Interpretation of Curvature and Torsion

Question

Dear all,

When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields

$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda.$

Usually, it is said that curvature is the responsible of the change of the direction of the vector under parallel transport through the two different paths.

However, in general there are three different effects in the transportation:

• Change of direction of the vector.
• Non-closure of the path (say, if one moves 1meter along each direction).
• Rotation about its own axis.

whilst the general commutator is

$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda- T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho.$

Is it possible to give a meaning to the curvature and torsion in term of these intuitive geometry or is not possible in general?

Thank you!

Answer 1

Here is my attempt to present the intuition behind torsion in an accessible way. Here is a similar, previous thread on MathOverflow.

In your question, you've described torsion in terms of its effect on parallel-transporting a vector along two different paths. The distinction between curvature and torsion may be more transparent if you think about scalars rather than vectors. Curvature effects vanish when you operate on a scalar, e.g., the mass of a hydrogen atom doesn't end up being different depending on which path you transport it along. But the covariant derivative does pick up an effect from the torsion when you compute the commutator of two derivatives acting on a scalar; the reason is that you're differentiating along two coordinate axes, and if there is torsion these axes themselves rotate as you move along.

Another nice way to distinguish between curvature and torsion is that nonvanishing torsion requires that the space have a detectable handedness to it, whereas curvature has no such handedness. E.g., in two dimensions, a bug living on a surface can never use measurements of curvature in the way we would use a magnetic compass to find north. In a real-world physical context, the experiment described at the end of 1 is looking for violations of the symmetry between left- and right-handedness.

Answer 2

Élie Cartan proposes such interpretations in his fundamental paper Sur les variétés à connexion affine et la theorie de la relativité généralisée (Ann. Ec. Norm. 40 (1923), 325–412 and 41 (1924), 1–25). (These are reprinted in his collected words, Partie III.) It may be a bit hard to follow, so, especially if you are into physics, you might want to consult Misner, Thorne, and Wheeler's discussion of this in their famous book Gravitation before you dive into Cartan's paper.