The Lorentz matrix defines the transformation of a four-vector between different frames of reference, such that $$ p^{'\mu} = \Lambda^{\mu}_{\ \ \nu}p^{\nu} $$ where in this example $p^{\mu}$ is the four-momentum.
1) Are Lorentz transformations of this form only valid for constant (not changing in magnitude) velocities?
I guess so, since $\gamma$ is a function of $v^2$. How can we transform between accelerating frames?
2) Is Lorentz invariance a law of nature?
Which physical quantities should we expect to be invariant (forces? charge?)?
3) What are the eigenvectors and the eigenvalues of the general Lorentz matrix?
I mean what is their physical significance? They do not change under Lorentz transformations?
(I know the ones for the boost in the z direction are something like the Doppler shifted frequencies, but what does this mean? They are the same in all frames? What about the eigenvalues for the boost in a random directiom matrix?)