Index Concordia

Maxwell wearing differential forms

Posted in Classical Electrodynamics, Geometry, Relativity by Index Guy on July 16, 2007

[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 \eta_{\mu\nu}.


Speculations on an eleven dimensional manifold

Posted in Classical Electrodynamics, Gauge Theory, Geometry by Index Guy on July 15, 2007

[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 \mathcal{M} with 3-form gauge potential A^{(3)} and an associated 4-form field strength F^{(4)} = \text{d}A^{(3)}. Let us first consider traditional, 4-dimensional Minkowski spacetime. In this spacetime, the gauge potential A is a 1-form and it couples to point particles with an action term of the form

S = \displaystyle\int_{\gamma}A,

with \gamma 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 \mathcal{M} can be integrated along a sort-off 3-dimensional spacetime trajectory called the worldvolume. The action term will look like

S = \displaystyle \int_{\Gamma}A^{(3)}.

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

Posted in Geometry, Relativity by Index Guy on July 13, 2007

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

Posted in Geometry, Relativity by Index Guy on July 11, 2007

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,

\Gamma_{\mu \nu}^{\mu} = \displaystyle \tilde{g}^{-1} \partial_{\nu}\tilde{g},

where \tilde{g} = \sqrt{\left|g\right|} is the square root of the absolute value of the metric tensor g_{\mu \nu}. 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.