Newton–Cartan theory

Newton–Cartan theory describes Newtonian mechanics geometrically using absolute time, a degenerate spatial inverse metric, and a compatible torsion-free affine connection. It is not a theory with a nondegenerate Lorentzian metric. Gravitationally accelerated particles are geodesics of that connection, whose curvature measures tidal gravity.

In Cartesian spatial coordinates and absolute time $t$, the gravitational potential $\Phi (t,x)$ determines an affine connection by the displayed coefficients, all others zero. The geodesic equation with affine time is $d^2{x}^i/d{t}^2=-{\partial }_i\Phi$. It preserves $dt$ and the spatial inverse metric $h^{ij}={\delta }^{ij}$, reproducing freely falling Newtonian mechanics in a curved spacetime connection.

Using $[{\nabla }_c,{\nabla }_d]V^a=R^a{}_{bcd}{V}^b$, the Newton–Cartan gravitational connection has only the displayed Hessian components and their antisymmetric partners in the last two indices. The geodesic deviation equation gives relative acceleration $-{\partial }_i{\partial }_j\Phi \delta x^j$. The Poisson equation for Newtonian gravity is therefore $R_{ab}=4\pi G\rho (dt{)}_a(dt{)}_b$. A spatially uniform gravitational acceleration has zero tidal curvature even if the connection coefficients do not vanish in the original coordinates.