[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 .
[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.
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!
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.