Einstein–Cartan theory differs from general relativity in two ways: This difference can be factored into 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.
In particular, to be able to describe spinors requires the inclusion of a spin structure, which suffices to produce such a geometry.
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.
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.
[7] Dennis Sciama[8] and Tom Kibble[9] independently revisited the theory in the 1960s, and an important review was published in 1976.
Since the Einstein–Cartan theory is purely classical, it also does not fully address the issue of quantum gravity.
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 (i.e., spacetime is assumed to have torsion in addition to curvature), and then the metric and torsion are varied independently.
The Lagrangian density for the gravitational field in the Einstein–Cartan theory is proportional to the Ricci scalar: where
for the gravitational field and matter vanishes: The variation with respect to the metric tensor
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.
Torsion is equal to its source term and can be replaced by a boundary or a topological structure with a throat such as a "wormhole".
Consequently, Einstein–Cartan theory is able to avoid the general-relativistic problem of the singularity at the Big Bang.
[18][19] The minimal coupling between torsion and Dirac spinors generates an effective nonlinear spin–spin self-interaction, which becomes significant inside fermionic matter at extremely high densities.
This scenario also explains why the present Universe at largest scales appears spatially flat, homogeneous and isotropic, providing a physical alternative to cosmic inflation.
Torsion allows fermions to be spatially extended instead of "pointlike", which helps to avoid the formation of singularities such as black holes, removes the ultraviolet divergence in quantum field theory, and leads to the toroidal ring model of electrons.
[20] According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular black hole.
In the Einstein–Cartan theory, instead, the collapse reaches a bounce and forms a regular Einstein–Rosen bridge (wormhole) to a new, growing universe on the other side of the event horizon; pair production by the gravitational field after the bounce, when torsion is still strong, generates a finite period of inflation.