Time-like solutions to the Lorentz force equation in time-dependent electromagnetic and gravitational fields

Caponio, Erasmo

doi:10.1016/j.jde.2003.09.001

Cited by 4 publications

(4 citation statements)

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“…on the space of all the (absolutely continuous) curves, non necessarily causal, which connect x 0 and x 1 in the interval [0, 1]. Concretely, Bartolo and Antonacci et al [2,5] studied the connectedness of the whole spacetime by means of critical points (non-necessarily causal) of this functional, and further results were obtained in posterior references (see, for example, [6,13,11,16,31] or the detailed acount in [30]). Remarkably, in [14,15] the authors were able to prove, under global hyperbolicity, the existence of at least one (uncontrolled) value of q/m such that a timelike connecting solution of the associated LFE does exist.…”

Section: Connectedness Through Solutions To the Lfementioning

confidence: 99%

“…Even though many, sometimes competitive, results have been obtained [5,12,13,17,11,29,30], and related mathematical problems studied [6,16,31], the full answer to the original problem has remained open:…”

Section: Introductionmentioning

confidence: 99%

“…Recently, there has been a renewed interest in the existence of solutions to the Lorentz force equation (LFE; equation (1) below) connecting two events x 0 ≪ x 1 of a spacetime M , for an (exact) electromagnetic field, F = dω. Even though many, sometimes competitive, results have been obtained [5,12,13,17,11,29,30], and related mathematical problems studied [6,16,31], the full answer to the original problem has remained open: Question (Q): Assume that M is globally hyperbolic, fix a charge-tomass ratio q/m, and let x 0 ≪ x 1 be any two fixed chronologically related events: must a timelike solution of the corresponding LFE, connecting the two events, exist?…”

Section: Introductionmentioning

confidence: 99%

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Connecting Solutions of the Lorentz Force Equation do Exist

Minguzzi

Sánchez

2006

Commun. Math. Phys.

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Recent results on the maximization of the charged-particle action I x0,x1 in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I x0,x1 over a given causal homotopy class C of curves connecting two causally related events x 0 ≤ x 1 . Action I x0,x1 is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ 0 on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C 2 ). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x 0 ≪ x 1 can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) in the nonexact case.As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.

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Section: Connectedness Through Solutions To the Lfementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Connecting Solutions of the Lorentz Force Equation do Exist

Minguzzi

Sánchez

2006

Commun. Math. Phys.

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“…Although the Kaluza-Klein formalism is very natural, and it provides powerful tools for studying existence and multiplicity results for causal solutions of the Lorentz force equation (see for instance [6][7][8]), the theory does not seem to be well suited to study phenomena depending on infinitesimal of second order, like bifurcation theory, and it is practically useless if one wants to relate the Morse theory for solutions of the Lorentz equation with the Morse theory of the corresponding geodesics. This observation is simple consequence of the fact that electromagnetic conjugate points along solutions of the Lorentz equation do not correspond necessarily to conjugate points along the corresponding Kaluza-Klein geodesics, due to the fact that distinct solutions of the Lorentz equation having common endpoints lift to KaluzaKlein geodesics with possibly distinct endpoints.…”

Section: Introductionmentioning

confidence: 99%

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

Piccione

Порталури

2005

Journal of Differential Equations

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We obtain a bifurcation result for solutions of the Lorentz equation in a semi-Riemannian manifold; such solutions are critical points of a certain strongly indefinite functionals defined in terms of the semi-Riemannian metric and the electromagnetic field. The flow of the Jacobi equation along each solution preserves the so-called electromagnetic symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution.We study electromagnetic conjugate instants with symplectic techniques, and we prove at first, an analogous of the semi-Riemannian Morse Index Theorem (see (Calculus of Variations, Prentice-Hall, Englewood Cliffs, NJ, USA, 1963)). By using this result, together with recent results on the bifurcation for critical points of strongly indefinite functionals (see (J. Funct. Anal. 162(1) (1999) 52)), we are able to prove that each non-degenerate and non-null electromagnetic conjugate instant along a given solution of the semi-Riemannian Lorentz force equation is a bifurcation point.

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Connecting Solutions of the Lorentz Force Equation do Exist

Minguzzi¹,

Sánchez

2006

Commun. Math. Phys.

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Due to a processing error the last paragraph of Sect. 3.1 on p. 358 was printed with an error. In addition, in Sect. 5.2 on pp. 365-366 the presentation of the Example was processed incorrectly. The second paragraph of Sect. 5.2 (two lines) must be removed and the last sentence in the same section must be replaced with 'Nevertheless, even though σ 0 maximizes in C 1 , it does not maximize in C 2 (nor in the causal homotopy class C), in agreement with our results.' The corrected paragraphs read as follows. p. 358:In particular, if x 0 x 1 the two points can be connected by means of a timelike geodesic (in fact, by one for each time like homotopy class in C x 0 ,x 1 , as will be apparent below). If x 1 ∈ E + (x 0 ) = J + (x 0 )\I + (x 0 ) then x 0 and x 1 can still be joined by a lightlike geodesic, but this case does not make sense for the LFE. One can also wonder for the connectedness of x 0 , x 1 by means of a geodesic even if they are not causally related, as in variational frameworks described below. Although this question has a geometrical interest (see for instance the survey [37]), it does not have a direct physical interpretation, nor equivalence for LFE.pp. 365-366: 5.2. A remarkable example. Lemma 5.1 does not forbid the existence of a lightlike geodesic σ which maximizes the functional on the closure of a timelike class C x 0 ,x 1 . However, in that case the maximizer on C x 0 ,x 1 ⊃ C x 0 ,x 1 does not coincide with σ , as the following example shows.Example. Let be a surface embedded in R 3 obtained by gluing the spherical cap2 r + z with a cylinder x 2 + y 2 = r 2 /4, z < − √ 3 2 r − z ,The online version of the original article can be found at http://dx

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Time-like solutions to the Lorentz force equation in time-dependent electromagnetic and gravitational fields

Cited by 4 publications

References 22 publications

Connecting Solutions of the Lorentz Force Equation do Exist

Connecting Solutions of the Lorentz Force Equation do Exist

A bifurcation result for semi-Riemannian trajectories of the Lorentz force equation

Connecting Solutions of the Lorentz Force Equation do Exist

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