Antonio Masiello

On the existence of geodesics on stationary Lorentz manifolds with convex boundary

Abstract In this paper we consider the problem of the existence and multiplicity for geodesics not touching the boundary of a stationary Lorentz manifold having convex boundary. A physical example of a stationary (and nonstatic) Lorentz manifold having convex boundary is the stationary, axisymmetric, asymptotically flat, gravitational field outside a rotating massive object, whenever its angular speed is small and its mean radius is close to the Schwarzschild radius.

On the energy functional on Finsler manifolds and applications to stationary spacetimes

In this paper we first study some global properties of the energy functional on a non-reversible Finsler manifold. In particular we present a fully detailed proof of the Palais–Smale condition under the completeness of the Finsler metric. Moreover, we define a Finsler metric of Randers type, which we call Fermat metric, associated to a conformally standard stationary spacetime. We shall study the influence of the Fermat metric on the causal properties of the spacetime, mainly the global hyperbolicity. Moreover, we study the relations between the energy functional of the Fermat metric and the Fermat principle for the light rays in the spacetime. This allows one to obtain existence and multiplicity results for light rays, using the Finsler theory. Finally the case of timelike geodesics with fixed energy is considered.

A Fermat principle for stationary space-times and applications to light rays

Abstract We present an extension of the classical Fermat principle in optics to stationary space-times. This principle is applied to study the light rays joining an event with a timelike curve. Existence and multiplicity results of light rays are proved. Moreover, Morse Relations relating the set of rays to the topology of the space-time are obtained, by using the number of conjugate points of the ray. The results hold also for stationary space-times with boundary, in particular the Kerr space-time outside the stationary limit surface.

Solitons and the electromagnetic field

In a recent paper [4], it has been introduced a Lorentz invariant equation in three space dimensions, having soliton like solutions. We recall that, roughly speaking, a soliton is a solution whose energy travels as a localized packet and which preserves this form of localization under small perturbations (see [6], [15], [13], [10]). The equation introduced in [4] is the Euler Lagrange equation of an action functional

The Fermat principle in general relativity and applications

In this paper we use a general version of Fermat’s principle for light rays in general relativity and a curve shortening method to write the Morse relations for light rays joining an event with a smooth timelike curve in a Lorentzian manifold with boundary. The Morse relations are obtained under the most general assumptions and one can apply them to have a mathematical description of the gravitational lens effect in a very general context. Moreover, Morse relations can be used to check if existing models are corrected.

Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric

We show that the index of a lightlike geodesic in a conformally standard stationary spacetime is equal to the index of its spatial projection as a geodesic of a Finsler metric associated to the spacetime. Moreover we obtain the Morse relations of lightlike geodesics connecting a point to an integral line of the standard timelike Killing vector field by using Morse theory on the associated Finsler manifold. To this end, we prove a splitting lemma for the energy functional of a Finsler metric. Finally, we show that the reduction to Morse theory of a Finsler manifold can be done also for timelike geodesics.

Geodesics on product Lorentzian manifolds ()

Abstract In this paper, using global variational methods, we prove existence and multiplicity results for geodesics joining two given events of a product Lorentzian manifold ℳ 0 × ℝ , where ℳ 0 is a complete Riemannian manifold.

Trajectories of charged particles in a region of a stationary spacetime

We study the trajectories for relativistic particles under the action of gravitational and electromagnetic stationary fields. We show how the topology of the spacetime influences the number of such trajectories. The results are applied to the Reissner–Nordstrom spacetime.

A Morse theory for massive particles and photons in general relativity

Abstract In this paper we develop a Morse theory for time-like geodesics parameterized by a constant multiple of proper time. The results are obtained using an extension to the time-like case of the relativistic Fermat principle, and techniques from Global Analysis on infinite dimensional manifolds. In the second part of the paper we discuss a limit process that allows to obtain also a Morse theory for light rays.

Some properties of the spectral flow in Semiriemannian geometry

Let ƒ(z) = ∫01 g(z)[z, z]ds be the action integral on a semiriemannian manifold (M, g) defined on the space of the curves z : [0, 1] → M joining two given points z0 and z1. The critical points of ƒ are the geodesics joining z0 and z1. Let s ϵ [0, 1]. We study the behavior, in dependence of s, of the eigenvalues of the Hessian form of ƒ evaluated at z, restricted to the interval [0, s]. A formula for the derivative of the eigenvalues is given and some applications are shown.