On gauge transformations in twistless-torsional Newton--Cartan geometry

Abstract

Twistless-torsional Newton--Cartan (TTNC) geometry exists in two variants, type I and type II, which differ by their gauge transformations. In TTNC geometry there exists a specific locally Galilei-invariant function, called by different names in existing literature, that we dub the `locally Galilei-invariant potential'. We show that in both types of TTNC geometry, there always exists a local gauge transformation that transforms the locally Galilei-invariant potential to zero. For type I TTNC geometry, we achieve this due to the corresponding equation for the gauge parameter taking the form of a Hamilton--Jacobi equation. In the case of type II TTNC geometry, we perform subleading spatial diffeomorphisms. In both cases, our arguments rigorously establish the existence of the respective gauge transformation also in case of only finite-degree differentiability of the geometric fields. This improves upon typical arguments for `gauge fixing' in the literature, which need analyticity. We consider two applications of our result. First, it generalises a classical result in standard Newton--Cartan geometry. Second, it allows to (locally) parametrise TTNC geometry in two new ways: either in terms of just the space metric and a unit timelike vector field, or in terms of the distribution of spacelike vectors and a positive-definite cometric.