Difference between covariant and contravariant levi-civita tensor?

[randombill]

The title says it all, basically I'm trying to figure out what the difference is between the two tensors (levi-civita) that are 3rd rank. Do they expand out in matrix form differently?

 

[arkajad]

The difference is in their transformation rules. As long as you are within orthonormal systems - you will not see the difference.

But, BTW, they are pseudo-tensors, not tensors.

P.S. The above applies if you were thinking about $$\epsilon_{ijk}$$ and not about $$\Gamma^\mu_{\nu\sigma}$$ which is an "object", not a "tensor".

 

[randombill]

The difference is in their transformation rules. As long as you are within orthonormal systems - you will not see the difference.

Can you show me what those rules are and do you mean orthonormal in terms of unit vectors in all coordinate systems such as cartesian, spherical, rectangular, etc?

 

[arkajad]

General rule:

$$\epsilon^{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|\epsilon^{ijk}$$$$\epsilon_{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|^{-1}\epsilon_{ijk}$$

So, for each coordinate system that you need, calculate the Jacobi determinant of the transformation from (or to) the Cartesian one. It may give you a factor in front different from 1.

But: you should distinguish between Levi-Civita symbol and Levi-Civita tensor.

For Levi-Civita tensor I was trying to guess what you mean.

Levi-Civita symbol is always the same (it is a tensor density, not a tensor).

 

[randombill]

General rule:

$$\epsilon^{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|\epsilon^{ijk}$$


$$\epsilon_{i'j'k'}=|\det (\frac{\partial x'}{\partial x})|^{-1}\epsilon_{ijk}$$

So, for each coordinate system that you need, calculate the Jacobi determinant of the transformation from (or to) the Cartesian one. It may give you a factor in front different from 1.

And this needs to be calculated whenever a change of coordinate systems takes place?

I guess I should also ask another question since this problem was really the reason why I started the thread, although my assumption of the topic for solving the problem was wrong. Below are 3 pictures from a book on WK Tungs group theory (problems and solutions) and its the chapter on Poincaré groups, but I'm having trouble understanding the tensor math involved at two steps.

The pictures are as follows,

Pic 1: The problem itself,
Pic 2: The solution and my question pertains to the part where it says:
"and so after renaming the indices we get". The part I have trouble seeing is what identities they used with the kronecker product in the previous step.
Pic 3: describes the identities used for the problem.
You will also need to rotate the second picture, sorry about that.

 

[arkajad]

Well, there is no "real tensor math" involved. All you need is how to play with Kronecker and Levi-Civita symbols. If you tell me, as an example, at which particular point you have a problem, I will try to help you.

 

[randombill]

If you tell me, as an example, at which particular point you have a problem, I will try to help you.

I did tell you the part, its where the book says "and so, after renaming the indices we get:"

I don't see how they got that equation after the previous equation. How did the 1/2 get canceled along with the minus sign disappearing as well as the left hand side of the equation getting simpler and the right hand side becoming "contracted"? I suspect that they used the 2nd identity from the 3rd picture somehow but I'm not sure what they mean by renaming indices and what the rules are for it.

 

[arkajad]

There you have summations over "dummy indices m,n. Whenever you have a repeating index - that means "summation" over this index. You may give it any name you wish (though different than the names of other indices in the same formula).

What you must know is that
$$\delta^i_m J^{mn}=J^{in}$$ etc.

You should also know that $$J^{mn}=-J^{nm}$$

Using this you will see that you get two exactly the same terms on the left. Thus 1/2 disappears.

Can you do it now? If you have still problem - that means you need to go back to the subject "dummy index", summations with Kronecker's delta's in the book or in Wikipedia, and do not skip anything around. Do all the exercises related to these concepts.

 

[randombill]

Thanks, I believe that answers my question. I was thinking along those lines too but I wasn't sure if those were dummy indices the book mentioned or...

But anyways thanks for the help. I don't think Wikipedia is useful by the way; except for rare exceptions when someone has already learned the material and serves as a reference, otherwise its just common knowledge.

 

[arkajad]

In http://en.wikipedia.org/wiki/Levi-Civita_symbol" Wiki has a bunch of useful formulas.