On the Lie subalgebra of Killing-Milne and Killing-Cartan vector fields in Newtonian space-time

TL;DR

This paper demonstrates that Killing-Milne and Killing-Cartan vector fields in Newtonian space-time form Lie subalgebras, extending the understanding of symmetries in Newtonian gravity with gauge transformations.

Contribution

It establishes that both types of Killing vector fields form Lie subalgebras, considering gauge transformations in Newtonian space-time.

Findings

Killing-Milne vector fields preserve Newtonian space-time structure.

Killing-Cartan vector fields preserve gravitational field.

Both form Lie subalgebras in Newtonian space-time.

Abstract

The Galilean (and more generally Milne) invariance of Newtonian theory allows for Killing vector fields of a general kind, whereby the Lie derivative of a field is not required to vanish but only to be cancellable by some infinitesimal Galilean (respectively Milne) gauge transformation. In this paper, it is shown that both the Killing-Milne vector fields, which preserve the background Newtonian space-time structure, and the Killing-Cartan vector fields, which in addition preserve the gravitational field, form a Lie subalgebra.