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Newton–Cartan theory is a geometrical re-formulation, as well as a generalization, of Newtonian gravity developed by Élie Cartan. 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 Kurt 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.

Geometric formulation of Poisson's equation

In Newton's theory of gravitation the Poisson equation reads

$$\Delta U = 4 \pi G \rho \,$$

where U is the gravitational potential, G is the gravitational constant and \rho is the mass density. The weak equivalence principle motivates a geometric version of the equation of motion for a point particle in the potential U

$$m_t \ddot{\vec x} = - m_g \nabla U$$

where m_t is the inertial mass and m_g the gravitational mass. Since, according to the weak equivalence principle m_t = m_g , the according equation of motion

$$\ddot{\vec x} = - \nabla U$$

doesn't 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

$$\frac{d^2 x^\lambda}{d s^2} + \Gamma_{\mu \nu}^\lambda \frac{d x^\mu}{d s}\frac{d x^\nu}{d s} = 0$$

represents the equation of motion of a point particle in the potential U. The resulting connection is

$$\Gamma_{\mu \nu}^{\lambda} = \gamma^{\lambda \rho} U_{, \rho} \Psi_\mu \Psi_\nu$$

with $$\Psi_\mu = \delta_\mu^0 and \gamma^{\mu \nu} = \delta^\mu_A \delta^\nu_B \delta^{AB} ( A, B = 1,2,3 )$$. 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 $$\Psi_\mu$$ and $$\gamma^{\mu \nu}$$ under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by

$$R^\lambda_{\kappa \mu \nu} = 2 \gamma^{\lambda \sigma} U_{, \sigma [ \mu}\Psi_{\nu]}\Psi_\kappa$$

where the brackets $$A_{[\mu \nu]} = \frac{1}{2!} [ A_{\mu \nu} - A_{\nu \mu} ]$$ mean the antisymmetric combination of the tensor $$A_{\mu \nu}$$ . The Ricci tensor is given by

$$R_{\kappa \nu} = \Delta U \Psi_{\kappa \nu} \,$$

which leads to following geometric formulation of Poisson's equation

$$R_{\mu \nu} = 4 \pi G \rho \Psi_\mu \Psi_\nu \,$$

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.[1] This lifting is considered to be useful for non-relativistic holographic models.[2]
References

^ C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)
^ Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [1]

Bibliography

Cartan, Elie (1923), Ann. Ecole Norm. 40: 325
Cartan, Elie (1924), Ann. Ecole Norm. 41: 1
Cartan, Elie (1955), OEuvres Complétes, III/1, Gauthier-Villars, pp. 659, 799

Chapter 1 of Ehlers, Jürgen (1973), "Survey of general relativity theory", in Israel, Werner, Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, ISBN 90-277-0369-8