====== Bianchi Identities ====== Intuitively, Bianchi identities state that "//the boundary of a boundary is zero//". For example, let's consider a disk: O The disk has a boundary, which is a circle. A circle has no boundary. Next, let's consider a line segment: | 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. For an extremely illuminating discussion see * [[https://link.springer.com/article/10.1007%2FBF01882731|The Boundary of a Boundary Principle - a unified approach]] by Arkady Kheyfets. In addition, good discussion can be found in * See chapter 15 in "Gravitation" by Misner Thorne and Wheeler and also * page 253 in Gauge fields, knots, and gravity by J. Baez ---- 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 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]])
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! page 96 in Gauge fields, knots, and gravity by John Baez
The boundary of a boundary is zero John Wheeler
The Bianchi identities have a close connection the [[theorems:noethers_theorems|Noether's second theorem]].