Editing Einstein–Cartan theory (section)

== Overview ==
Einstein–Cartan theory differs from general relativity in two ways:

: (1) it is formulated within the framework of Riemann–Cartan geometry, which possesses a locally gauged Lorentz symmetry, while general relativity is formulated within the framework of Riemannian geometry, which does not;

: (2) an additional set of equations are posed that relate torsion to spin.

This difference can be factored into

:: general relativity (Einstein–Hilbert) → general relativity (Palatini) → '''Einstein–Cartan'''

by first reformulating general relativity onto a Riemann–Cartan geometry, replacing the Einstein–Hilbert action over Riemannian geometry by the Palatini action over Riemann–Cartan geometry; and second, removing the zero torsion constraint from the Palatini action, which results in the additional set of equations for spin and torsion, as well as the addition of extra spin-related terms in the Einstein field equations themselves.

The theory of general relativity was originally formulated in the setting of [[Riemannian geometry]] by the [[Einstein–Hilbert action]], out of which arise the [[Einstein field equations]]. At the time of its original formulation, there was no concept of Riemann–Cartan geometry. Nor was there a sufficient awareness of the concept of [[gauge symmetry]] to understand that Riemannian geometries do not possess the requisite structure to embody a locally gauged [[Lorentz symmetry]], such as would be required to be able to express continuity equations and conservation laws for rotational and boost symmetries, or to describe [[spinors]] in curved spacetime geometries. The result of adding this infrastructure is a Riemann–Cartan geometry. In particular, to be able to describe spinors requires the inclusion of a [[spin structure]], which suffices to produce such a geometry.

The chief difference between a Riemann–Cartan geometry and Riemannian geometry is that in the former, the [[affine connection]] is independent of the metric, while in the latter it is derived from the metric as the [[Levi-Civita connection]], the difference between the two being referred to as the [[contorsion tensor|contorsion]]. In particular, the antisymmetric part of the connection (referred to as the [[torsion tensor|torsion]]) is zero for Levi-Civita connections, as one of the defining conditions for such connections.

Because the contorsion can be expressed linearly in terms of the torsion, it is also possible to directly translate the Einstein–Hilbert action into a Riemann–Cartan geometry, the result being the [[Palatini action]] (see also [[Palatini variation]]). It is derived by rewriting the Einstein–Hilbert action in terms of the affine connection and then separately posing a constraint that forces both the torsion and contorsion to be zero, which thus forces the affine connection to be equal to the Levi-Civita connection. Because it is a direct translation of the action and field equations of general relativity, expressed in terms of the Levi-Civita connection, this may be regarded as the theory of general relativity, itself, transposed into the framework of Riemann–Cartan geometry.

Einstein–Cartan theory relaxes this condition and, correspondingly, relaxes general relativity's assumption that the affine connection have a vanishing antisymmetric part ([[torsion tensor]]). The action used is the same as the Palatini action, except that the constraint on the torsion is removed. This results in two differences from general relativity:

: (1) the field equations are now expressed in terms of affine connection, rather than the Levi-Civita connection, and so have additional terms in Einstein's field equations involving the contorsion that are not present in the field equations derived from the Palatini formulation;

: (2) an additional set of equations are now present which couple the torsion to the intrinsic angular momentum ([[Particle spin|spin]]) of matter, much in the same way in which the affine connection is coupled to the energy and momentum of matter.

In Einstein–Cartan theory, the torsion is now a variable in the [[principle of stationary action]] that is coupled to a curved spacetime formulation of spin (the [[spin tensor]]). These extra equations express the torsion linearly in terms of the spin tensor associated with the matter source, which entails that the torsion generally be non-zero inside matter.

A consequence of the linearity is that outside of matter there is zero torsion, so that the exterior geometry remains the same as what would be described in general relativity. The differences between Einstein–Cartan theory and general relativity (formulated either in terms of the Einstein–Hilbert action on Riemannian geometry or the Palatini action on Riemann–Cartan geometry) rest solely on what happens to the geometry inside matter sources. That is: "torsion does not propagate". Generalizations of the Einstein–Cartan action have been considered which allow for propagating torsion.<ref>{{cite journal | last=Neville | first=Donald E. | date=15 February 1980 | title=Gravity theories with propagating torsion | journal=Physical Review D | volume=21 | issue=4 | issn=0556-2821 | doi=10.1103/physrevd.21.867 | pages=867–873 | bibcode=1980PhRvD..21..867N }}</ref>

Because Riemann–Cartan geometries have Lorentz symmetry as a local gauge symmetry, it is possible to formulate the associated conservation laws. In particular, regarding the metric and torsion tensors as independent variables gives the correct generalization of the conservation law for the total (orbital plus intrinsic) angular momentum to the presence of the gravitational field.