koenigs metrics, only type I can result in Eisenhart lifts which satisfy the weak energy condition. However, some physically viable metrics with hidden symmetries are presented.

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Ricci-flat spacetimes admitting higher rank Killing tensors

Cited by 16 publications

References 25 publications

Null lifts and projective dynamics

Null lifts and projective dynamics

Eisenhart lift for higher derivative systems

Eisenhart lift of 2-dimensional mechanics

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