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'''Newton–Cartan theory''' is a geometrical re-formulation, as well as a generalization, of [[Newtonian gravity]] developed by [[Élie Cartan]].  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 [[Kurt 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.


==Geometric formulation of Poisson's equation==


In Newton's theory of gravitation, [[Poisson's equation]] reads
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:<math>
\Delta U = 4 \pi G \rho \,
</math>
where <math>U</math> is the gravitational potential, <math>G</math> is the gravitational constant and <math>\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> U </math>
:<math>
m_t \ddot{\vec x} = - m_g \nabla U
</math>
where <math>m_t</math> is the inertial mass and <math>m_g</math> the gravitational mass. Since, according to the weak equivalence principle <math> m_t = m_g </math>, the according equation of motion
:<math>
\ddot{\vec x} = - \nabla U
</math>
doesn't 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>
\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>U</math>. The resulting connection is
:<math>
\Gamma_{\mu \nu}^{\lambda} = \gamma^{\lambda \rho} U_{, \rho} \Psi_\mu \Psi_\nu
</math>
with <math>\Psi_\mu = \delta_\mu^0 </math> and <math>\gamma^{\mu \nu} = \delta^\mu_A \delta^\nu_B \delta^{AB}</math> (<math> 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> \Psi_\mu</math> and <math> \gamma^{\mu \nu} </math> under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by
:<math>
R^\lambda_{\kappa \mu \nu} = 2 \gamma^{\lambda \sigma} U_{, \sigma [ \mu}\Psi_{\nu]}\Psi_\kappa
</math>
where the brackets <math> A_{[\mu \nu]} = \frac{1}{2!} [ A_{\mu \nu} - A_{\nu \mu} ] </math> mean the antisymmetric combination of the tensor <math> A_{\mu \nu} </math>. The [[Ricci tensor]] is given by
:<math>
R_{\kappa \nu} = \Delta U \Psi_{\kappa \nu} \,
</math>
which leads to following geometric formulation of Poisson's equation
:<math>
R_{\mu \nu} = 4 \pi G \rho \Psi_\mu \Psi_\nu \,
</math>
 
==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.<ref>C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)</ref> This lifting is considered to be useful for non-relativistic [[Holographic principle|holographic]] models.<ref>Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [http://lanl.arxiv.org/abs/0806.2867]</ref>
 
==References==
 
{{Reflist}}
 
==Bibliography==
 
*{{Citation
| last=Cartan
| first=Elie
| year=1923
| journal=Ann. Ecole Norm.
| volume=40
| page=325
}}
*{{Citation
| last=Cartan
| first=Elie
| year=1924
| journal=Ann. Ecole Norm.
| volume=41
| page=1
}}
*{{Citation
| last=Cartan
| first=Elie
| year=1955
| title=OEuvres Complétes
| volume=III/1
| pages=659, 799
| publisher=Gauthier-Villars
}}
 
*Chapter 1 of {{Citation
| last=Ehlers
| first = Jürgen
| author-link=Jürgen Ehlers
| contribution=Survey of general relativity theory
| editor-last=Israel
| editor-first=Werner
| title=Relativity, Astrophysics and Cosmology
| year=1973
| publisher=D. Reidel
| pages=1–125
| isbn=90-277-0369-8
}}
 
{{DEFAULTSORT:Newton-Cartan theory}}
[[Category:Theories of gravitation]]

Latest revision as of 00:52, 7 December 2014


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