# [Physics] Galilean transformation in relativity

general-relativityinertial-framesrelative-motionspecial-relativity

Assume flat spacetime in a general relativistic framework (or special relativity for that matter) and two observers $A$ and $B$, with non-vanishing velocity relative to each other. We know that they are both moving in their own (global) inertial frames $I_A,I_B$ and these physical frames are related by a Lorentz transformation, namely a Lorentz boost. This must be so, because for examples, one light ray has the same velocity in both of these frames.

Starting from the frame of $I_A$, what is the physical interpretation of the coordinate system we get, if we make a Galileian transformation, i.e. a Galileian boost?

We might do this by, say, using the spatial part of the relative velocity $v_{BA}^i$ of $A$ and $B$. Maybe there is a better choice than $v_{BA}^i$ $-$ I'm especially (but not necessarily only) interested in a choice of the transformation parameters, such that the spatial centre of the new coordinate system coincides with the worldline of observer $B$. I assume this is possible, because both transformations transform the world line of $A$ into a straight line. Maybe then the velocity vector will not be normed any longer, but this seems to be just an issue of proper time. Is this a system which doesn't correspond to any free physical observer because the time evolution in this system is unnatural? If so, what might be a cause/force on a physical observer to move through spacetime according to this and how does he experience the world around him (the observers $A$, $B$ and light rays)?

Suppose I start off with myself and a friend every meter, making a grid along the x-axis. Then I start moving with velocity v along the x-axis, and at the same time (as measured by an observer at rest) all my friends start moving too. After a while we will see each other moving, but our separation will not be one meter as measured by us, but longer by a factor $1/\sqrt{1-v^2}$ (since the separation between the observers will still be 1m in the rest frame, where it is Lorentz contracted by a factor $\sqrt{1-v^2}$).