A Note on the Kaluza-Klein Theory

TL;DR

This paper critically examines the Kaluza-Klein theory, revealing fundamental issues with its assumptions about four-dimensional spacetime and electromagnetic variables, and suggests abandoning it for alternative unified theories.

Contribution

The paper identifies a core mathematical problem in Kaluza-Klein theory related to non-integrable vector spaces, challenging its validity as a unified framework.

Findings

The four-dimensional metric and electromagnetic potential are not generally integrable.

The derived four-dimensional field equations are not well-defined on a submanifold.

No satisfactory solutions are found within the Kaluza-Klein formalism.

Abstract

We show that the Kaluza-Klein theory contains a fundamental problem: The four-dimensional metric tensor and the electromagnetic potential vector assumed in the Kaluza-Klein theory belong to four-dimensional vector spaces that are not integrable in general, resulting that the four-dimensional physical variables and the corresponding field equations derived from the five-dimensional Einstein field equation (i.e., the four-dimensional Einstein field equation and the Maxwell equations) are not defined on a four-dimensional submanifold. That is, the four-dimensional spacetime assumed in the Kaluza-Klein theory does not exist. No satisfactory solutions are found within the Kaluza-Klein formalism. Perhaps the best approach to fix the problem is giving up the Kaluza-Klein theory and looking for a new unified scheme for gravitational and electromagnetic interactions in the framework of a spacetime with extra dimensions, as having already been explored in some literature.