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advanced_tools:bianchi_identities [2017/05/07 13:14]
jakobadmin
advanced_tools:bianchi_identities [2019/01/16 14:35] (current)
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 ====== Bianchi Identities ====== ====== Bianchi Identities ======
  
-<​blockquote>​ 
-The boundary of a boundary is zero 
-<​cite>​John Wheeler</​cite>​ 
-</​blockquote>​ 
  
-<tabbox Why is it interesting?> ​ 
  
-The Bianchi identities have a close connection the [[http://​wiki.physicsinsider.com/​doku.php?​id=noethers_theorems#​tab__researcher|Noether'​s second theorem]].+<tabbox Intuitive> ​
  
-<tabbox Layman?> ​+Intuitively,​ Bianchi identities state that "//the boundary of a boundary is zero//"​. ​
  
-<note tip> + 
-Explanations in this section should contain ​no formulasbut instead colloquial things ​like you would hear them during ​coffee break or at cocktail party. +For example, let's consider a disk: O 
-</​note>​+ 
 +The disk has a boundary, which is a circle. A circle has no boundary.  
 + 
 +Nextlet's consider a line segment: | 
 + 
 +It is one-dimensional ​like the circle but has boundary: the two endpoints. But then again, this boundary (=the two endpoints) don't have boundary.
   ​   ​
-<​tabbox ​Student+<​tabbox ​Concrete 
 +For an extremely illuminating discussion see
  
-<note tip> +  * [[https://​link.springer.com/article/​10.1007%2FBF01882731|The Boundary of a Boundary Principle - a unified approach]] by Arkady Kheyfets.
-In this section things should be explained by analogy and with pictures and, if necessary, some formulas. +
-</note> +
-  +
-<tabbox Researcher> ​+
  
-<note tip> +In additiongood discussion can be found in
-The motto in this section is: //the higher the level of abstractionthe better//. +
-</​note>​+
  
---> Common Question 1#+  * See chapter 15 in "​Gravitation"​ by Misner Thorne and Wheeler and also 
 +  * page 253 in Gauge fields, knots, and gravity by J. Baez
  
-  +----
-<--+
  
---> Common Question 2#+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
  
    
-<-- +<tabbox Abstract> ​ 
-   +In general relativity, the Bianchi identity  
-<​tabbox ​Examples+$$ \nabla R = \nabla \nabla \theta =0  $$ 
 +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?​ 
 +<​blockquote>​ 
 +When we get to gauge theories we will see that Maxwell'​s equations are a special case of the Yang-Mills equations, which describe not only electromagnetism but also the strong and weak nuclear forces. A generalization of the identity $d^2=0$, the Bianchi identity, implies conservation of "​charge"​ in all these theories - although these theories have different kinds of "​charge"​. Similarly, we will see when we get to general relativity that due to the Bianchi identity, Einstein'​s equations for gravity automatically imply local conservation of energy and momentum!
  
---Example1#+<​cite>​ page 96 in Gauge fields, knots, and gravity ​ by John Baez</​cite>​ 
 +</​blockquote>
  
-  +<​blockquote>​ 
-<--+The boundary of a boundary is zero 
 +<​cite>​John Wheeler</​cite>​ 
 +</​blockquote>​
  
---> Example2:+The Bianchi identities have a close connection the [[theorems:noethers_theorems|Noether'​s second theorem]].
- +
-  +
-<-- +
-   +
-<tabbox History> ​+
  
 </​tabbox>​ </​tabbox>​
  
  
advanced_tools/bianchi_identities.1494155691.txt.gz · Last modified: 2017/12/04 08:01 (external edit)