The classification of general affine connections in Newton--Cartan geometry: Towards metric-affine Newton--Cartan gravity

Abstract

We give a full classification of general affine connections on Galilei manifolds in terms of independently specifiable tensor fields. This generalises the well-known case of (torsional) Galilei connections, i.e. connections compatible with the metric structure of the Galilei manifold. Similarly to the well-known pseudo-Riemannian case, the additional freedom for connections that are not metric-compatible lies in the covariant derivatives of the two tensors defining the metric structure (the clock form and the space metric), which however are not fully independent of each other.

20+2+2 pages (main text, references, appendix), to appear in Classical and Quantum Gravity. v2: added references, corrected typos. v3: extended discussion, added references

References in corpus (5)

• The teleparallel equivalent of general relativity
• Review on Non-Relativistic Gravity
• Teleparallel Newton--Cartan gravity
• The Non-Relativistic Geometric Trinity of Gravity
• Generalized Newton-Cartan Geometries for Particles and Strings

Cited by in corpus (3)

• The Non-Relativistic Geometric Trinity of Gravity
• The Newtonian limit of orthonormal frames in metric theories of gravity
• On gauge transformations in twistless-torsional Newton--Cartan geometry