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

Geometric formulation of Poisson's equation
In Newton's theory of gravitation, Poisson's 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 {\vec \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 corresponding equation of motion

\ddot{\vec x} = - {\vec \nabla} U $$ no longer contains 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}\Psi_{\nu} \, $$ which leads to following geometric formulation of Poisson's equation

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

More explicitly, if the roman indices i and j range over the spatial coordinates 1, 2, 3, then the connection is given by

\Gamma^i_{00} = U_{,i} $$ the Riemann curvature tensor by

R^i_{0j0} = -R^i_{00j} = U_{,ij} $$ and the Ricci tensor and Ricci scalar by

R = R_{00} = \Delta U $$ 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.