Consequences of Kaluza-Klein Covariance

TL;DR

This paper explores how the broader coordinate transformation group in 5D Kaluza-Klein theory affects physical quantities, revealing implications for unification theories and the nature of mass, cosmological constants, and wave mechanics.

Contribution

It demonstrates the physical consequences of 5D covariance on 4D quantities through transformations of the canonical metric, highlighting their impact on cosmology, mass, and quantum behavior.

Findings

The 4D cosmological constant can depend on the fifth coordinate.

Test particle mass relates to its Compton wavelength under 5D transformations.

Dynamics can exhibit wave-mechanical features with a small mass quantum.

Abstract

The group of coordinate transformations for 5D noncompact Kaluza-Klein theory is broader than the 4D group for Einstein's general relativity. Therefore, a 4D quantity can take on different forms depending on the choice for the 5D coordinates. We illustrate this by deriving the physical consequences for several forms of the canonical metric, where the fifth coordinate is altered by a translation, an inversion and a change from spacelike to timelike. These cause, respectively, the 4D cosmological 'constant' to become dependent on the fifth coordinate, the rest mass of a test particle to become measured by its Compton wavelength, and the dynamics to become wave-mechanical with a small mass quantum. These consequendes of 5D covariance -- whether viewed as positive or negative -- help to determine the viability of current attempts to unify gravity with the interactions of particles.