Lorentz Transform on Covariant Vector (Lahiri QFT 1.5)

[BucketOfFish]

Homework Statement

Given that $x_\mu x^\mu = y_\mu y^\mu$ under a Lorentz transform ($x^\mu \rightarrow y^\mu$, $x_\mu \rightarrow y_\mu$), and that $x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu$, show that $x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu$.

Homework Equations

$$g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma$$

The Attempt at a Solution

So this isn't actually a homework problem, it's just an exercise in Lahiri and Pal that I was looking at. Seems like this thing would be really simple, but I can't work it out for some reason.

I get, for example, to the point where $x_\mu x^\mu = g_{\mu\nu}x^\mu x^\nu = g_{\rho\sigma}\Lambda^\rho{}_\mu\Lambda^\sigma{}_\nu x^\mu x^\nu = y_\mu y^\mu$. Then, switching labels and using the definition of $y^\mu$, we get that $y_\mu = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu$. But I go in circles from there.

Part of my confusion is that I don't really know what the relation is between $\Lambda_\mu{}^\nu$ and $\Lambda^\mu{}_\nu$. Seems like I'm missing something really obvious. Can anyone help?

 

[stevendaryl]

Homework Statement

Given that $x_\mu x^\mu = y_\mu y^\mu$ under a Lorentz transform ($x^\mu \rightarrow y^\mu$, $x_\mu \rightarrow y_\mu$), and that $x^\mu \rightarrow y^\mu = \Lambda^\mu{}_\nu x^\nu$, show that $x_\mu \rightarrow y_\mu = \Lambda_\mu{}^\nu x_\nu$.

Homework Equations

$$g_{\rho\sigma} = g_{\mu\nu}\Lambda^\mu{}_\rho\Lambda^\nu{}_\sigma$$

The Attempt at a Solution

So this isn't actually a homework problem, it's just an exercise in Lahiri and Pal that I was looking at. Seems like this thing would be really simple, but I can't work it out for some reason.

I get, for example, to the point where $x_\mu x^\mu = g_{\mu\nu}x^\mu x^\nu = g_{\rho\sigma}\Lambda^\rho{}_\mu\Lambda^\sigma{}_\nu x^\mu x^\nu = y_\mu y^\mu$. Then, switching labels and using the definition of $y^\mu$, we get that $y_\mu = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu$. But I go in circles from there.

Part of my confusion is that I don't really know what the relation is between $\Lambda_\mu{}^\nu$ and $\Lambda^\mu{}_\nu$. Seems like I'm missing something really obvious. Can anyone help?

[edit]
I see that you were actually sort of close, except that you made a slight mistake.

y_\mu = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu

should be
y_\rho = g_{\rho\mu}\Lambda^\mu{}_\nu x^\nu

At this point, you can operate on both sides by \Lambda^\rho{}_\lambda.

 

[Fredrik]

Part of my confusion is that I don't really know what the relation is between $\Lambda_\mu{}^\nu$ and $\Lambda^\mu{}_\nu$.

$$\Lambda_{\mu}{}^\nu =g_{\mu\rho}\Lambda^\rho{}_\sigma g^{\sigma\nu}.$$ What does this tell you about a combo like $\Lambda_\rho{}^\mu \Lambda^\rho{}_\nu$?

You may also find my post in this thread useful: https://www.physicsforums.com/threads/einstein-notation-notes.770129/#post-4847943