What is the Hodge dual and how does it work?

[AlbertEi]

Hi,

I'm trying to get my head around the Hodge dual and how it exactly works. In the book "Gauge Fields, Knots and Gravity" by John Baez and Javier P. Muniain they define:

\begin{equation}
\omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol}
\end{equation}

for two p-forms. This implies that:

\begin{equation}
\omega \wedge * \mu = * \mu \wedge \omega
\end{equation}

Therefore, if we consider a vector space with basis dx, dy, dz, and

\begin{equation}
\omega = \omega_x \mathrm{d}x
\end{equation}

\begin{equation}
*\mu = \mu_y \mathrm{d} y
\end{equation}

Then the definition by Baez and Muniain yields:

\begin{equation}
\omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = \mu_y \omega_x \mathrm{d} y \wedge \mathrm{d}x
\end{equation}

However, if I would try to calculate the above equation using 'anti-commuting' property of the wedge product, then I get:

\begin{equation}
\omega \wedge * \mu = \omega_x \mu_y \mathrm{d}x \wedge \mathrm{d} y = - \omega_x \mu_y \mathrm{d}y \wedge \mathrm{d} x \neq * \mu \wedge \omega
\end{equation}

So clearly, I am going wrong somewhere, but I can't see where I'm going wrong.

 

[martinbn]

$\omega$ and $\mu$ have to be p-forms. In your example one is 1-form, the other is a 2-form.

 

[AlbertEi]

Thanks for your reply, I will think about it over the next couple of days. I've the feeling that in my example they are both 1-forms so they should satisfy that equation.

 

[atyy]

\begin{equation}
\omega \wedge * \mu = \langle \omega , \mu \rangle \mathrm{vol}
\end{equation}

for two p-forms. This implies that:

\begin{equation}
\omega \wedge * \mu = * \mu \wedge \omega
\end{equation}

If you interchange $\omega$ and $\mu$, shouldn't the second line be $\omega \wedge * \mu = \mu \wedge * \omega$ ?

Winitzki gives something like that in the first equation at the top of the right column on p4 of https://sites.google.com/site/winitzki/linalg.

 

[micromass]

Thanks for your reply, I will think about it over the next couple of days. I've the feeling that in my example they are both 1-forms so they should satisfy that equation.

You have made $\ast \eta$ a $1$-form. Thus $\eta$ is then a $n-1$-form.
However, you also made $\omega$ a $1$-form. This is incompatible.
Both $\omega$ and $\eta$ should be $1$-forms.

 

[AlbertEi]

Ok, thank micromass, that makes sense.

 

[AlbertEi]

atty, yeah but in other sources I have also seen

\begin{equation}
(*\omega) \wedge \mu = \langle \omega , \mu \rangle \mathrm{vol}
\end{equation}

so I think they are equivalent (although it seems that everybody is using different conventions so I might be wrong). However, what micromass said made sense, so I think that is where I went wrong.

 

[Mandelbroth]

I'm trying to get my head around the Hodge dual and how it exactly works.

Are you looking for a definite, mathematically rigorous way of thinking of it, or are you looking for intuition? Intuition is often good for wrapping your head around something, but I get the feeling that you aren't looking for that.

 

[AlbertEi]

Yeah, it would be nice to have a more rigorous way of thinking about it. If you have any good sources where I can learn this stuff then that would be great (I thought Sean Carroll's book was quite disappointing regarding differential forms, so now I have the aforementioned book from which I try to learn it). Thanks

Edit: FYI I study physics and am mainly interested in their application in gauge theory and general relativity.

 

[WannabeNewton]

I would recommend Nakahara for a physics oriented viewpoint (it is primarily gauge theory) as well as Frankel. I personally don't know of any GR text that treats differential forms in a completely modern (i.e. index free) way. Wald's GR text has a nice appendix on differential forms but it is index based hence rather classical in nature.

If you want something on the pure math side, Spivak and Kobayashi are the reigning kings.

Links: https://www.amazon.com/dp/0471157333/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098705/?tag=pfamazon01-20
https://www.amazon.com/dp/0750306068/?tag=pfamazon01-20
https://www.amazon.com/dp/1107602602/?tag=pfamazon01-20

 

[AlbertEi]

Thanks WannabeNewton for your recommendations!

 

[arkajad]

Another very good book is by Marian Fecko "Differential Geometry and Lie Groups for Physicists". In particular the Hodge dual is discussed there.