Why are rotated parallel axes still parallel?

[unified]

TL;DR Summary

In relativity, angles are coordinate dependent. If this is true, then why are rotated parallel axes still parallel?

I tried posting this on the physicsstackexchange, but wasn't making any progress in understanding what's going on.
Suppose the axes in two coordinate systems S, S' are parallel. Now, suppose I rotate S through some angle $\theta$ and also rotate S' through the same angle $\theta$ It's not clear to me that the rotated axes will remain parallel. In relativity, angles are coordinate dependent. For example, if S and S' are parallel and S' moves at velocity $\left| v \right| \hat x$ relative to S, and someone in S' places a bar at an angle of $\theta$ with respect to the axis x', then this angle will not be the same as the angle relative to the x axis. If someone in S then places a bar at an angle $\theta$ with respect to x, then am I correct that this bar won't be parallel to the bar in S' since an observer in S measures the bars at different angles relative to the x axis? However, these bars are just rotated axes, both rotated at the same angle $\theta$, but they're not parallel.

 

[PeterDonis]

It's not clear to me that the rotated axes will remain parallel.

Is it a problem if they don't? Is there some reason why you think they should?

 

[Ibix]

Now, suppose I rotate S through some angle $\theta$ and also rotate S' through the same angle $\theta$ It's not clear to me that the rotated axes will remain parallel.

I was just wondering who is measuring the rotation angle. Is the quoted text meant to mean that the S frame's $x$ axis makes an angle $\theta$ with its un-rotated original as measured in S, and the S' frame's $x'$ axis makes the same angle $\theta$ with its un-rotated original as measured in S'? If so, I agree with Peter, and I don't immediately see that the rotated axes would still be parallel. At least not in general - it should hold for rotations in the plane perpendicular to the frames' travel directions.

 

[unified]

Yes, that's correct. My motivation for asking this question is that the popular derivation of the formula for a generalized Lorentz transformation uses a rotation of the axes in this way, and it's assumed that they remain parallel.

Let me clarify.

Suppose S and S' have axes aligned with origins coinciding at t = t' = 0, and S' moves with velocity $\left| v \right|$ at an angle of $\theta$ relative to the x axis as measured in frame S. Then, to derive the Lorentz transformation from S to S', the usual solution is to rotate the x-y axes so that as measured in S, the rotated axis $\bar x$ makes an angle of $\theta$ with respect to the x axis. Also, rotate the x'-y' axes so that as measured in S', the rotated axis $\bar x'$ makes an angle of $\theta$ with respect to the x' axis. Now, assuming $\bar S$ and $\bar S'$ have parallel axes, then since $\bar S'$ moves with velocity $\left| v \right|$ $\hat {\bar x}$, we can use Lorentz transformation formula that applies in this case, and then transform back to the original reference frames using the inverse rotation matrices.

I can't understand why $\bar S$ and $\bar S'$ have parallel axes.

 

[PeterDonis]

the popular derivation of the formula for a generalized Lorentz transformation uses a rotation of the axes in this way

It does? How so? Please give a specific reference.

 

[A.T.]

If someone in S then places a bar at an angle $\theta$ with respect to x, then am I correct that this bar won't be parallel to the bar in S' since an observer in S measures the bars at different angles relative to the x axis?

The Lorentz Transformation just applies a scale factor to the spatial dimensions along the relative motion of the two frames. So if two physical bars are parallel in one inertial frame, they are also parallel in every inertial frame.

But if you are talking about the angle of a physical bar to a coordinate axis, that depends on how you defined the frame axes, and can change even for Galilean Transformations.