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Connecting Solutions of the Lorentz Force Equation do Exist

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An Erratum to this article was published on 05 August 2006

Abstract

Recent results on the maximization of the charged-particle action in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of over a given causal homotopy class of curves connecting two causally related events x 0≤x 1. Action is proved to admit a maximum on , and also one in the adherence of each timelike homotopy class C. Moreover, the maximum σ 0 on is timelike if 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 corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m≠0) in the non-exact 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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Authors and Affiliations

  1. Departamento de Matemáticas,  , Plaza de la Merced 1–4, 37008, Salamanca, Spain

    E. Minguzzi

  2. INFN, Piazza dei Caprettari 70, 00186, Roma, Italy

    E. Minguzzi

  3. Departamento de Geometría y Topología Facultad de Ciencias,  , Avda. Fuentenueva s/n, 18071, Granada, Spain

    M. Sánchez

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  1. E. Minguzzi
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  2. M. Sánchez
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Correspondence to E. Minguzzi.

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Communicated by G.W. Gibbons

An erratum to this article is available at http://dx.doi.org/10.1007/s00220-006-0064-7.

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Minguzzi, E., Sánchez, M. Connecting Solutions of the Lorentz Force Equation do Exist. Commun. Math. Phys. 264, 349–370 (2006). https://doi.org/10.1007/s00220-006-1547-2

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