Null Killing Vector Dimensional Reduction and Galilean Geometrodynamics

TL;DR

This paper develops a covariant geometric framework for Einstein solutions with lightlike Killing vectors, revealing a Galilean structure with a Newtonian connection and analyzing the associated dimensional reduction and symmetries.

Contribution

It introduces a covariant theory for lightlike Killing vectors, deriving equations of motion and symmetries within a Galilean geometric setting.

Findings

Establishment of a Galilean, torsionless connection from lightlike Killing vector reduction

Identification of a non-separable quasi-Maxwell field in the affine connection

Derivation of a complete set of equations of motion and an associated action

Abstract

The solutions of Einstein's equations admitting one non-null Killing vector field are best studied with the projection formalism of Geroch. When the Killing vector is lightlike, the projection onto the orbit space still exists and one expects a covariant theory with degenerate contravariant metric to appear, its geometry is presented here. Despite the complications of indecomposable representations of the local Euclidean subgroup, one obtains an absolute time and a canonical, Galilean and so-called Newtonian, torsionless connection. The quasi-Maxwell field (Kaluza Klein one-form) that appears in the dimensional reduction is a non-separable part of this affine connection, in contrast to the reduction with a non-null Killing vector. One may define the Kaluza Klein scalar (dilaton) together with the absolute time coordinate after having imposed one of the equations of motion in order to prevent the emergence of torsion. We present a detailed analysis of the dimensional reduction using moving frames, we derive the complete equations of motion and propose an action whose variation gives rise to all but one of them. Hidden symmetries are shown to act on the space of solutions.