# Bianchi Identities

## Intuitive

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.

## Concrete

• See chapter 15 in "Gravitation" by Misner Thorne and Wheeler and also
• page 253 in Gauge fields, knots, and gravity by J. Baez

## Abstract

The motto in this section is: the higher the level of abstraction, the better.

## Why is it interesting?

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 Noether's second theorem.

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