Fabio Giannoni

On the existence of multiple geodesics in static space-times

Abstract In this paper we study the problem of the existence of geodesies in static space-times which are a particular case of Lorentz manifolds. We prove multiplicity results about geodesies joining two given events and about periodic trajectories having a prescribed period.

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.

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.

Periodic solutions of prescribed energy for a class of Hamiltonian systems with singular potentials

Abstract : Hamiltonian systems of ordinary differential equations model the motion of a discrete mechanical system when no frictional forces are present. A basic property of such systems is that energy is conserved. Therefore solutions of Hamiltonian systems lie on surfaces of fixed energy. The main result of this paper is a fairly general criterion for such a surface to possess a periodic solution.

Homoclinic orbits on compact manifolds

Abstract Under a suitable assumption on the potential energy V, we prove the existence of a homoclinic orbit of the equation Dt(x′(t)) + grad V(x(t)) = 0, x ϵ M, where M is a compact Riemannian manifold. Our assumption is satisfied, for instance, if the function V has a unique nondegenerate maximum point.

New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court

Abstract: A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non-vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates – blackholes or naked singularities – is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed.

Periodic bounce trajectories with a low number of bounce points

Abstract In this paper we study the existence of a periodic trajectory with prescribed period, which bounoes against the boundary of an open subset of ℝ N , in presence of a potential field. We prove the existence of periodic solutions with at most N + 1 bounce points.

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.

On the multiplicity of homoclinic orbits on Riemannian manifolds for a class of second order Hamiltonian systems

AbstractWe prove the existence of infinitely many homoclinic orbits on a Riemannian manifold (possibly non-compact), for a class of second order Hamiltonian systems of the form: