Parallel transport, the solution

Posted by on Jan 27, 2013 in Science | 0 comments

So far, in previous posts, I’ve discussed how spaces with an irreducible curvature yield a failure in parallel transport around a closed path, how the usual derivative of a vector fails to yield another proper vector, and how the covariant derivative solves this problem. In this post, I want to show how the covariant derivative can solve the problem of parallel transport along a path. The idea is the essence of simplicity. Suppose you have a vector, . You want to transport this along some arbitrary curve in some arbitrary manifold of some arbitrary dimension. Let a differential element...

Read More

Parallel transport and curvature

Posted by on Jan 13, 2013 in Science | 0 comments

Parallel transport and curvature

In the most basic physics, we learn that the most basic equations are generally vector equations. This is true of Newton’s equations involving force, , and acceleration, , and equally true of the Maxwell equations that involve vectors for the electric and magnetic fields, , and . Moreover, we learn things like acceleration is proportional to force, force is proportional to charge, rotational acceleration is proportional to torque, and in consideration of the fact that all of these quantities are vectors, that the resultant vector is in the same direction as the vector it is...

Read More