advanced_tools:bianchi_identities
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advanced_tools:bianchi_identities [2018/05/03 11:53] jakobadmin [Intuitive] |
advanced_tools:bianchi_identities [2019/01/16 14:35] (current) jakobadmin [Abstract] |
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| <tabbox Intuitive> | <tabbox Intuitive> | ||
| - | Intuitively, Bianchi identities state that "the boundary of a boundary is zero". | + | Intuitively, Bianchi identities state that "//the boundary of a boundary is zero//". |
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| + | For example, let's consider a disk: O | ||
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| + | The disk has a boundary, which is a circle. A circle has no boundary. | ||
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| + | Next, let's consider a line segment: | | ||
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| + | It is one-dimensional like the circle but has a boundary: the two endpoints. But then again, this boundary (=the two endpoints) don't have a boundary. | ||
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| <tabbox Concrete> | <tabbox Concrete> | ||
| + | For an extremely illuminating discussion see | ||
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| + | * [[https://link.springer.com/article/10.1007%2FBF01882731|The Boundary of a Boundary Principle - a unified approach]] by Arkady Kheyfets. | ||
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| + | In addition, good discussion can be found in | ||
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| * See chapter 15 in "Gravitation" by Misner Thorne and Wheeler and also | * See chapter 15 in "Gravitation" by Misner Thorne and Wheeler and also | ||
| * page 253 in Gauge fields, knots, and gravity by J. Baez | * page 253 in Gauge fields, knots, and gravity by J. Baez | ||
| - | + | ---- | |
| - | <tabbox Abstract> | + | |
| - | <note tip> | + | Bianchi identities express the fact that the boundary of a boundary is always zero. Mathematically this follows by applying Stoke's theorem twice. This is discussed explicitly in the book No-Nonsense Electrodynamics by Schwichtenberg |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | |
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| + | <tabbox Abstract> | ||
| + | In general relativity, the Bianchi identity | ||
| + | $$ \nabla R = \nabla \nabla \theta =0 $$ | ||
| + | roughly says "that the sum over a closed two-dimensional surface of rotations induced by Riemannian curvature is equal to zero. [...] Geometrically this means that the density of the moment of rotation induced by Riemannian curvature is equal to zero automatically." ([[https://link.springer.com/article/10.1007%2FBF01882731|Source]]) | ||
| <tabbox Why is it interesting?> | <tabbox Why is it interesting?> | ||
| <blockquote> | <blockquote> | ||
advanced_tools/bianchi_identities.1525341215.txt.gz · Last modified: 2018/05/03 09:53 (external edit)
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