Suppose there is an extra compact dimension, and suppose

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Suppose there is an extra compact spacial dimension in addition to familiar space dimensions x, y, and z. Let us suppose that matter is some kind of 3 dimensional surface moving in these 4 spacial dimensions, 4-space, in some cyclical manner. Let us suppose that from the shape of the surface we can get physics. Assume as an observer we ourselves are some complicated 3 dimensional surface in 4-space. Now as observers we only notice the dimensions x, y, and z. At a given instant matter is "distributed" about in 4-space so that in 3-space at a given instant matter is but a 2 dimensional surface in 3-space. Matter is nowhere and in time everywhere.

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Kaluza-Klein theory:
It was originally formulated as a way to geometrically unify electromagnetism with gravity. Solving the Einstein field equations in 4+1 dimensions (where one of the spatial dimensions is periodic) gives the 10 original EFEs for 3+1 dimensions along with Maxwell’s equations. There’s also an extra scalar equation that predicts the existence of a hypothetical particle called the dilaton which has never been observed. KK theory has fallen somewhat out of favor because it only treats fermions in an ad hoc manner, but it is an important jumping off point for many modern beyond-Standard-Model theories that incorporate dimensional compactification.

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