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advanced_tools:parallel_transport [2018/04/14 11:18] aresmarrero ↷ Links adapted because of a move operation |
advanced_tools:parallel_transport [2018/04/14 11:30] aresmarrero [Intuitive] |
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====== Parallel Transport ====== | ====== Parallel Transport ====== | ||
+ | //see also [[advanced_tools:connections]] // | ||
<tabbox Intuitive> | <tabbox Intuitive> | ||
- | <note tip> | + | <blockquote>Parallel transport on a sphere can best be understood by imagining the sphere to be rolling on a flat |
- | Explanations in this section should contain no formulas, but instead colloquial things like you would hear them during a coffee break or at a cocktail party. | + | surface. Suppose there are straight lines and curved lines drawn on the flat surface in wet ink. And |
- | </note> | + | suppose there are arrows spaced frequently along the line, all pointing, say, to the lower left. The term |
+ | "parallel transport" is based on this picture. Since the arrows in the plane are all parallel, the arrows | ||
+ | printed onto the sphere exemplify parallel transport along the printed curve on the sphere. | ||
+ | |||
+ | Given a | ||
+ | curve on the sphere and an arrow representing a direction at the start of the ruve, one can place the | ||
+ | sphere on the plane so that the starting point is the point of tangency. | ||
+ | Imagine there is an arrow at each point in the plane drawn in ink and parallel to the original arrow. | ||
+ | If the sphere is rolled along the plane so that it prints the curve on the sphere onto the plane, the arrows | ||
+ | from the plane will also be printed onto the sphere. The latter arrows give a path in the bundle of | ||
+ | directions that starts with the initial arrow and lies above the given curve. | ||
+ | |||
+ | When a straight line is printed onto the rolling sphere (or onto any other curved surface), the curve | ||
+ | that it forms on the surface is called a [[equations:geodesic_equation|geodesic]]. | ||
+ | |||
+ | {{ :advanced_notions:paralleltransport.png?nolink |}} | ||
+ | |||
+ | Parallel transport can be described for a curved path made up of geodesic segments without | ||
+ | reference to rolling. It can be carried out by maintaining a constant angle between the transported | ||
+ | arrows and the tangents along successive geodesics. From a perspective above the surface of the sphere, | ||
+ | however, parallel transport along a geodesic may seem anything but parallel. The arrows may appear to | ||
+ | rotate. When an arbitrary curved line is printed onto the sphere, the rotation of the arrows as viewed | ||
+ | from above the surface may appear to be even more chaotic. | ||
+ | |||
+ | Parallel transport provides a way of making quantitative and explicit the intuitive difference between | ||
+ | a curved surface and a flat one. When an arrow is carried by parallel transport around a closed path in | ||
+ | the plane, the directions of the arrow at the start and at the finish coincide. | ||
+ | Parallel transport around a closed path on a curved surface, however, may not lead to such | ||
+ | coincidence. If there is a change in the direction of the arrow when it completes a single circuit of a | ||
+ | closed path, the angle between the final direction and the initial one is called the angular excess of the | ||
+ | path. It follows from the way parallel transport is defined that the angular excess does not depend on the | ||
+ | initial direction of the arrow. | ||
+ | |||
+ | |||
+ | |||
+ | <cite>Fiber Bundles and Quantum Theory by Bernstein and Phillips</cite></blockquote> | ||
| | ||
<tabbox Concrete> | <tabbox Concrete> |