Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic space–times

Caponio, Erasmo; Minguzzi, E.

doi:10.1016/s0393-0440(03)00073-1

Cited by 9 publications

(29 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [11], the authors prove that, given a globally hyperbolic M endowed with an exact electromagnetic field, there exists a time-like solution to Eq. (1) joining any two chronologically related points on M: Such result has been extended in [12] to time-like solutions having fixed charge-to-mass ratio q m in a suitable neighborhood of 0AR: These results generalize the standard result in Lorentzian geometry that chronologically related points in a globally hyperbolic manifold can be connected by a time-like geodesic. Their proofs are based on the causal structure of the trivial bundle M Â R endowed with the Kaluza-Klein metric.…”

Section: Introduction and Statement Of The Resultssupporting

confidence: 55%

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

Caponio

2004

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

Section: Introduction and Statement Of The Resultssupporting

confidence: 55%

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

Caponio

2004

Journal of Differential Equations

Self Cite

View full text Add to dashboard Cite

show abstract

“…Indeed this allowed to prove the existence of solutions having an a priori fixed charge-to-mass ratio in most cases. The results in [29] improve those in [17]; the best achieved result is then:…”

Section: Known Results On (Q)mentioning

confidence: 88%

“…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%

“…-In Section 4, problem (A) on the existence of local maximizers is solved, as explained above. Even though we follow the approach in [17,29], the proof is rewritten completely. In fact, apart from some simplifications, several new technicalities appear when causal homotopy classes are considered, in both the Kaluza-Klein fiber bundle (Subsection 4.1) and the limit process on curves of the homotopy class (Subsection 4.2, Lemma 4).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Connecting Solutions of the Lorentz Force Equation do Exist

Minguzzi

Sánchez

2006

Commun. Math. Phys.

View full text Add to dashboard Cite

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.

show abstract

“…This result is proved for instance in [3, Prop. 3.61,3.62,3.64,3.68] (for the globally hyperbolic case see also [18,6,13]). I give explicitly the proof for the causal case as we will use it.…”

Section: Proof If the Setmentioning

confidence: 99%

On the causal properties of warped product spacetimes

Minguzzi¹

2007

Class. Quantum Grav.

View full text Add to dashboard Cite

It is shown that the warped product spacetime P = M × f H, where H is a complete Riemannian manifold, and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. For instance, it is shown that if diam H = +∞, the direct product spacetime P = M × H is causally simple if and only if (M, g) is causally simple, the Lorentzian distance on M is continuous and any two causally related events at finite distance are connected by a maximizing geodesic. Similar conditions are found for the causal continuity property. Some new results concerning the behavior of the Lorentzian distance on distinguishing, causally continuous, and causally simple spacetimes are obtained. Finally, a formula which gives the Lorentzian distance on the direct product in terms of the distances on the two factors (M, g) and (H, h) is obtained.

show abstract

Solutions to the Lorentz force equation with fixed charge-to-mass ratio in globally hyperbolic space–times

Cited by 9 publications

References 2 publications

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

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

Connecting Solutions of the Lorentz Force Equation do Exist

On the causal properties of warped product spacetimes

Contact Info

Product

Resources

About