Null Killing Vector Dimensional Reduction and Galilean Geometrodynamics

TL;DR

This paper analyzes Einstein gravity reduced along a null Killing vector, revealing that the orbit space carries a degenerate metric and admits a Galilean, torsionless connection, with the quasi-Maxwell (Kaluza–Klein) field absorbed into the affine connection. By employing moving frames and a careful Weyl rescaling, the authors derive a complete set of reduced equations of motion and propose an action principle in $d+1$ dimensions that yields these equations except for one, while also identifying hidden symmetries acting on solution spaces. The work establishes a covariant Newton–Cartan-type structure in $d$ dimensions, characterized by an absolute time $u$ and a degenerate metric, and clarifies how dilatons and frame choices control torsion and gauge degrees of freedom. It also shows why Maxwell-type degrees of freedom effectively disappear in this null reduction and discusses the implications for hidden $SO(2)$-type dualities in the space of solutions. The results pave the way for further inclusion of matter, fermions, and connections to nonrelativistic limits and string-inspired contexts, highlighting the foundational role of null-Killing reductions in Galactic geometrodynamics.

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.