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

Geometric formulation of Poisson's equation

In Newton's theory of gravitation, Poisson's equation reads

[math]\displaystyle{ \Delta U = 4 \pi G \rho \, }[/math]

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

[math]\displaystyle{ m_t \, \ddot{\vec x} = - m_g {\vec \nabla} U }[/math]

where [math]\displaystyle{ m_t }[/math] is the inertial mass and [math]\displaystyle{ m_g }[/math] the gravitational mass. Since, according to the weak equivalence principle [math]\displaystyle{ m_t = m_g }[/math], the according equation of motion

[math]\displaystyle{ \ddot{\vec x} = - {\vec \nabla} U }[/math]

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

[math]\displaystyle{ \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 }[/math]

represents the equation of motion of a point particle in the potential [math]\displaystyle{ U }[/math]. The resulting connection is

[math]\displaystyle{ \Gamma_{\mu \nu}^{\lambda} = \gamma^{\lambda \rho} U_{, \rho} \Psi_\mu \Psi_\nu }[/math]

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

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

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

[math]\displaystyle{ R_{\kappa \nu} = \Delta U \Psi_{\kappa}\Psi_{\nu} \, }[/math]

which leads to following geometric formulation of Poisson's equation

[math]\displaystyle{ R_{\mu \nu} = 4 \pi G \rho \Psi_\mu \Psi_\nu }[/math]

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

[math]\displaystyle{ \Gamma^i_{00} = U_{,i} }[/math]

the Riemann curvature tensor by

[math]\displaystyle{ R^i_{0j0} = -R^i_{00j} = U_{,ij} }[/math]

and the Ricci tensor and Ricci scalar by

[math]\displaystyle{ R = R_{00} = \Delta U }[/math]

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.

Bibliography

• Cartan, Élie (1923), "Sur les variétés à connexion affine et la théorie de la relativité généralisée (Première partie)", Annales Scientifiques de l'École Normale Supérieure 40: 325, doi:10.24033/asens.751, http://archive.numdam.org/article/ASENS_1923_3_40__325_0.pdf 
• Cartan, Élie (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (Première partie) (Suite)", Annales Scientifiques de l'École Normale Supérieure 41: 1, doi:10.24033/asens.753, http://archive.numdam.org/article/ASENS_1924_3_41__1_0.pdf 
• Cartan, Élie (1955), Œuvres complètes, III/1, Gauthier-Villars, pp. 659, 799 
• Renn, Jürgen; Schemmel, Matthias, eds. (2007), The Genesis of General Relativity, 4, Springer, pp. 1107–1129  (English translation of Ann. Sci. Éc. Norm. Supér. #40 paper)
• 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