The 5D-4D correspondence for the magnetic field

TL;DR

This paper explores the 5D-4D correspondence in gravitational and electromagnetic theories, deriving relations between 5D gauge fields and 4D electromagnetic potentials in cosmological and black hole models.

Contribution

It introduces an algorithm to express magnetic fields in 5D theories using gauge potentials and scalar fields, applicable to cosmological and Schwarzschild-like models.

Findings

Effective 4D electromagnetic potential derived from 5D gauge fields.

New terms appear in 5D field equations when scalar fields depend on coordinates.

Algorithm effectively relates 5D gauge and scalar fields to 4D magnetic fields.

Abstract

5D theory is an alternative model for understanding gravitational and electromagnetic interactions together. In this work we used the correspondence between 5D Einstein field equations with cosmological constant and the 4D Einstein equations with sources. We started with the principal fiber bundle P(M/F,U(1)) metric and studied the case when the gauge potential Aa corresponds to the magnetic field: Aa =(0, 0, A3, 0). We identified the base-space with the space-time of two models of universe: the Robertson-Walker - like model and the Schwarzschild - like model. With this ansatz, we followed an algorithm that permits to express the magnetic field in terms of the gauge potential Aa and the scalar field F. This algorithm seems to work very well if the scalar field is time dependent. When we analyzed its dependence on other coordinates, new terms it comes out on the 5D field equations. We obtained the translation between the effective 4D electromagnetic potential and the 5D gauge and scalar fields.