On the ADM decomposition of the 5-D Kaluza-Klein model

TL;DR

This paper reformulates the 5-D Kaluza-Klein model using ADM variables, demonstrating the consistency of the approach, and explores its implications for Hamiltonian and Ashtekar formulations, confirming the physical and geometrical coherence of the model.

Contribution

It provides a consistent ADM decomposition of the 5-D Kaluza-Klein model, clarifying the role of cylindricity and enabling Hamiltonian analysis including gauge components.

Findings

KK reduction commutes with ADM slicing

Time component of gauge vector arises from geometric constraints

Ensures the consistency and physical meaning of KK model

Abstract

Our purpose is to recast 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 details 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 can be the first step for the Ashtekar reformulation of the KK scheme. Moreover we show how the time component of the gauge 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 wants to take into account the compactification scenario: this result enforces the physical meaning of KK model.