# Galilean transformation in 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)?

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This is easiest to answer by a rotational analogy. Suppose you don't rotate a block, but shear it. This is not a symmetry--- so the block is under stress. The same is true if you Galilean boost a material instead of Lorentz boosting it. You must have many things filling space-time in order to see what is going on, because if you just look a point observer and move it, there is no difference on the world line itself between a relativistic boost and a galilean one. The Galilean one only adds a shear to the motion when you consider several parallel observers.

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 $\sqrt{1-v^2}$).

This situation is interesting, because if we are connected by weak springs, the springs will be at tension if we start moving, since the distance is a little longer in our frame.

A physical system in which this behavior is natural is the motion of electrons down a wire. At whatever speed they propagate, their density cannot Lorentz contract, because the wire has to stay neutral and the protons are not moving, so Lorentz contracted for those electrons which are moving, which means that the distance between the moving electrons increases in their frame. To find the moving electron gas in a classical Drude-like model of a metal, you Galilean shear the electrons from rest, you don't Lorentz boost the electrons.

The system of electrons in a metal is really not invariant under boosts, because the nuclei aren't, so this isn't the best picture quantum mechanically.

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Ron, I did not get your explanation about a wire with current. A neutral wire with current looks charged in a moving RF. –  Vladimir Kalitvianski Nov 8 '11 at 15:41
@Vladimir: Yes--- but in the frame of the nuclei, the moving electrons are sheared to maintain a fixed distance between each other, to neutralize the field of the protons. If you take objects and move them to so that their distance is fixed in the rest frame, so that they don't Lorentz contract, you have made a mock-Galilean transformation. It isn't quite a Galilean transformation because time still dilates, there's not much you can do about that. –  Ron Maimon Nov 8 '11 at 17:59

The Lorentz transformations connect the observation results of some phenomenon in two reference frames moving relatively to each other and having the same scales and clocks.

Any change in scales and/or clocks used in another reference frame will spoil the Lorentz transformation, but using differently scaled "meters" and "seconds" in different RFs is still possible (meters versus feet, for example). I think the Galilean transformations correspond to transition (there is no physical transition but result recalculation) in a RF with clocks tuned differently (different "units").

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