# How to define the lower indexed tensor

1. Sep 28, 2007

### ehrenfest

1. The problem statement, all variables and given/known data
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.
2. Relevant equations

3. The attempt at a solution

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• ###### zwiebach142.jpg
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Last edited: Sep 28, 2007
2. Sep 29, 2007

### 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.

3. Sep 29, 2007

### 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?

Last edited: Sep 29, 2007
4. Sep 29, 2007

### Jimmy Snyder

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

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

Last edited: Sep 29, 2007
5. Sep 29, 2007

### ehrenfest

I see.

Fixed it.

6. Sep 29, 2007

### 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.