Newton–Cartan theory

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}{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 \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^\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 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^i_{00} = U_{,i} the Riemann curvature tensor by : R^i_{0j0} = -R^i_{00j} = 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.

References

Bibliography

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References

1. Malament, David B.. (2012-04-02). "Topics in the Foundations of General Relativity and Newtonian Gravitation Theory". University of Chicago Press.
2. (1985). "Bargmann structures and Newton-Cartan theory". Physical Review D.
3. (2009). "AdS/CFT duality for non-relativistic field theory". Journal of High Energy Physics.