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We consider a version of Kaluza-Klein theory where the cylinder condition isnot 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-mattertheories. We extend the usual discussion by considering five-dimensionalmetrics with off-diagonal terms. We replace the condition of cylindricity bythe requirement that physics in four-dimensional space-time should remaininvariant under changes of coordinates in the five-dimensional bulk. Thisinvariance does not eliminate physical effects from the extra dimension butseparates them from spurious geometrical ones. We use the appropriate splittingtechnique to construct the most general induced energy-momentum tensor,compatible with the required invariance. It generalizes all previous results inthe literature. In addition, we find two four-vectors, J_{m}^{mu} andJ_{e}^{mu}, induced by off-diagonal metrics, that separately satisfy the usualequation of continuity in 4D. These vectors appear as source-terms in equationsthat closely resemble the ones of electromagnetism. These are Maxwell-likeequations for an antisymmetric tensor {F-hat}_{mu nu} that generalizes theusual electromagnetic one. This generalization is not an assumption, butfollows naturally from the dimensional reduction. Thus, if {F-hat}_{mu nu}could be identified with the electromagnetic tensor, then the theory wouldpredict the existence of classical magnetic charge and current. The splittingformalism used allows us to construct 4D physical quantities fromfive-dimensional ones, in a way that is independent on how we choose ourspace-time coordinates from those of the bulk.Comment: New title, editorial changes made as to match the version to appear  in International Journal of Modern Physics

THE EFFECTIVE ENERGY-MOMENTUM TENSOR IN KALUZA–KLEIN GRAVITY WITH LARGE EXTRA DIMENSIONS AND OFF-DIAGONAL METRICS