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.

In Newton–Cartan theory, one starts with a smooth four-dimensional manifold ${\displaystyle M}$ and defines two (degenerate) metrics. A temporal metric ${\displaystyle t_{ab}}$ with signature ${\displaystyle (1,0,0,0)}$, used to assign temporal lengths to vectors on ${\displaystyle M}$ and a spatial metric ${\displaystyle h^{ab}}$ with signature ${\displaystyle (0,1,1,1)}$. One also requires that these two metrics satisfy a trasversality (or "orthogonality") condition, ${\displaystyle h^{ab}t_{bc}=0}$. Thus, one defines a classical spacetime as an ordered quadruple ${\displaystyle (M,t_{ab},h^{ab},\nabla )}$, where ${\displaystyle t_{ab}}$ and ${\displaystyle h^{ab}}$ are as described, ${\displaystyle \nabla }$ 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 ${\displaystyle (M,g_{ab})}$, where ${\displaystyle g_{ab}}$ is a smooth Lorentzian metric on the manifold ${\displaystyle M}$.

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