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 and expounded in the following few years.

The theory relaxes the assumption that the affine connection has vanishing antisymmetric part (torsion tensor), so that the torsion can be coupled to the intrinsic angular momentum (spin) of matter, much in the same way in which the curvature is coupled to the energy and momentum of matter. In fact, the spin of matter in curved spacetime requires that torsion is not constrained to be zero but is a variable in the principle of stationary action.

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.

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.

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