This paper proposes a clean, off‑shell definition of the Weyl tensor for Galilean geometry — the geometric setting that describes non‑relativistic or “slow‑speed” limits of spacetime. The Weyl tensor is the part of spacetime curvature that stays the same if you rescale distances (a conformal transformation). In relativity it encodes tidal forces and free gravitational degrees of freedom. The authors show how to carry a similar, conformally invariant object over to the Galilean case without assuming any field equations (“off‑shell”), and they give matching definitions of its electric and magnetic parts relative to observers.

To get there they start from the familiar Lorentzian (relativistic) Weyl tensor and take the Galilean limit. That limit keeps the speed of light explicit while sending it to infinity in a controlled way. Using this procedure they arrive at a Galilean Weyl tensor that transforms correctly under conformal rescalings of the Galilean structure. They also define electric and magnetic parts of this tensor with respect to particular timelike observers, in analogy with the 1+3 split used in relativity.

A key conceptual point is about the magnetic part. For the magnetic part to vanish you need observers with special kinematic properties. In Newton–Cartan gravity (the standard geometric form of Newtonian gravity) the field equations guarantee such observers exist. But the authors point out that other Galilean‑invariant theories might not supply them. They therefore propose that the existence of observers for which the off‑shell magnetic part is zero should be taken as a necessary condition for calling a Galilean theory “Newtonian”. This proposal echoes Trautman’s earlier Newtonian condition in traditional Newton–Cartan gravity.

As a secondary but concrete technical result, the paper shows there is a unique Galilean connection that is invariant under Galilean boosts and can be built from just a Galilean structure plus a choice of Coriolis field (a field that encodes rotation‑type effects). This remains true even when the clock one‑form is not closed. That is notable because the standard way to build boost‑invariant connections usually introduces an extra “mass gauge” field. Here no extra structure is needed.