Newton–Cartan Theory

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'' $t_$ with signature $(1, 0, 0, 0)$, used to assign temporal lengths to vectors on $M$ and a ''spatial metric'' $h^$ with signature $(0, 1, 1, 1)$. One also requires that these two metrics satisfy a transversality (or "orthogonality") condition, $h^t_=0$. Thus, one defines a ''classical spacetime'' as an ordered quadruple $(M, t_, h^, \nabla)$, where $t_$ and $h^$ 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 $(M, g_)$, where $g_$ 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 $U$
:$m_t \, \ddot = - m_g U$
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
:$\ddot = - U$
does not 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
:$\frac + \Gamma_^\lambda \frac\frac = 0$
represents the equation of motion of a point particle in the potential $U$. The resulting connection is
:$\Gamma_^ = \gamma^ U_ \Psi_\mu \Psi_\nu$
with $\Psi_\mu = \delta_\mu^0$ and $\gamma^ = \delta^\mu_A \delta^\nu_B \delta^$ ($A, B = 1,2,3$). 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 $\gamma^$ under Galilei-transformations. The Riemann curvature tensor in inertial system coordinates of this connection is then given by
:$R^\lambda_ = 2 \gamma^ U_\Psi_\Psi_\kappa$
where the brackets A_ = \frac A_ - A_ mean the antisymmetric combination of the tensor $A_$. The Ricci tensor is given by
:$R_ = \Delta U \Psi_\Psi_ \,$
which leads to following geometric formulation of Poisson's equation
:$R_ = 4 \pi G \rho \Psi_\mu \Psi_\nu$

More explicitly, if the roman indices ''i'' and ''j'' range over the spatial coordinates 1, 2, 3, then the connection is given by
:$\Gamma^i_ = U_$
the Riemann curvature tensor by
:$R^i_ = -R^i_ = U_$
and the Ricci tensor and Ricci scalar by
:$R = R_ = \Delta U$
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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* (English translation of Ann. Sci. Éc. Norm. Supér. #40 paper)
*Chapter 1 of

Theories of gravity
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