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[1][2] and Kurt Friedrichs[3] and later developed by Dautcourt,[4] Dixon,[5] Dombrowski and Horneffer, Ehlers, Havas,[6] Künzle,[7] Lottermoser, Trautman,[8] 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 $\displaystyle{ M }$ 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 $\displaystyle{ M }$.

Geometric formulation of Poisson's equation

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

$\displaystyle{ \frac{d^2 x^\lambda}{d s^2} + \Gamma_{\mu \nu}^\lambda \frac{d x^\mu}{d s}\frac{d x^\nu}{d s} = 0 }$

represents the equation of motion of a point particle in the potential $\displaystyle{ U }$. 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 $\displaystyle{ \gamma^{\mu \nu} = \delta^\mu_A \delta^\nu_B \delta^{AB} }$ ($\displaystyle{ A, B = 1,2,3 }$). 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

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

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

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

the Riemann curvature tensor by

$\displaystyle{ R^i_{0j0} = -R^i_{00j} = U_{,ij} }$

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.

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

References

1. 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
2. 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
3. Friedrichs, K. O. (1927), "Eine Invariante Formulierung des Newtonschen Gravitationsgesetzes und der Grenzüberganges vom Einsteinschen zum Newtonschen Gesetz", Mathematische Annalen 98: 566–575, doi:10.1007/bf01451608
4. Dautcourt, G. (1964), "Die Newtonische Gravitationstheorie als strenger Grenzfall der allgemeinen Relativitätstheorie", Acta Physica Polonica 65: 637–646
5. Dixon, W. G. (1975), "On the uniqueness of the Newtonian theory as a geometric theory of gravitation", Communications in Mathematical Physics 45 (2): 167–182, doi:10.1007/bf01629247, Bibcode1975CMaPh..45..167D
6. Havas, P. (1964), "Four-dimensional formulations of Newtonian mechanics and their relation to the special and general theory of relativity", Reviews of Modern Physics 36 (4): 938–965, doi:10.1103/revmodphys.36.938, Bibcode1964RvMP...36..938H
7. Künzle, H. (1976), "Covariant Newtonian limts of Lorentz space-times", General Relativity and Gravitation 7 (5): 445–457, doi:10.1007/bf00766139, Bibcode1976GReGr...7..445K
8. Trautman, A. (1965), Deser, Jürgen; Ford, K. W., eds., Foundations and current problems of general relativity, 98, Englewood Cliffs, New Jersey: Prentice-Hall, pp. 1–248
9. Duval, C.; Burdet, G.; Künzle, H. P.; Perrin, M. (1985). "Bargmann structures and Newton-Cartan theory". Physical Review D 31 (8): 1841–1853. doi:10.1103/PhysRevD.31.1841. PMID 9955910. Bibcode1985PhRvD..31.1841D.
10. Goldberger, Walter D. (2009). "AdS/CFT duality for non-relativistic field theory". Journal of High Energy Physics 2009 (3): 069. doi:10.1088/1126-6708/2009/03/069. Bibcode2009JHEP...03..069G.