Maslov index and Morse theory for the relativistic Lorentz force equation

Abstract

We study the Jacobi equation for fixed endpoints solutions of the Lorentz force equation on a Lorentzian manifold. The flow of the Jacobi equation along each solution preserves the so-called twisted symplectic form, and the corresponding curve in the symplectic group determines an integer valued homology class called the Maslov index of the solution. We introduce the notion of F-conjugate plane for each conjugate instant; the restriction of the spacetime metric to the F-conjugate plane is used to compute the Maslov index, which is given by a sort of algebraic count of the conjugate instants. For a stationary Lorentzian manifold and an exact electromagnetic field admitting a potential vector field preserving the flow of the Killing vector field, we introduce a constrained action functional having finite Morse index and whose critical points are fixed endpoints solution of the Lorentz force equation. We prove that the value of this Morse index equals the Maslov index and we prove the Morse relations for the solutions of the Lorentz force equation in a static spacetime.