Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

Chanda, Sumanto; Gibbons, G. W.; Guha, Partha; Maraner, Paolo; Werner, Marcus C.

doi:10.1063/1.5098869

Cited by 32 publications

(29 citation statements)

References 18 publications

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“…The Jacobi metric has so far been considered for 'natural' Lagrangian systems [1,2,3], 'natural' Lagrangian systems subject to magnetic-like interactions [12], relativistic particles in a static spacetime [8,9,10] and relativistic particles in stationary spacetimes [11,12]. In all these cases (12) reproduces the already known result.…”

Section: Relation To Known Casesmentioning

confidence: 85%

“…It should be noted that this formal derivation of the Jacobi line element is not a rigorous prove that the paths in configuration space of the general Lagrangian system (6) are the geodesics of the energy dependent Finsler metric (12). This statement can be proved more simply a posteriori, by showing that, with an appropriate choice of parameterisation, the geodesic equations associated with (12) are identical to the equations of motion for (6).…”

Section: Line Elementmentioning

confidence: 91%

“…The Jacobi line element for a general Lagrangian system that can be expanded as a power series in theẋ i , can also be derived by the method used in [10] and [12] to obtain the Jacobi metric for timelike geodesics in static and stationary spacetimes respectively. The idea is to use the conservation of energy to rewrite the Jacobi Lagrangian, i.e.…”

Section: Appendix A: the Jacobi Metric As A Finsler Metricmentioning

confidence: 99%

“…The Jacobi metric generalises to the spatial paths of relativistic massive particles moving in static spacetimes [8,9,10]. It further extends to the spatial paths of dynamical systems subject to velocity-dependent interactions, such as classical charged particles in a magnetic background or relativistic particles in a stationary spacetime [11,12], with the important difference that the corresponding metric is no longer Riemannian, but a Finsler metric of Randers type [13]. More generally, in [14] it has been proved that the solutions of the Euler-Lagrange equations of a strongly convex autonomous Lagrangian system are the geodesics of an associated energy dependent Finsler function [15,16].…”

Section: Introductionmentioning

confidence: 99%

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On the Jacobi metric for a general Lagrangian system

Maraner

2019

Journal of Mathematical Physics

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An explicit expression for the Jacobi metric for a general Lagrangian system is obtained as a series expansion in the square root of the kinetic energy of the system and the corresponding geodesics are described in terms of an appropriate non-linear connection and the associated curvature. In the limit of low kinetic energies the trajectories of motion of any Lagrangian system are very well approximated by the geodesics of an energy dependent Randers metric or, equivalently, by the paths in configuration space of a representative point subject to electromagnetic-and gravitational-like force fields. For higher kinetic energy values the trajectories of motion are instead the geodesics of a general energy dependent Finsler metric, corresponding also to the paths in configuration space of a representative point subject to a hierarchy of a potentially infinite number of covariant force fields that generalise the electromagnetic and gravitational ones. Some general implications of these findings are discussed.

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Section: Relation To Known Casesmentioning

confidence: 85%

Section: Line Elementmentioning

confidence: 91%

Section: Appendix A: the Jacobi Metric As A Finsler Metricmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On the Jacobi metric for a general Lagrangian system

Maraner

2019

Journal of Mathematical Physics

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“…To this end, we will apply the definition of deflection angle given by Ono et al [64]. In order to use the GB theorem, we shall apply the Jacobi metric method [70,71]. For stationary spacetime, the corresponding Jacobi metric is the Jacobi-Maupertuis Randers-Finsler metric (JMRF).…”

Section: Introductionmentioning

confidence: 99%

Finite-Distance Gravitational Deflection of Massive Particles by the Kerr-like Black Hole in the Bumblebee Gravity Model

Li¹,

Овгун²

2019

Preprint

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In this paper, we study the weak gravitational deflection angle of relativistic massive particles by the Kerr-like black hole in the bumblebee gravity model. In particular, we focus on weak field limits and calculate the deflection angle for a receiver and source at a finite distance from the lens. To this end, we use the Gauss-Bonnet theorem of a two-dimensional surface defined by a generalized Jacobi metric. The spacetime is asymptotically non-flat due to the existence of a bumblebee vector field. Thus the deflection angle is modified and can be divided into three parts: the surface integral of the Gaussian curvature, the path integral of a geodesic curvature of the particle ray and the change in the coordinate angle. In addition, we also obtain the same results by defining the deflection angle. The effects of the Lorentz breaking constant on the gravitational lensing are analyzed. We then consider the finite-distance correction for the deflection angle of massive particles.

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Harmonic functions and gravity localization

De Luca,

De Ponti,

Mondino

et al. 2023

J. High Energ. Phys.

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In models with extra dimensions, matter particles can be easily localized to a ‘brane world’, but gravitational attraction tends to spread out in the extra dimensions unless they are small. Strong warping gradients can help localize gravity closer to the brane. In this note we give a mathematically rigorous proof that the internal wave-function of the massless graviton is constant as an eigenfunction of the weighted Laplacian, and hence is a power of the warping as a bound state in an analogue Schrödinger potential. This holds even in presence of singularities induced by thin branes.We also reassess the status of AdS vacuum solutions where the graviton is massive. We prove a bound on scale separation for such models, as an application of our recent results on KK masses. We also use them to estimate the scale at which gravity is localized, without having to compute the spectrum explicitly. For example, we point out that localization can be obtained at least up to the cosmological scale in string/M-theory solutions with infinite-volume Riemann surfaces; and in a known class of $$ \mathcal{N} $$ N = 4 models, when the number of NS5- and D5-branes is roughly equal.

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Jacobi-Maupertuis Randers-Finsler metric for curved spaces and the gravitational magnetoelectric effect

Cited by 32 publications

References 18 publications

On the Jacobi metric for a general Lagrangian system

On the Jacobi metric for a general Lagrangian system

Finite-Distance Gravitational Deflection of Massive Particles by the Kerr-like Black Hole in the Bumblebee Gravity Model

Harmonic functions and gravity localization

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