Index manipulations

1. May 28, 2015

spaghetti3451

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

Prove the following: $(\partial_{\mu} \phi)^{2} = \dot{\phi}^{2} - (\nabla \phi)^{2}$.

2. Relevant equations

3. The attempt at a solution

$(\partial_{\mu} \phi)^{2}$
$= (\partial_{\mu} \phi)(\partial_{\mu} \phi)$
$= (\partial_{0} \phi)(\partial_{0} \phi) + (\partial_{1} \phi)(\partial_{1} \phi) + (\partial_{2} \phi)(\partial_{2} \phi) + (\partial_{3} \phi)(\partial_{3} \phi)$

Am I on the right track?

2. May 28, 2015

DEvens

Nearly there. But you seem to be missing a minus sign. And the definitions of ()^2 has a small issue.

Look in your course material or text or whatever you are using. There should be something called a metric. For the purposes of this question it looks like the metric should be a diagonal square matrix, zeroes off axis, and (1, -1, -1, -1) on the diagonal.

3. May 28, 2015

spaghetti3451

Is this correct?

$(\partial_{\mu} \phi)^{2}$
$= (\partial_{\mu}) (\phi \partial^{\mu} \phi)$
$= (\partial_{0} \phi) (\partial^{0} \phi) + (\partial_{1} \phi) (\partial^{1} \phi) + (\partial_{2} \phi) (\partial^{2} \phi) + (\partial_{3} \phi) (\partial^{3} \phi)$

4. May 28, 2015

DEvens

Not quite. $\partial_{\mu} \phi$ is a vector. The index is "down." So to get the "square" of such a vector you need the inner product.

$g^{\mu \nu} \partial_{\mu} \phi \partial_{\nu} \phi$

5. May 28, 2015

nrqed

Yes, this is absolutely correct (I disagree with Devens on that point) . But it is true that it may also be written as $g^{\mu \nu} \partial_\mu \phi \partial_\nu \phi$ although here we are not doing general relativity so we would usually write it as $\eta^{\mu \nu} \partial_\mu \phi \partial_\nu \phi$. But you don't need that notation if you know what $\partial^0, \partial_0, \partial^i$ and $\partial_i$ mean.

6. May 29, 2015

DEvens

I see that I forgot to say what was actually wrong. My bad. It's this line. This is taking the derivative $(\partial_{\mu})$ of $(\phi \partial^{\mu} \phi)$. That is, this is not the same as the previous line. It should be $(\partial_{\mu} \phi) ( \partial^{\mu} \phi)$.

7. May 29, 2015

nrqed

Sorry, I had misunderstood your point. I thought you were referring to the last line, I had actually ignored the second line, assuming that it was a typo. I agree completely with you.

Regards,

Patrick