Killing Reduction of 5-Dimensional Spacetimes

TL;DR

This paper generalizes Geroch's reduction method for 5-dimensional spacetimes with a Killing vector, deriving a 4D gravity theory coupled to vector and scalar fields, and provides an alternative action principle consistent with Einstein's equations.

Contribution

It extends the geometric reduction technique to 5D spacetimes, connecting vacuum Einstein equations to a 4D gravity theory with additional fields, and introduces an alternative action formulation.

Findings

Derived a 4D gravity coupled to vector and scalar fields from 5D vacuum Einstein equations.

Established an alternative action principle leading to the same field equations.

Generalized Geroch's reduction method for higher-dimensional spacetimes.

Abstract

In a 5-dimensional spacetime ($M,g_{ab}$) with a Killing vector field $\xi ^a$ which is either everywhere timelike or everywhere spacelike, the collection of all trajectories of $\xi ^a$ gives a 4-dimensional space $S$. The reduction of ($M,g_{ab}$) is studied in the geometric language, which is a generalization of Geroch's method for the reduction of 4-dimensional spacetime. A 4-dimensional gravity coupled to a vector field and a scalar field on $S$ is obtained by the reduction of vacuum Einstein's equations on $M$, which gives also an alternative description of the 5-dimensional Kaluza-Klein theory. Besides the symmetry-reduced action from the Hilbert action on $M$, an alternative action of the fields on $S$ is also obtained, the variations of which lead to the same fields equations as those reduced from the vacuum Einstein equation on $M$.