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

Arian L. von Blanckenburg, Philip K. Schwartz

TL;DR

Twistless-torsional Newton–Cartan geometry admits two gauge variants, Type I and II, distinguished by how the Bargmann form transforms. The authors prove that in both variants a local gauge transformation can set the locally Galilei-invariant potential hat_phi to zero, using a Hamilton-Jacobi equation in Type I and a spacelike-diffeomorphism construction in Type II, under finite differentiability. This result generalises the classical Newton–Cartan fact that one can locally choose a unit timelike vector with vanishing Newton–Coriolis form, and it yields two local parametrisations of TTNC data: by (h,v) or by (Sigma,gamma). These parametrisations pave the way for action formulations of TTNC gravity via variational completion and may illuminate connections to TTNC field equations.

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.

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

TL;DR

Twistless-torsional Newton–Cartan geometry admits two gauge variants, Type I and II, distinguished by how the Bargmann form transforms. The authors prove that in both variants a local gauge transformation can set the locally Galilei-invariant potential hat_phi to zero, using a Hamilton-Jacobi equation in Type I and a spacelike-diffeomorphism construction in Type II, under finite differentiability. This result generalises the classical Newton–Cartan fact that one can locally choose a unit timelike vector with vanishing Newton–Coriolis form, and it yields two local parametrisations of TTNC data: by (h,v) or by (Sigma,gamma). These parametrisations pave the way for action formulations of TTNC gravity via variational completion and may illuminate connections to TTNC field equations.

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.
Paper Structure (10 sections, 6 theorems, 29 equations)

This paper contains 10 sections, 6 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Twistless-torsional Newton--Cartan geometry
  3. Gauge transformations of the locally Galilei-invariant potential
  4. Type I TTNC geometry
  5. Type II TTNC geometry
  6. Application: parametrisation of TTNC geometries
  7. Parametrisation of TTNC geometries up to gauge
  8. Interlude: compatible connections
  9. Parametrisation of standard Newton--Cartan geometry
  10. Conclusion

Key Result

Theorem 1

Let $(M,\tau,h)$ be a Galilei manifold satisfying the twistless torsion condition $\tau \wedge \mathrm{d}\tau = 0$, and let $\boldsymbol{\tilde{a}}$ be a Bargmann form on it. Then locally there exist a gauge-equivalent Bargmann form ${\boldsymbol{a}}$ and a unit timelike vector field $v$ such that t

Theorems & Definitions (8)

  • Theorem 1
  • proof : Proof of \ref{['thm:Bargmann_gauge_v_0']}
  • Theorem 2
  • Theorem 3
  • Corollary 4
  • proof
  • Theorem 5
  • Theorem 6