Valentino Lacquaniti

GEOMETRY AND MATTER REDUCTION IN A 5D KALUZA–KLEIN FRAMEWORK

In this paper we consider the Kaluza–Klein field equations in the presence of a generic 5D matter tensor which is governed by a conservation equation due to 5D Bianchi identities. Following a previous work, we provide a consistent approach to matter where the problem of huge massive modes is removed, without relaxing the compactification hypotheses; therefore we perform the dimensional reduction either for metric fields and for matter, thus identifying a pure 4D tensor term, a 4D vector term and a scalar one. Hence we are able to write down a consistent set of equations for the complete dynamics of matter and fields; with respect to the pure Einstein–Maxwell system we now have two additional scalar fields: the usual dilaton one plus a scalar source term. Some significant scenarios involving these terms are discussed and perspectives for cosmological applications are suggested.

DYNAMICS OF MATTER IN A COMPACTIFIED KALUZA–KLEIN MODEL

A long-standing problem in Kaluza–Klein models is the description of matter dynamics. Within the 5D model, the dimensional reduction of the geodesic motion for a 5D free test particle formally restores electrodynamics, but the reduced 4D particle shows a charge–mass ratio that is upper-bounded, such that it cannot fit in with any kind of elementary particle. At the same time, from the quantum dynamics viewpoint, there is the problem of the generation of huge massive modes. We present a criticism against the 5D geodesic approach and face the hypothesis that in Kaluza–Klein space the geodesic motion does not deal with the real dynamics of the test particle. We propose a new approach: starting from the conservation equation for the 5D matter tensor, within the Papapetrou multipole expansion, we prove that the 5D dynamical equation differs from the 5D geodesic one. Our new equation provides right coupling terms without bounding and in such a scheme the tower of massive modes is removed.

EFFECTIVE POTENTIAL APPROACH TO THE MOTION OF MASSIVE TEST PARTICLES IN KALUZA–KLEIN GRAVITY

Effective potential for a class of static solutions of Kaluza-Klein equations with three-dimensional spherical symmetry is studied. Test particles motion is analyzed. In attempts to read the obtained results with the experimental data, particular attention is devoted to the Schwarzschilds limit of the four dimensional counterpart of these electromagnetic-free solutions. Massive particles stable circular orbits in particular are studied, and a comparison between the well known results if the Schwarzschilds case and those found for the static higher-dimensional case is performed. A modification of the circular stable orbits is investigated in agreement with the experimental constraints.

ON THE ADM DECOMPOSITION OF THE 5-D KALUZA–KLEIN MODEL

Our purpose is to recast the KK model in terms of ADM variables. We examine and solve the problem of the consistency of this approach, with particular care about the role of the cylindricity hypothesis. We show in detail how the KK reduction commutes with the ADM slicing procedure and how this leads to a well-defined and unique ADM reformulation. This allows us to consider the Hamiltonian formulation of the model and moreover it can be viewed as the first step for the Ashtekar reformulation of the KK scheme. Moreover, we show how the time component of the gage vector arises naturally from the geometrical constraints of the dynamics; this is a positive check for the autoconsistency of the KK theory and for an Hamiltonian description of the dynamics which will take into account the compactification scenario; this result enforces the physical meaning of the KK model.

Massive test particle motion in 5-dimensional electromagnetic-free Kaluza-Klein theory

A class of static, vacuum solutions of (free-electromagnetic) Kaluza-Klein equations with three-dimensional spherical symmetry is studied. In order to explore the dynamic in such spacetimes, geodesic equations are obtained and the effective potential for massive test particles is analysed. Particular attention is devoted to the properties of the four-dimensional counterpart of these solutions in their Schwarzschild limit. A modification of the circular stable orbits compared with the Schwarzschild case is investigated.

Hamiltonian Formulation of 5-dimensional Kaluza-Klein Theory

We analyze the consistency of the ADM approach to KK model; we prove that KK reduction commute with ADM splitting. This leads to a well defined Hamiltonian; we provide the outcome. The electromagnetic constraint is derived from a geometrical one and this result enforces the physical meaning of KK model. Moreover we study the role of the extra scalar field we have in our model; classical hints from geodesic motion and cosmological solutions suggest that the scalar field can be an alternative time variable in the relational point of view.

Particles and Fields within a Unification Scheme

We discuss properties of particles and fields in a multi-dimensional space-time, where the geometrization of gauge interactions can be performed. For instance, in a 5-dimensional Kaluza-Klein manifold we argue that the motion of charged spinning bodies is obtain in a Papapetrou-like formulation. As far as spinors are concerned, we outline how the gauge coupling can be recognized by a proper dependence on extra-coordinates and by the dimensional reduction procedure.

HAMILTONIAN FORMULATION OF THE 5-D KALUZA-KLEIN MODEL AND TEST-PARTICLE MOTION

We examine the ADM reformulation of the 5-D KK model: the dimensional reduction is provided to commute with the ADM splitting and we show how the time component of the gauge vector is given by combination of the Lagrangian multipliers for the 5-D gravitational field. We consider 5D particles motion and after dimensional reduction the definition of charge is recovered within electrodynamic coupling. A time-varying fine structure constant is recognized because an extra scalar field is present in the 4-D theory.

ON MATTER COUPLING IN 5D KALUZA-KLEIN MODEL

We analyze some unphysical features of the geodesic approach to matter coupling in a compactified Kaluza-Klein scenario, like the q/m puzzle and the huge massive modes. We propose a new approach, based on Papapetrou multipole expansion, that provides a new equation for the motion of a test particle. We show how this equation provides right couplings and does not generate huge massive modes.