Applications of Wedge-Product and Differential Forms

[davidge]

Hi everyone. In reading some popular textbooks I noticed that in (maybe) most of GR and SR we don't encounter situations where we can use wedge-product and differential forms. However, these things are presented to us in most of the textbooks. But... if most of the books present them, it means that they are important and useful, I think... So maybe what I've read on GR and SR is "nothing" compared to the existing material. Thus I would like to know some applications of the wedge-product and the differential forms. Can someone write down a list containing some topics where they are useful (or even essential)?

 

[fresh_42]

Hi everyone. In reading some popular textbooks I noticed that in (maybe) most of GR and SR we don't encounter situations where we can use wedge-product and differential forms. However, these things are presented to us in most of the textbooks. But... if most of the books present them, it means that they are important and useful, I think... So maybe what I've read on GR and SR is "nothing" compared to the existing material. Thus I would like to know some applications of the wedge-product and the differential forms. Can someone write down a list containing some topics where they are useful (or even essential)?

Stokes theorem.
Volume calculations.
Poincaré's Lemma.

Every time you write a $d$ in front of an object, you probably deal with a differential form. Therefore pretty much the entire physics can be expressed by differential forms. The wedge product together with the tensor product are the universal concepts to deal with vector spaces that allow a multiplication of some kind. In the end they only represent a coordinate free way of dealing with differentiation. Historically and naturally in physics, equations are expressed by coordinates. The general principles behind are often differential forms.

 

[robphy]

Electromagnetism
Curvature , Bianchi identities
Mechanics (angular momentum, torque)

 

[Orodruin]

Stokes theorem.

I would like to underline this. As a physics student, the divergence theorem, curl theorem, Green's formula in the plane, and the line integral of a gradienr are things you will be unable to get anywhere without. They are all just special cases of Stokes' theorem and whenever you are applying any of them you are likely using differential forms without realising it.

 

[vanhees71]

A textbook, where Cartan calculus is used in GR is Misner, Thorne, Wheeler.

 

[davidge]

Thanks to everyone who replied here. I committed a mistake. I would actually ask for p-forms instead of differential forms!
I would like to see a explicit example on General Relativity where one need to use p-forms to solve a particular problem and another example where one need to use wedge-product.

 

[haushofer]

What's the difference? Doesn't the p simply indicate the rank of the differential form?

As an example, apart from the ones already mentioned: a Lagrangian can be seen as an integrand and as such as a 4-form in four dimensions. If you use Wald's "black hole entropy as a Noether charge of diffeomorphisms", you can use nice geometric identities linking Lie- and exterior derivatives of differential forms.

 

[davidge]

What's the difference? Doesn't the p simply indicate the rank of the differential form?

I thought a p-form was a totally anti-symmetric tensor

 

[robphy]

What's the difference? Doesn't the p simply indicate the rank of the differential form?

I thought a p-form was a totally anti-symmetric tensor

Maybe the distinction is analogous to a vector vs. a vector-field?
Maybe you are interested in the exterior-algebraic properties alone... and not about exterior-calculus.

You might want to poke around here:

http://physics.oregonstate.edu/coursewikis/GDF/book/gdf/start
http://physics.oregonstate.edu/coursewikis/GGR/book/ggr/start
which are preprint versions of
Tevian Dray's book " Differential Forms and the Geometry of General Relativity": http://physics.oregonstate.edu/coursewikis/DFGGR/bookinfo/

 

[davidge]

Maybe the distinction is analogous to a vector vs. a vector-field?

Yes
Thank you. I will read the content on these pages.

 

[PeterDonis]

I thought a p-form was a totally anti-symmetric tensor

So is a differential form. The term p-form, as Haushofer said, simply draws attention to the rank, p, of the tensor.