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?


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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.