Differential Forms & the Star Operator

[leon1127]

I am reading some books about differential forms. I don't quite understand what is the geometrical meaning of star (hodge) operator. Can anyone give me a hand please?

Leon

 

[Chris Hillman]

In d dimensions, the Hodge star sets up a duality between p-forms at (d-p)-forms, via
\alpha \wedge \beta = ({}^\star\alpha , \beta) \, \omega
where \omega is the volume form (a basis form in the one dimensional space of d-forms), and where \alpha is a p-form, and where \beta, \; {}^\star\alpha are (d-p)-forms.

So for example, when d=4, we can describe a current using a one-form J or a dual three-form *J, and given a two-form describing the EM field, F, we can take its dual to find another two-form, *F. Half of Maxwell's equations are then seen to be trivial: dF = 0 automatically since F=dA. The other half are then d {}^\star F = 4 \pi \, {}^\star J.

The book Gravitation by Misner, Thorne, and Wheeler contains some nice discussion. For example, from the simple two-form F = -q/r^2 \, dt \wedge dr (which describes a Coloumb-type purely radial electrostatic field), we obtain the simple two-form {}^\star F= q \sin(\theta) d\theta \wedge d\phi, which is suitable for integration over a sphere. The duality becomes even more vivid if you use the coframe
\sigma^{\hat{0}} = -dt, \; \sigma^{\hat{1}} = dr, \; \sigma^{\hat{2}} = r \, d\theta, \; \sigma^{\hat{3}} = r \sin(\theta) \, d\phi[/tex]<br /> Then we obtain<br /> F = q/r^2 \, \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} , \; {}^\star F = q/r^2 \, \sigma^{\hat{2}} \wedge \sigma^{\hat{3}}<br /> which we could have obtained even faster using <br /> {}^\star \left( \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} \right) =\sigma^{\hat{2}} \wedge \sigma^{\hat{3}}<br /> However, the &quot;coordinate coframe&quot; is usually more convenient when we are actually integrating. Note that I am talking about Minkowski spacetime here, not a curved spacetime. The volume form is simply the wedge product of the four covectors making up the coframe. However, this can be readily generalized to work on curved spacetimes.<br /> <br /> (This example is potentially misleading, since in general, F will be a &quot;rank two&quot; two-form, meaning that it contains both dt \wedge dr and d\theta \wedge d\phi terms. A rank one exterior form is often called a &quot;simple&quot; form. Don&#039;t confuse this &quot;rank&quot; with &quot;second rank tensor&quot;! In our example, our &quot;rank one&quot; or &quot;simple&quot; two-form F corresponds to an antisymmetric &quot;second rank&quot; tensor.)<br /> <br /> Another good book is Burke, Applied Differential Geometry.