Newton–Cartan theory

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

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

${\displaystyle \Delta U=4\pi G\rho \,}$

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

${\displaystyle m_{t}\,{\ddot {\vec {x}}}=-m_{g}{\vec {\nabla }}U}$

where ${\displaystyle m_{t}}$ is the inertial mass and ${\displaystyle m_{g}}$ the gravitational mass. Since, according to the weak equivalence principle ${\displaystyle m_{t}=m_{g}}$, the according equation of motion

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

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

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

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

${\displaystyle \Gamma _{\mu \nu }^{\lambda }=\gamma ^{\lambda \rho }U_{,\rho }\Psi _{\mu }\Psi _{\nu }}$

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

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

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

${\displaystyle R_{\kappa \nu }=\Delta U\Psi _{\kappa }\Psi _{\nu }\,}$

which leads to following geometric formulation of Poisson's equation

${\displaystyle R_{\mu \nu }=4\pi G\rho \Psi _{\mu }\Psi _{\nu }}$

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

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

the Riemann curvature tensor by

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

and the Ricci tensor and Ricci scalar by

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

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.

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.[9] This lifting is considered to be useful for non-relativistic holographic models.[10]

• 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