General Lorentz Transformation Explained: Visualize and Grasp It!

[Sang-Hyeon Han]

Hi guys, I'm reading a book 'the theoretical minimum: special relativity and classical field theory'. In chapter 1.3, author explains the general Lorentz transformation.

He said "Suppose you have two frames in relative motion along some oblique direction, not along any of the coordinate axes. It would be easy to make the primed axes line up with the unprimed axes by performing a sequence of rotations. After doing those rotations, you would again have uniform motion in the x direction. The general Lorentz transformation—where two frames are related to each other by an arbitrary angle in space, and are moving relative to each other in some arbitrary direction—is equivalent to:

1. A rotation of space to align the primed axes with the unprimed axes.
2. A simple Lorentz transformation along the new x axis.
3. A second rotation of space to restore the original orientation of the
unprimed axes relative to the primed axes. "

My head is stuck with that paragraph and I can't understand how he can do that. I can't visualize it in my head. Could anyone make me grasp it?

 

[Ibix]

If the velocity you want to boost in isn't the x direction, then you rotate your axes (you can imagine a little construction of rods pointing in three directions that you imagine rotating) so that the new x direction does point in the direction you want to boost. Then you boost. Then you rotate your axes back to how they were before.