## Maxwell wearing differential forms

*[The title of this post is not intended to offend the memory of the great Scottish physicist J. C. Maxwell.]*

Let us try to write the four Maxwell equations using the language of differential forms, exterior differentiation and the Hodge star duality operation. We work over the flat four-dimensional Minkowski spacetime with flat metric .

## Speculations on an eleven dimensional manifold

[This post is inspired by exercise 11 on chapter 2 of *Spacetime and Geometry*. I am not completely sure that my analysis is correct, so readers are encouraged to point out my mistakes.]

We consider an 11-dimensional spacetime with 3-form gauge potential and an associated 4-form field strength . Let us first consider traditional, 4-dimensional Minkowski spacetime. In this spacetime, the gauge potential is a 1-form and it couples to point particles with an action term of the form

with being the 1-dimensional trajectory of the 0-dimensional particle through spacetime, the *worldline*. Analogously we can say that our 3-form gauge potential on can be integrated along a sort-off 3-dimensional spacetime trajectory called the *worldvolume*. The action term will look like

This would imply that the object that the gauge potential will couple to will be 2-dimensional, a membrane.

## Some (trivial) consequences of metric-compatibility

If you were a bit nervous that I used two identities in the previous post without proof, well here I am to make you happy!

## An identity concerning the Christoffel connection

One of my summer readings is Sean Carroll’s book on general relativity. It is good to teach the basics of GR; I hope it will serve its purpose when I start the fall with Warren. Anyway, I came upon the following identity regarding the Christoffel connection,

,

where is the square root of the absolute value of the metric tensor . It is one of those expressions that is preceded by “It is straightforward to show that…”. Since I was a bit doubtful, I decided to try my hand at it.

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