Girish Mahajan (Editor)

Newton–Cartan theory

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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.

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Geometric formulation of Poisson's equation

In Newton's theory of gravitation, Poisson's equation reads

Δ U = 4 π G ρ

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 x → ¨ = − m g ∇ 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 according equation of motion

x → ¨ = − ∇ 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

d 2 x λ d s 2 + Γ μ ν λ d x μ d s d x ν d s = 0

represents the equation of motion of a point particle in the potential U . The resulting connection is

Γ μ ν λ = γ λ ρ U , ρ Ψ μ Ψ ν

with Ψ μ = δ μ 0 and γ μ ν = δ A μ δ B ν δ A B ( 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 Ψ μ and γ μ ν under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by

R κ μ ν λ = 2 γ λ σ U , σ [ μ Ψ ν ] Ψ κ

where the brackets A [ μ ν ] = 1 2 ! [ A μ ν − A ν μ ] mean the antisymmetric combination of the tensor A μ ν . The Ricci tensor is given by

R κ ν = Δ U Ψ κ Ψ ν

which leads to following geometric formulation of Poisson's equation

R μ ν = 4 π G ρ Ψ μ Ψ ν


More explicitly, if the roman indices i and j range over the spatial coordinates 1, 2, 3, then the connection is given by

Γ 00 i = U , i

the Riemann curvature tensor by

R 0 j 0 i = − R 00 j i = U , i j

and the Ricci tensor and Ricci scalar by

R = R 00 = Δ 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

Newton–Cartan theory Wikipedia
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