Antisymmetric 4-Tensor: Hodge/Duality Transformation Explained

[yukcream]

What is Hodge or duality trnsformation? I just know it is an operation to transform tensor of rank p into rank (n-p), where n is the dimension? I want to know more about it especially how it works in general relativity,can anyone give me some introductory reading for me? Thank you

yukyuk

 

[Berislav]

Here's a wikipedia article about it:
http://en.wikipedia.org/wiki/Hodge_star_operator

I heard that the first reference is often called the 'Bible of GR'.

 

[pervect]

I think the Hodge duality transformation is the most easily understood in terms of Clifford algebras.

There is some introductory reading about Clifford algebras

here

If you are already familiar with the standard vector dot products and wedge products, it should be fairly easy reading. If you are not already somewhat familiar with the wedge product, it may not be so easy.

Anyway, suppose you have three non-collinear vectors in a 4-d space (since we are talking about relativity). (Note that this collection of three vectors is really a three-form. I'm not sure if you are familiar with three-forms or not. The Clifford algebra article will describe three-forms in more detail if you are not already familiar with them).

There is one and only one vector that's orthogonal to all three vectors (the three-form) - you can think of it as the time vector that's associated with the volume element defined by the three non-collinear vectors.

We can make the length of the vector proportional to the volume of the pareallel piped spanned by the three vectors.

This vector is the "hodge dual". In the language of forms, it associates a 1-form with every three-form. (I called it a vector before, but it's not really a vector, its the dual of a vector, a 1-form).

You can do similar transformations with other n-forms.

 

[robphy]

Some suggestions [that worked for me].

First, recognize that the [antisymmetric] cross-product of two vectors, which is most naturally visualized as an oriented plane, can be thought of as a vector in 3-dim Euclidean space... with the help of the Hodge dual operation.

Next, study Hodge duality in electromagnetism.
For example,
http://farside.ph.utexas.edu/teaching/jk1/lectures/node22.html
http://farside.ph.utexas.edu/teaching/jk1/lectures/node23.html

I'd suggest these books:

Schouten - Tensor Analysis for Physicists


Bamberg & Sternberg - A Course in Mathematics for Students of Physics


Burke - Applied Differential Geometry


Schutz - Geometrical Methods of Mathematical Physics


In GR, the Hodge dual shows up when discussing curvature tensors.
Exercise: The Riemann curvature tensor has two pairs of antisymmetric indices. By dualizing each pair, one gets the "double-dual" of Riemann. Take its [nontrivial] trace. What do you get?
The answer is in MTW - Gravitation