arXiv subjects

Ricardo Gallego Torromé

Publications and source records attributed to Ricardo Gallego Torromé.

41 records · Page 3Linked to original sources

arXiv2012

On Fermat's principle for causal curves in time oriented Finsler spacetimes

In this work, a version of Fermat's principle for causal curves with the same energy in time orientable Finsler spacetimes is proved. We calculate the secondvariation of the {\it time arrival functional} along a geodesic in terms of the index form associated with the Finsler spacetime Lagrangian. Then the character of the critical points of the time arrival functional is investigated and a Morse index theorem in the context of Finsler spacetime is presented.

math.DG↗

arXiv2012

Some Rigidity Conditions on Berwald Structures

This thesis contains an introduction to the method of average in Finsler geometry. The method is applied to Berwald spaces, obtaining geodesic rigidity conditions. We prove that the Levi-Civita connection of any Riemannian metric affine equivalent to the Berwald metric leaves invariant the indicatrix of th Finsler metric F. A converse result also holds: if (M,F) is a Finsler structure such that there is a Riemannian connection whose Levi-Civita leaves invariant by parallel transport the indicatrix of the Finsler structure, then the structure (M,F) is Berwald. As an application we obtain a necessary condition for a structure to be Landsberg but not Berwald.

math.DG↗

arXiv2012

Averaged dynamics of ultra-relativisitc charged particles beams

In this thesis, we consider the suitability of using the charged cold fluid model in the description of ultra-relativistic beams. The method that we have used is the following. Firstly, the necessary notions of kinetic theory and differential geometry of second order differential equations are explained. Then an averaging procedure is applied to a connection associated with the Lorentz force equation. The result of this averaging is an affine connection on the space-time manifold. The corresponding geodesic equation defines the averaged Lorentz force equation. We prove that for ultra-relativistic beams described by narrow distribution functions, the solutions of both equations are similar. This fact justifies the replacement of the Lorentz force equation by the simpler {\it averaged Lorentz force equation}. After this, for each of these models we associate the corresponding kinetic model, which are based on the Vlasov equation and {\it averaged Vlasov equation} respectively. The averaged Vlasov equation is simpler than the original Vlasov equation. This fact allows us to prove that the differential operation defining the averaged charged cold fluid equation is controlled by the {\it diameter of the distribution function}, by powers of the {\it energy of the beam} and by the time of evolution $t$. We show that the Vlasov equation and the averaged Vlasov equation have similar solutions, when the initial conditions are the same. Finally, as an application of the {\it averaged Lorentz force equation} we re-derive the beam dynamics formalism used in accelerator physics from the Jacobi equation of the averaged Lorentz force equation.

math-ph↗

arXiv2012

On the tensorial properties of the generalized Jacobi equation

The generalized Jacobi equation is a differential equation in local coordinates that describes the behavior of infinitesimally close geodesics with an arbitrary relative velocity. In this note we study some transformation properties for solutions to this equation. We prove two results. First, under any affine coordinate changes we show that the tensor transformation rule maps solutions to solutions. As a consequence, the generalized Jacobi equation is a tensor equation when restricted to suitable Fermi coordinate systems along a geodesic. Second, in dimensions n\ge 3, we explicitly show that the transformation rule does not in general preserve solutions to the generalized Jacobi equation.

math-ph↗

arXiv2006

A Finslerian version of 't Hooft Deterministic Quantum Models

Using the Finsler structure living in the phase space associated to the tangent bundle of the configuration manifold, deterministic models at the Planck scale are obtained. The Hamiltonian function are constructed directly from the geometric data and some assumptions concerning time inversion symmetry. The existence of a maximal acceleration and speed is proved for Finslerian deterministic models. We investigate the spontaneous symmetry breaking of the orthogonal symmetry SO(6N) of the Hamiltonian of a deterministic system. This symmetry break implies the non-validity of the argument used to obtain Bell's inequalities for spin states. It is introduced and motivated in the context of Randers spaces an example of simple 't Hooft model with interactions.

math-ph↗