Lorentz Invariants and Field Strength Tensor Fuv

[jc09]

Homework Statement

The Field strength tensor F^uv encodes the electric and magnetic fields via:
E_i=-cF⁰ⁱ, B_i=-1/2 e_ijkF^jk, i=1,2,3 Show that E^2-c^2B^2 and cE.B are invariant under lorentze transformations, by writing them explicitly as invariant contractions using the tensors F^uv and e_uvab

Homework Equations

The Attempt at a Solution

What does write the explicitly as invariant contractions mean?

 

[fzero]

The Attempt at a Solution

What does write the explicitly as invariant contractions mean?

A Lorentz invariant quantity can't have any free index. A quantity like A_{\mu\nu\rho}v^\mu}B^{\nu\rho} would be invariant, but A_{\mu\nu\rho}B^{\nu\rho} would not.

 

[jc09]

ok so for the first one E^2-c^2B^2 how would I start the problem. I know the equation for E and B are given in terms of the metric tensor so I would write them out but what would be the next step

 

[fzero]

Well what sort of contractions can you form from F^{\mu\nu} and \epsilon_{\mu\nu ab}? Note that the quantities you're supposed to obtain are quadratic in the fields.

 

[jc09]

I know I can contract F^uvU_b=V^a but not sure how to proceed at all

 

[fzero]

I know I can contract F^uvU_b=V^a but not sure how to proceed at all

The expression you've written has free indices \mu, \nu and b on the LHS and an index a on the RHS, so it is incorrect. Also U and V are undefined and have nothing to do with the question. You do not need any other tensors other than F^{\mu\nu} and \epsilon_{\mu\nu ab}. Try to form contractions that have no free indices. For example, \epsilon_{\mu\nu ab} \epsilon^{\mu\nu ab} is a Lorentz invariant, but it's not one of the ones you're looking for.

 

[jc09]

so does e^uvxy=e_uve_xy make more sense? are these correct? I know these are still not what I used in the question but is the idea right here. Then F^uvF_uv

 

[jc09]

so does e^uvxy=e_uve_xy make more sense? are these correct? I know these are still not what I used in the question but is the idea right here. Then F^uvF_uv

sorry that isn't right I think maybe this is e^uve_uve^xye_xy

 

[fzero]

\epsilon^{\mu\nu a b} is the Levi-Civita symbol, you might want to brush up on what it means here: http://en.wikipedia.org/wiki/Levi-Civita_symbol You can't write \epsilon^{\mu\nu a b} =\epsilon_{\mu\nu} \epsilon_{ a b} for any definition of \epsilon_{\mu\nu}, the RHS does not have all of the symmetries that the LHS does.

I wrote the expression with 2 \epsilon's to illustrate a Lorentz-invariant combination, don't get hung up on trying to rewrite that. You mentioned F^{\mu\nu}F_{\mu\nu}. Why don't you try writing that in terms of E and B?

 

[jc09]

hi so if I write F^uvF_uv in terms of E and B I get F^uvF_uv=F_olF^ol+F^ijF_ij=cE.B is this correct?

Then to get the other one I can write 1/2e_uvxyF^uvF^xy in terms of E and B

 

[fzero]

hi so if I write F^uvF_uv in terms of E and B I get F^uvF_uv=F_olF^ol+F^ijF_ij=cE.B is this correct?

No, that's not right. Remember that E_i=-cF⁰ⁱ, B_i=-1/2 e_ijkF^jk, i=1,2,3.

Then to get the other one I can write 1/2e_uvxyF^uvF^xy in terms of E and B

That's one of the invariants that you need. You should see what expression that gives in terms of E and B.

 

[jc09]

No, that's not right. Remember that E_i=-cF⁰ⁱ, B_i=-1/2 e_ijkF^jk, i=1,2,3.



That's one of the invariants that you need. You should see what expression that gives in terms of E and B.

So for the first bit if I write F^uvF_uv=F⁰ⁱF_0i+F^jkF_jk is that better. sorry I'm having quite a lot of trouble with these

 

[fzero]

So for the first bit if I write F^uvF_uv=F⁰ⁱF_0i+F^jkF_jk is that better. sorry I'm having quite a lot of trouble with these

That part was fine, the part that was wrong was what you wrote for the E and B expression. You should use the relationships between E, B and the components of F^uv to explicitly compute that, don't just try to guess.

 

[jc09]

so for F⁰ⁱF_0i does that equal -2/c E.B then to finish I add on the second part of it?

 

[jc09]

ah wait sorry I see the error of my ways I had written down the second matrix wrong so I had B's where there were meant to be E's etc. Think I have the first one now.