Properties of Lorentz Transformations

This animation shows four properties of Lorentz transformations in Minkowski space-time: a proper rotation, a boost, a four-screw and a null rotation. First, rapidity is set to a constant value and the rotation is animated through 2pi about the z-axis. Second, the rotation angle is set to zero and the change through various rapidity values is animated for a z-boost. Third, both rotation angles and rapidity values are animated at the same time to show a four-screw. For a faithful representation and direct computation of the observables as a point cloud (preserving the bundle inner product space,) we decompose the transformation matrix before normalizing the vector of eigenvalues and reconstructing a new instance of the transformation that takes the Hopf bundle into itself, while leaving the transformations of the Argand plane intact. This normalization is necessary since all null vectors in the bundle have unit length, but have zero Lorentz length. Finally, animate a real number over a range of values for visualizing the effect of a null rotation on the horizontal sections of the bundle. Unlike a four-screw, a null rotation takes the bundle into itself and so normalization of the transform is not needed. Again, we asserted that the transformation on the Argand plane remains the same after conversion from spin-vectors to quaternions and vice versa.

[@TheMemesofDestruction]

I smell inversion. ^.^