Newton–Cartan theory explained

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 and Kurt Friedrichs and later developed by Dautcourt, Dixon, Dombrowski and Horneffer, Ehlers, Havas, Künzle, Lottermoser, Trautman, 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

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(d ^{2} x ^{λ})/(d s ^{2}) + \Gamma ^{λ} _{\mu \nu} (d x ^{\mu})/(d s) (d x ^{\nu})/(d s) = 0 represents the equation of motion of a point particle in the potential U . The resulting connection is\Gamma ^{λ} _{\mu \nu} = \gamma ^{λ} U _{,} \Psi _{\mu} \Psi _{\nu} with \Psi _{\mu} = \delta ^{0} _{\mu} and \gamma ^{\mu} = \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} under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given byR ^{λ} _{\kappa \mu \nu} = 2 \gamma ^{λ} U _{,} \Psi _{\nu]} \Psi _{\kappa} where the brackets A _{[\mu} = (1)/(2 !) [ A _{\mu} - A _{\nu} ] mean the antisymmetric combination of the tensor A _{\mu} . The Ricci tensor is given byR _{\kappa} = \Delta U \Psi _{\kappa} \Psi _{\nu} which leads to following geometric formulation of Poisson's equationR _{\mu} = 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 byR ^{i} _{0 j 0} = - R ^{i} _{00 j} = U _{,ij} and the Ricci tensor and Ricci scalar byR = 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.^[1] This lifting is considered to be useful for non-relativistic holographic models.^[2]

Bibliography

• • • • (English translation of Ann. Sci. Éc. Norm. Supér. #40 paper)
• Chapter 1 of

Notes and References

1. 10.1103/PhysRevD.31.1841. 9955910. Bargmann structures and Newton-Cartan theory. Physical Review D. 31. 8. 1841–1853. 1985. Duval. C.. Burdet. G.. Künzle. H. P.. Perrin. M.. 1985PhRvD..31.1841D.
2. 0806.2867. 10.1088/1126-6708/2009/03/069. AdS/CFT duality for non-relativistic field theory. Journal of High Energy Physics. 2009. 3. 069. 2009. Goldberger. Walter D.. 2009JHEP...03..069G. 118553009 .