Four Dimensionality in Non-compact Kaluza-Klein Model

TL;DR

The paper investigates a five-dimensional, non-compact spacetime in which four-dimensionality arises as a consequence of a stability requirement that traps matter on a 4D submanifold. Using a stability condition that sets $P_5=0$ (hence $T_5^5=T_5^{\alpha}=0$), the authors derive a RS-like metric split with $g_{\alpha\beta}=\lambda^2(x^5)\eta_{\alpha\beta}$ and $g_{55}=-1$, and argue that conformal invariance in the 4D submanifold selects exactly four extended dimensions. The trapping mechanism is made explicit through the extrinsic curvature, yielding a wall-like solution $\lambda=\cosh(EX^5)$ that localizes fields near the 4D shell, with a width $\Delta \sim 1/E$. When scalar fields are included, the coupling to gravity fixes the conformal value $\xi=1/6$, reinforcing the 4D conformal structure and producing a 4D-like scalar equation in the trapped geometry. Overall, the work provides a stability-driven route to 4D localization without compactification, connecting RS-type trapping, conformal invariance, and matter confinement in a five-dimensional framework.

Abstract

Five dimensional model with extended dimensions investigated. It is shown that four dimensionality of our world is the result of stability requirement. Extra component of Einstein equations giving trapping solution for matter fields coincides with the one of conditions of stability.