**Subject:** Torsion and Connection

**Date:** Wed, 25 Mar 2009 07:08:20 EDT

It may be objected that the connection is not a tensor whereas the angular energy momentum density is a tensor, so one cannot be proportional to the other. The answer is that the torsion is a tensor because it is the antisymmetric difference of two connections and so torsion can be proportional to angular energy / momentum density. Using the antisymmetry of the connection torsion may be written formally as twice the connection. The connection does not transform as a tensor, but the antisymmetric difference of two connections transforms as a tensor, the torsion tensor. The reason is that the inhomogeneous terms in the coordinate transformations of the two connections cancel out. This is shown in detail by Carroll, chapters one to three.

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