‘Kaluza-Klein Gravity’

Kaluza attempted to unify Einstein’s theory of gravity with Maxwell’s theory
of electromagnetism by adding a fifth dimension to the universe. Kaluza assumed that there would be no five-dimensional matter, that the mathematical
structure of general relativity would be extended to five dimensions without
changes, and that the quantities would have no dependence on the fifth coordinate. A consequence of his work was that an empty five-dimensional
universe implied the presence of four-dimensional electromagnetic radiation
and Maxwell’s equations. However, the third assumption, called the cylinder
condition, was considered unnatural. Klein and others attempted to modify Kaluza’s theory in a manner that allowed the cylinder condition to be
dropped. Some of this work led to modern string theory as one possible
grand unification theory.
In this paper, we will review Kaluza’s theory and discuss directions in which
the cylinder condition has been removed. The direction depends in part
on how one interprets the fifth dimension. Klein assumed the fifth dimension was lengthlike. This required that the fifth dimension was compact.
This leads to theories in which higher-dimensional matter must be present.
The STM hypothesis (where “STM” stands for “Space-Time-Matter”), on
the other hand, assumes the fifth dimension is masslike. This leads to noncompactified five dimensional theories of gravity in which, when one removes
the cylinder condition, four dimensional massive matter is induced by the
geometry of a five-dimensional empty universe.