How to define the lower indexed tensor

[ehrenfest]

Homework Statement

Can someone help me with QC 8.4?
I am unsure
how to define the lower indexed tensor here. I have worked with upper and lower indices before but the relationship between the two has always just been given to me.

Let me know if you want to help but think that the attachment is too small to read.

Homework Equations

The Attempt at a Solution

 

[Jimmy Snyder]

I can't see the attempt yet. However, this problem is a forced march. Just multiply both sides of equation (8.51) by two copies of $\eta$ to lower the indices.

 

[ehrenfest]

I see:

$$\eta_{\beta \nu} \eta_{\alpha \mu} \epsilon ^{\mu \nu} = -\eta_{\alpha \mu} \eta_{\beta \nu} \epsilon ^{\nu \mu}$$

And then 2 Minkowski metrics is just the identity matrix.

Is my Einstein notation correct?

 

[Jimmy Snyder]

And then 2 Minkowski metrics is just the identity matrix.

Instead of multiplying the metrics together, use them to lower the indices on $\epsilon$.
For instance
$$\eta_{\beta\nu}\eta_{\alpha\mu}\epsilon^{\mu\nu} = \eta_{\alpha\mu}{\epsilon^{\mu}}_{\beta}$$

Is my Einstein notation correct?

Yes, but the equation you wrote is wrong. It is missing a minus sign on the r.h.s.

 

[ehrenfest]

Instead of multiplying the metrics together, use them to lower the indices on $\epsilon$.
For instance
$$\eta_{\beta\nu}\eta_{\alpha\mu}\epsilon^{\mu\nu} = \eta_{\alpha\mu}{\epsilon^{\mu}}_{\alpha}$$


Yes, but the equation you wrote is wrong. It is missing a minus sign on the r.h.s.

I see.

Yes, but the equation you wrote is wrong. It is missing a minus sign on the r.h.s.

Fixed it.

 

[Jimmy Snyder]

I'm sorry Ehrenfest, my post #4 which you quoted is incorrect. I have edited it. Please use the edited post, not the one that you quoted.