The Effective Energy-Momentum Tensor in Kaluza-Klein Gravity With Large Extra Dimensions and Off-Diagonal Metrics

TL;DR

This paper develops a generalized framework for Kaluza-Klein gravity with large extra dimensions and off-diagonal metrics, deriving invariant 4D energy-momentum tensors and Maxwell-like equations that suggest the possibility of magnetic monopoles.

Contribution

It introduces a coordinate-invariant splitting method for off-diagonal 5D metrics, extending previous results and deriving Maxwell-like equations with potential magnetic monopole solutions.

Findings

Derived a general invariant 4D energy-momentum tensor from 5D metrics.

Found two conserved 4-vectors analogous to electric and magnetic currents.

Obtained Maxwell-like equations predicting classical magnetic charge and current.

Abstract

We consider a version of Kaluza-Klein theory where the cylinder condition is not imposed. The metric is allowed to have explicit dependence on the "extra" coordinate(s). This is the usual scenario in brane-world and space-time-matter theories. We extend the usual discussion by considering five-dimensional metrics with off-diagonal terms. We replace the condition of cylindricity by the requirement that physics in four-dimensional space-time should remain invariant under changes of coordinates in the five-dimensional bulk. This invariance does not eliminate physical effects from the extra dimension but separates them from spurious geometrical ones. We use the appropriate splitting technique to construct the most general induced energy-momentum tensor, compatible with the required invariance. It generalizes all previous results in the literature. In addition, we find two four-vectors, J_{m}^{mu} and J_{e}^{mu}, induced by off-diagonal metrics, that separately satisfy the usual equation of continuity in 4D. These vectors appear as source-terms in equations that closely resemble the ones of electromagnetism. These are Maxwell-like equations for an antisymmetric tensor {F-hat}_{mu nu} that generalizes the usual electromagnetic one. This generalization is not an assumption, but follows naturally from the dimensional reduction. Thus, if {F-hat}_{mu nu} could be identified with the electromagnetic tensor, then the theory would predict the existence of classical magnetic charge and current. The splitting formalism used allows us to construct 4D physical quantities from five-dimensional ones, in a way that is independent on how we choose our space-time coordinates from those of the bulk.