Interpretation of Curvature and Torsion - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T03:15:25Z http://mathoverflow.net/feeds/question/103316 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/103316/interpretation-of-curvature-and-torsion Interpretation of Curvature and Torsion Dox 2012-07-27T14:50:55Z 2012-07-27T20:38:02Z <p>Dear all,</p> <p>When dealing with General Relativity one uses the Levi-Civita connection with is torsion-free. Thus the commutator of the covariant derivatives yields</p> <p>$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda.$</p> <p>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.</p> <p>However, in general there are three different effects in the transportation:</p> <ul> <li>Change of direction of the vector.</li> <li>Non-closure of the path (say, if one moves 1meter along each direction).</li> <li>Rotation about its own axis.</li> </ul> <p>whilst the general commutator is</p> <p><code>$[\nabla_\mu,\nabla_\nu]V^\rho = R_{\mu\nu}{}^\rho{}_\lambda V^\lambda- T^\lambda{}_{\mu\nu}\nabla_\lambda V^\rho.$</code></p> <p>Is it possible to give a meaning to the curvature and torsion in term of these intuitive geometry or is not possible in general?</p> <p>Thank you!</p> http://mathoverflow.net/questions/103316/interpretation-of-curvature-and-torsion/103324#103324 Answer by Robert Bryant for Interpretation of Curvature and Torsion Robert Bryant 2012-07-27T15:23:40Z 2012-07-27T15:23:40Z <p>Élie Cartan proposes such interpretations in his fundamental paper <em>Sur les variétés à connexion affine et la theorie de la relativité généralisée</em> (Ann. Ec. Norm. <strong>40</strong> (1923), 325–412 and <strong>41</strong> (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 <em>Gravitation</em> before you dive into Cartan's paper.</p> http://mathoverflow.net/questions/103316/interpretation-of-curvature-and-torsion/103330#103330 Answer by Ben Crowell for Interpretation of Curvature and Torsion Ben Crowell 2012-07-27T17:03:06Z 2012-07-27T20:38:02Z <p><a href="http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.8" rel="nofollow">Here</a> is my attempt to present the intuition behind torsion in an accessible way. <a href="http://mathoverflow.net/questions/20493/what-is-torsion-in-differential-geometry-intuitively" rel="nofollow">Here</a> is a similar, previous thread on MathOverflow.</p> <p>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 <em>does</em> 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.</p> <p>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 <a href="http://www.lightandmatter.com/html_books/genrel/ch05/ch05.html#Section5.8" rel="nofollow">1</a> is looking for violations of the symmetry between left- and right-handedness.</p>