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While learning General Relativity, we always use the Levi-Civita connection, which has the special property of being torsion-less.
My question is: Do we need to use a torsion-less connection to study General Relativity? If so, why is it essential? What does a torsion-full connection change physics-wise?

Extra: Are there any models in theoretical physics that use torsion-full connections?

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Yes, it is one of the mathematical assumptions when describing Einstein's theory. If you drop the torsion-less connection requirement, you obtain a more general theory, the Einstein-Cartan one.

https://en.wikipedia.org/wiki/Torsion_tensor to see how the Bianchi identities change in the presence of torsion.

An Einstein-Cartan theory is an intermediary step to supergravity theories. We also have the Poincare gauge theories of Hehl.

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  • $\begingroup$ Thanks for the answer! Could you please explain the reason that we need a torsion-less connection? $\endgroup$ – TheQuantumMan Oct 30 '17 at 15:28
  • $\begingroup$ See the links by Qmechanic to see how a non-vanishing (but small) torsion would check against experimental results/predictions. $\endgroup$ – DanielC Oct 30 '17 at 15:30
  • $\begingroup$ Sure. Thanks! By the way, do supergravity theories use torsion-full connections? $\endgroup$ – TheQuantumMan Oct 30 '17 at 16:10

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