Clarification of labeling of Lorentz transformation

Question

I read a labeling of Lorentz matrix labeling $\Lambda^\mu_\nu$. To be more specific, it's used as $x^\mu\rightarrow \Lambda^\mu_\nu x^\nu$. I want to ask:

1. What does the indices $\mu$, $\nu$ mean in $\Lambda^\mu_\nu$?

2. What does $x^\mu\rightarrow \Lambda^\mu_\nu x^\nu$ mean? Are $x^\mu$ and $x^\nu$ in the same frame?

Answer 1

1. Τhe indices $\,\mu,\nu\,$ take the values $\,0,1,2,3$. The value $\,0\,$ corresponds to the time coordinate $x^0=c\,t\,$ while the values $\,1,2,3\,$ correspond to the space coordinates $\:\:x^1=\mathrm x,\:\:\:x^2=\mathrm y\:$ and $x^3=\mathrm z$.
2. Equation $\,x^\mu\rightarrow \Lambda^\mu_\nu x^\nu\,$ is better to expressed as $\,x'^\mu=\Lambda^\mu_\nu x^\nu\,$ and is the Lorentz transformation of the space-time position 4-vector from an inertial frame $\,\rm S\,$ to another inertial frame $\,\rm S'\,$ moving with constant velocity 3-vector with respect to the unprimed one.