Help with problem in MIT open course general relativity

[DavidL070949]

Hi! Apologies in advance for a somewhat long post.

First, by some background: Long ago in what seems like a galaxy far away, I got a Ph.D. in physics (statistical physics) before switching gears and going into law. The nature of my grad school was such that there wasn't any coursework - you got an office, and advisor and were expected to produce a thesis in a reasonable number of years. Great for learning to be independent but it did result in a number of holes in my education.

Fast forward a number of decades, post-retirement, my interest in physics got rekindled and I began to work my way through the MIT open course on general relativity on my own. Great course! But it can at times be a bit tough to do this solo and I've gotten stuck on Problem set 3, #3(b). There's a set of worked out solutions that I found online. Unfortunately, it contains errors, including (I think) one in the work on this problem, which deals with electromagnetic fields as seen by an observer with 4-velocity U.

I've attached the page with the problem in question. It seems like one way to address this is to plug in the formulas for E and B into the formula given for the generating field tensor and show that the rhs reduces to the lhs, which is what I did. From the problem, it seems like one is supposed to get to the place where the Levi-Civita symbol identity given with the problem should come into play. But the expression I got to only has one index in that part to sum over and the identity needs two. The other way to do this would be to contract the non-field tensor portions of the E and B 4-field formulas with both sides and show that the rhs reduces to the target field. That hasn't gotten me very far either. So, I'm stuck and would appreciate any help/insight anyone can add.

 

[robphy]

But the expression I got to only has one index in that part to sum over and the identity needs two. The other way to do this would be to contract the non-field tensor portions of the E and B 4-field formulas with both sides and show that the rhs reduces to the target field.

Can you show what you did? (Use MathJax.)
Here's the expression from 3b.
$F^{\alpha \beta}=U^\alpha E_{\vec{U}}^\beta-E_{\vec{U}}^\alpha U^\beta+\epsilon^{\alpha \beta}{ }_{\gamma \delta} U^\gamma B_{\vec{U}}^\delta$

 

[PeterDonis]

@DavidL070949 please post your math directly in the thread using LaTeX. There is a LaTeX Guide link at the bottom left of each post window.

 

[DavidL070949]

Thanks. Next task is definitely to figure out how to use LaTeX.

 

[DavidL070949]

And, got it no more attachments.

To the very helpful person who asked me to post my work, I'm working on learning enough about LaTex to do that properly. That may take a bit, but I wanted to thank you in advance for the prompt offer to help.

 

[PeterDonis]

no more attachments

If there's a link to what you're referencing, that's much better, yes.

 

[robphy]

And, got it no more attachments.

To the very helpful person who asked me to post my work, I'm working on learning enough about LaTex to do that properly. That may take a bit, but I wanted to thank you in advance for the prompt offer to help.

If you right-click on the equation in my post, you can see the $\LaTeX$-code used to create it.

 

[Sagittarius A-Star]

As delimiters, you can use double $ or double # on each side. See in below LaTex Guide link under "Delimiting your LaTeX code".