Palatini Formalism of 5-Dimensional Kaluza-Klein Theory

TL;DR

This paper develops a Palatini formalism for 5-dimensional Kaluza-Klein theory, deriving 4D gravity coupled to vector and scalar fields, aligning with traditional Kaluza-Klein results.

Contribution

It introduces a Palatini approach to 5D Kaluza-Klein theory and derives equivalent 4D field equations through Killing reduction.

Findings

Derived 4D gravity coupled to vector and scalar fields from 5D Palatini formalism.

Established equivalence with standard Kaluza-Klein field equations.

Provided a new geometric perspective on dimensional reduction in gravity theories.

Abstract

The Einstein field equations can be derived in $n$ dimensions ($n>2$) by the variations of the Palatini action. The Killing reduction of 5-dimensional Palatini action is studied on the assumption that pentads and Lorentz connections are preserved by the Killing vector field. A Palatini formalism of 4-dimensional action for gravity coupled to a vector field and a scalar field is obtained, which gives exactly the same fields equations in Kaluza-Klein theory.