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 $t_$ with signature $(1,0,0,0)$ , used to assign temporal lengths to vectors on $M$ and a spatial metric $h^$ with signature $(0,1,1,1)$ . One also requires that these two metrics satisfy a transversality (or "orthogonality") condition, $h^t_=0$ . Thus, one defines a classical spacetime as an ordered quadruple $(M,t_,h^,\nabla )$ , where $t_$ and $h^$ 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 $(M,g_)$ , where $g_$ 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 $U$ $m_\,{\ddot {\vec }}=-m_{\vec {\nabla }}U$ where $m_$ is the inertial mass and $m_$ the gravitational mass. Since, according to the weak equivalence principle $m_=m_$ , the according equation of motion

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

${\frac x^{\lambda }}}}+\Gamma _{\mu \nu }^{\lambda }{\frac }}{\frac }}=0$ 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 $\Psi _{\mu }=\delta _{\mu }^$ and $\gamma ^{\mu \nu }=\delta _^{\mu }\delta _^{\nu }\delta ^$ ($A,B=1,2,3$ ). 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

$R_{\kappa \mu \nu }^{\lambda }=2\gamma ^{\lambda \sigma }U_{,\sigma [\mu }\Psi _{\nu ]}\Psi _{\kappa }$ where the brackets $A_{[\mu \nu ]}={\frac }[A_{\mu \nu }-A_{\nu \mu }]$ 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

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

$R_^=-R_^=U_{,ij}$ and the Ricci tensor and Ricci scalar by

$R=R_=\Delta U$ 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.