Newton–Cartan theory

Newton–Cartan theory

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.

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Geometric formulation of Poisson's equation

In Newton's theory of gravitation the Poisson equation reads


\Delta U = 4 \pi G \rho \,

where U is the gravitational potential, G is the gravitational constant and ρ 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 \nabla U

where mt is the inertial mass and mg the gravitational mass. Since, according to the weak equivalence principle mt = mg, the according equation of motion


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

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


\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 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^\mu_A \delta^\nu_B \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 Ψμ and γμν under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by


R^\lambda_{\kappa \mu \nu} = 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 Aμν. The Ricci tensor is given by


R_{\kappa \nu} = \Delta U \Psi_{\kappa \nu} \,

which leads to following geometric formulation of Poisson's equation


R_{\mu \nu} = 4 \pi G \rho \Psi_\mu \Psi_\nu \,

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.[1] This lifting is considered to be useful for non-relativistic holographic models.[2]

References

  1. ^ C. Duval, G. Burdet, H. P. Künzle, and M. Perrin, Bargmann structures and Newton-Cartan theory, Phys. Rev. D 31, 1841–1853 (1985)
  2. ^ Walter D. Goldberger, AdS/CFT duality for non-relativistic field theory, JHEP03(2009)069 [1]

Bibliography

  • Cartan, Elie (1923), Ann. Ecole Norm. 40: 325 
  • Cartan, Elie (1924), Ann. Ecole Norm. 41: 1 
  • Cartan, Elie (1955), OEuvres Complétes, III/1, Gauthier-Villars, pp. 659, 799 
  • 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 

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