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
[itex] \sigma^{\hat{0}} = -dt, \; \sigma^{\hat{1}} = dr, \; \sigma^{\hat{2}} = r \, d\theta, \; \sigma^{\hat{3}} = r \sin(\theta) \, d\phi[/tex]
Then we obtain
$F = q/r^2 \, \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} , \; {}^\star F = q/r^2 \, \sigma^{\hat{2}} \wedge \sigma^{\hat{3}}$
which we could have obtained even faster using
${}^\star \left( \sigma^{\hat{0}} \wedge \sigma^{\hat{1}} \right) =\sigma^{\hat{2}} \wedge \sigma^{\hat{3}}$
However, the "coordinate coframe" 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.

(This example is potentially misleading, since in general, F will be a "rank two" 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 "simple" form. Don't confuse this "rank" with "second rank tensor"! In our example, our "rank one" or "simple" two-form F corresponds to an antisymmetric "second rank" tensor.)

Another good book is Burke, Applied Differential Geometry.

 

[tacman]

hard Euler first introduced the concept of differential forms in the 18th century, but it wasn't until the 20th century that the star operator was introduced by Hermann Weyl and later refined by mathematicians such as Élie Cartan and Hermann Weyl. The star operator, also known as the Hodge star operator, is a fundamental tool in the study of differential forms and plays a crucial role in differential geometry and topology.

The geometrical meaning of the star operator is best understood in the context of differential forms. Differential forms are mathematical objects that represent geometric quantities, such as length, area, and volume, in a coordinate-independent way. They are used to describe various mathematical concepts in a way that is invariant under coordinate transformations. This makes them incredibly useful in differential geometry, where coordinate systems can vary and traditional vector calculus methods can become complicated.

The star operator is a map that takes a differential form of degree k and returns a differential form of degree n-k, where n is the dimension of the underlying manifold. In simpler terms, it converts a k-form into an (n-k)-form. This may seem like a simple mathematical operation, but it has profound geometric implications.

One way to think about the star operator is as a duality map. It allows us to relate forms of different degrees to each other and provides a way to "rotate" forms in a higher-dimensional space. This is particularly useful in understanding the concept of orientation, where the star operator can be used to define a positive or negative orientation on a manifold.

Another important application of the star operator is in the definition of the Hodge dual. The Hodge dual of a differential form is a new form that represents the same geometric quantity but with the opposite orientation. This is crucial in many areas of mathematics, such as in Maxwell's equations in electromagnetism, where the Hodge dual is used to relate electric and magnetic fields.

In summary, the star operator is a powerful tool in the study of differential forms and has important geometric implications. It allows us to relate forms of different degrees, define orientation, and play a crucial role in various mathematical theories. I hope this explanation has helped you better understand the geometrical meaning of the star operator.