Symplectic scattering approach to gravitational lensing

In this paper we apply the symplectic formalism to describe light deflection in general relativity as a scattering process. In the first part of the paper we present gravitational lensing in the action principle and in the Hamiltonian formulations. After defining the Souriau's symplectic scattering and introducing the mathematical properties in the infinitesimal case, we show that this formalism is an appropriate tool to study the deflection of a light signal in a gravitational field, not only in determining the scattering angle, but also the corresponding redshift and the Shapiro time delay. As examples we apply the symplectic scattering to lensing in weak fields and in the Swiss Cheese cosmological model. The conditions for the applicability of the infinitesimal scattering are derived. Finally we discuss its relation with the non infinitesimal case, we propose an extension of the results of the infinitesimal scattering and examine some properties of the non infinitesimal scattering in the hamiltonian formalism paying attention to the definition of photon sphere.