NewtonCartan

Newton-Cartan geometry, or Newton-Cartan theory, is a geometric reformulation of Newtonian gravity that presents gravity as a feature of spacetime geometry rather than a force in flat space. It was developed in the early 20th century by Élie Cartan and later refined by others, notably Trautman, to provide a coordinate-free description of Newtonian physics compatible with a geometric view of spacetime.

The framework models spacetime as a four-dimensional differentiable manifold equipped with two degenerate structures: a nowhere-vanishing one-form t_a that selects universal time, and a spatial contravariant metric h^{ab} that projects onto space. These obey t_a h^{ab} = 0 and h^{ab} t_b = 0. A torsion-free affine connection ∇ is chosen to be compatible with both structures, meaning ∇_a t_b = 0 and ∇_a h^{bc} = 0. In this setting, freely falling trajectories are geodesics of ∇, and gravity is encoded in the connection coefficients rather than in curvature of a nondegenerate metric. In convenient coordinates, the gravitational potential Φ appears through components such as Γ^i_{00} ∝ ∂^i Φ, and the field equation reduces to Poisson’s equation ∇^2 Φ = 4πGρ for mass density ρ, reproducing Newtonian gravity in the appropriate limit.

Historically, Newton-Cartan theory provides a coordinate-independent formulation of Newtonian gravity and serves as a bridge to the relativistic theory. It has become foundational in studies of nonrelativistic limits of general relativity, nonrelativistic holography, and condensed-mmatter systems with Galilean symmetry. Extensions include torsionful variants and the Bargmann framework, which embeds Newton-Cartan geometry in a higher-dimensional spacetime and clarifies its relation to the nonrelativistic limit.

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