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 G. Dautcourt, W. G. Dixon, P. Havas, H. Künzle, Andrzej 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

${\frac {d^{2}x^{\lambda }}{ds^{2}}}+\Gamma _{\mu \nu }^{\lambda }{\frac {dx^{\mu }}{ds}}{\frac {dx^{\nu }}{ds}}=0$

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 $\gamma ^{\mu \nu }=\delta _{A}^{\mu }\delta _{B}^{\nu }\delta ^{AB}$ ($A,B=1,2,3$). 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

$R_{\kappa \mu \nu }^{\lambda }=2\gamma ^{\lambda \sigma }U_{,\sigma [\mu }\Psi _{\nu ]}\Psi _{\kappa }$

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

$\Gamma _{00}^{i}=U_{,i}$

the Riemann curvature tensor by

$R_{0j0}^{i}=-R_{00j}^{i}=U_{,ij}$

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.

Bibliography

• Cartan, Élie (1923), "Sur les variétés à connexion affine et la théorie de la relativité généralisée (Première partie)" (PDF), Annales Scientifiques de l'École Normale Supérieure, 40: 325, doi:10.24033/asens.751
• Cartan, Élie (1924), "Sur les variétés à connexion affine, et la théorie de la relativité généralisée (Première partie) (Suite)" (PDF), Annales Scientifiques de l'École Normale Supérieure, 41: 1, doi:10.24033/asens.753
• Cartan, Élie (1955), Œuvres complètes, vol. III/1, Gauthier-Villars, pp. 659, 799
• Renn, Jürgen; Schemmel, Matthias, eds. (2007), The Genesis of General Relativity, vol. 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 (ed.), Relativity, Astrophysics and Cosmology, D. Reidel, pp. 1–125, ISBN 90-277-0369-8