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My question originates from what is done in the book on Quantum Field Theory book by Mark Srednicki on page 21 (if anyone has it).

So say you have an inertial frame that is represented in the coordinates $x^{\mu}$. Any other coordinates $\bar{x}^{\mu}$ will also be represented by an inertial frame if they are related in the following way.

\begin{equation} \bar{x}^{\mu}=\Lambda^{\mu}{}_{\nu}x^{\nu}+a^{\mu} \end{equation}

Where $\Lambda^{\mu}{}_{\nu}$ is a Lorentz transformation matrix and $a^{\mu}$ is a translation vector.

This is all well and good and I understand it, it is just a transformation on the coordinates. What I don't fully quite grasp is the notation of the matrix $\Lambda^{\mu}{}_{\nu}$. I know that something like $x^{\mu}$ represents a contravariant vector and something like $x_{\mu}$ represents a covariant vector, and both of these are rank 1 tensors. When it comes to matrices though I get a bit confused.

I know the basic rule of raising and lowering indices, but I guess I don't know it in a very robust manner. For example, I know that $\Lambda^{\mu}{}_{\nu}x^{\nu}$ MUST yield a rank 1 tensor $y^{\mu}$ since the $\nu$ indices "cancel." But I don't know the difference between the tensors $\Lambda^{\mu}{}_{\nu}$, $\Lambda_{\nu\mu}$ and $\Lambda^{\nu\mu}$. (I am aware that for this example, only the first one makes sense but I don't understand the conceptual differences between these three terms.)

I hope all this makes sense!

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    $\begingroup$ "I know the basic rule of raising and lowering indices" If you do, why don't you know the difference between $\Lambda_{\mu\nu}$,${\Lambda_\mu}^\nu$ and $\Lambda^{\mu\nu}$? $\endgroup$
    – ACuriousMind ♦
    Commented Mar 1, 2016 at 20:19
  • $\begingroup$ may I suggest this answer of mine, where I explain the basics of Lorentz transformations and raising/lowering indices? $\endgroup$
    – AccidentalFourierTransform
    Commented Mar 1, 2016 at 20:21
  • $\begingroup$ ACuriousMind, What I meant by that is, I know of only the rule, not the origin of the rule. My knowledge is limited in this subject. @AccidentalFourierTransform I will look at that right now! Thanks! $\endgroup$
    – user41178
    Commented Mar 1, 2016 at 20:25
  • $\begingroup$ Related: physics.stackexchange.com/q/158309/2451 , physics.stackexchange.com/q/169762/2451 , physics.stackexchange.com/q/237270/2451 and links therein. $\endgroup$
    – Qmechanic ♦
    Commented Mar 1, 2016 at 21:01
  • $\begingroup$ I read through your post @AccidentalFourierTransform and I think I understand much better. So say we have a matrix $A$, and we multiply it by a vector $x^{\mu}$, if this matrix transforms the vector from contravariant to covariant, then we denote the matrix $A$ as $A_{\mu\nu}$, but if the vector simply transforms the vector to another frame, but keeps it contravariant, we denote it $A^{\nu}_{\mu}$, so the result would be $x^{\nu}$. Is this correct? It is simply a way of denoting what kind of tensor $A$ is? $\endgroup$
    – user41178
    Commented Mar 1, 2016 at 21:02

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