Newton–Cartan theory
Newton–Cartan theory (or geometrized Newtonian gravitation) is a geometrical re-formulation, as well as a generalization, of Newtonian gravity first introduced by Élie Cartan and Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski and Horneffer, Ehlers, Havas, Künzle, Lottermoser, Trautman, and others. In this re-formulation, the structural similarities between Newton's theory and Albert Einstein's general theory of relativity are readily seen, and it has been used by Cartan and Friedrichs to give a rigorous formulation of the way in which Newtonian gravity can be seen as a specific limit of general relativity , and by Jürgen Ehlers to extend this correspondence to specific solutions of general relativity. Classical spacetimes. In Newton–Cartan theory, one starts with a smooth four-dimensional manifold formula_1 and defines "two" (degenerate) metrics. A "temporal metric" formula_2 with signature formula_3, used to assign temporal lengths to vectors on formula_1 and a "spatial metric" formula_5 with signature formula_6. One also requires that these two metrics satisfy a transversality (or "orthogonality") condition, formula_7. Thus, one defines a "classical spacetime" as an ordered quadruple formula_8, where formula_2 and formula_5 are as described, formula_11 is a metrics-compatible covariant derivative operator; and the  metrics satisfy the orthogonality condition. One might say that a classical spacetime is the analog of a relativistic spacetime formula_12, where formula_13 is a smooth Lorentzian metric on the manifold formula_1. Geometric formulation of Poisson's equation. In Newton's theory of gravitation, Poisson's equation reads where formula_16 is the gravitational potential, formula_17 is the gravitational constant and formula_18 is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential formula_19 where formula_21 is the inertial mass and formula_22 the gravitational mass. Since, according to the  weak equivalence principle formula_23, the according equation of motion does not contain anymore a reference to the mass of the particle. Following the idea that the solution of the equation then is a property of the curvature of space, a connection is constructed so that the geodesic equation represents the equation of motion of a point particle in the potential formula_16. The resulting connection is with formula_28 and formula_29 (formula_30). The connection has been constructed in one inertial system but can be shown to be valid in any inertial system by showing the invariance of formula_31 and formula_32 under Galilei -transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by where the brackets formula_34 mean the antisymmetric combination of the tensor formula_35. The Ricci tensor is given by which leads to following geometric formulation of Poisson's equation More explicitly, if the roman indices "i" and "j" range over the spatial coordinates 1, 2, 3, then the connection is given by the Riemann curvature tensor by and the Ricci tensor and Ricci scalar by where all components not listed equal zero. Note that this formulation does not require introducing the concept of a metric : the connection alone gives all the physical information. Bargmann lift. It was shown that four-dimensional Newton–Cartan theory of gravitation can be reformulated as Kaluza–Klein reduction of five-dimensional Einstein gravity along a null-like direction. This lifting is considered to be useful for non-relativistic holographic models.