nLab Newton-Cartan structure

Idea

A Newton-Cartan structure on a 4-manifold gives the manifold the structure of a non-relativistic Newtonian spacetime.

Definition

Definition

Let $M$ be a 4-manifold. A Newton-Cartan structure consists of

• A degenerate symmetric bivector $h : M \to S^2 (TM)$ of signature $(0, +, +, +)$.
• A 1-form $\theta: M \to T^*M$ that lies in the kernel of $h$ viewed as a map $T^*M \to TM$.
• A torsion-free affine connection $\nabla$ such that $\nabla h = 0$ and $\nabla \theta = 0$.

Note that unlike Riemannian metrics, the connection is not uniquely specified by $h$ and $\theta$.

Related concepts

• pseudo-Riemannian metric