In that sense "rest frame" should be applicable for a single particle or a rigid body, not for a group of particles or compound materials, e.g. the floating boat on a river flow is not in the "rest frame".
That's only true if the boost (it's not a general Lorentz transform being discussed here but a boost) is in the direction of the photon's momentum. Indeed, a boost has two space-like and two light-like eigenvectors. For simplicity take a boost in ##x## direction as mentioned in #1. It's representing matrix is given byYes, in fact the energy-momentum of a photon is an eigenvector of the Lorentz Transformation; and, the Doppler red and blue-shift factors are the eigenvalues!
Yes.In that sense "rest frame" should be applicable for a single particle or a rigid body
If your boat is moving rigidly then it would have a rest frame. The river flow would clearly not be rigid.e.g. the floating boat on a river flow is not in the "rest frame".
Which is precisely why the term "center of momentum frame" is so common in the particle physics literature.In these strict sense rest frame could be applicable for a few kind of single elementary particles
More precisely, it excludes bodies in which internal forces are propagated faster than light. But it is perfectly possible to have rigid motions of bodies, under appropriate conditions.The theory of relativity excludes rigid bodies.