Einstein–Cartan theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein–Cartan theory is the simplest Poincaré gauge theory.
Overview
Einstein–Cartan theory differs from general relativity in two ways: 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; an additional set of equations are posed that relate torsion to spin. This difference can be factored intoby 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. In particular, the antisymmetric part of the connection 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, then is also possible to directly translate the Einstein–Hilbert action into a Riemann–Cartan geometry, the result being the Palatini action. 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. 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: 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; an additional set of equations are now present which couple the torsion to the intrinsic angular momentum 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. 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 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.
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 angular momentum to the presence of the gravitational field.
History
The theory was first proposed by Élie Cartan in 1922 and expounded in the following few years. Albert Einstein became affiliated with the theory in 1928 during his unsuccessful attempt to match torsion to the electromagnetic field tensor as part of a unified field theory. This line of thought led him to the related but different theory of teleparallelism.Dennis Sciama and Tom Kibble independently revisited the theory in the 1960s, and an important review was published in 1976.
Einstein–Cartan theory has been historically overshadowed by its torsion-free counterpart and other alternatives like Brans–Dicke theory because torsion seemed to add little predictive benefit at the expense of the tractability of its equations. Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of quantum gravity. In the Einstein–Cartan theory, the Dirac equation becomes nonlinear and therefore the superposition principle used in usual quantization techniques would not work. Recently, interest in Einstein–Cartan theory has been driven toward cosmological implications, most importantly, the avoidance of a gravitational singularity at the beginning of the universe. The theory is considered viable and remains an active topic in the physics community.
Field equations
The Einstein field equations of general relativity can be derived by postulating the Einstein–Hilbert action to be the true action of spacetime and then varying that action with respect to the metric tensor.The field equations of Einstein–Cartan theory come from exactly the same approach,
except that a general asymmetric affine connection is assumed rather than the symmetric Levi-Civita connection
,
and then the metric and torsion are varied independently.
Let represent the Lagrangian density of matter and represent the Lagrangian density of the gravitational field. The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the Ricci scalar:
where is the determinant of the metric tensor, and is a physical constant involving the gravitational constant and the speed of light. By Hamilton's principle, the variation of the total action for the gravitational field and matter vanishes:
The variation with respect to the metric tensor yields the Einstein equations:
where is the Ricci tensor and is the canonical stress–energy–momentum tensor.
The Ricci tensor is no longer symmetric because the connection contains a nonzero torsion tensor; therefore, the right-hand side of the equation cannot be symmetric either, implying that must include an asymmetric contribution that can be shown to be related to the spin tensor. This canonical energy–momentum tensor is related to the more familiar symmetric energy–momentum tensor by the Belinfante–Rosenfeld procedure.
The variation with respect to the torsion tensor yields the Cartan spin connection equations
where is the spin tensor. Because the torsion equation is an algebraic constraint rather than a partial differential equation, the torsion field does not propagate as a wave, and vanishes outside of matter. Therefore, in principle the torsion can be algebraically eliminated from the theory in favor of the spin tensor, which generates an effective "spin–spin" nonlinear self-interaction inside matter.