The Lorentz group has one subgroup of 3D rotations, and another for inertial reference frame boosts where time effectively rotates into a spatial dimension.
The spacial rotations are in yellow, while the boosts are in red. The hyperbolas can cover all of spactime while the rotations are limited to a circle about the origin. Out at infinity, there will be two points that approach the yellow circle. They split so that one set of red points can be there are the creation and annihilation of the yellow points, and another set can greet the yellow points when they are furthest apart, about to change directions.
for the circle: for the hyperbola:
Two inertial observers, looking at the same collection of events, will see significantly different animations depending on their velocies so long as the difference in speed is a significant fraction of the speed of light.
The events in yellow move at a nice steady rate. The line in blue represents a boost along the x axis only. In the tx complex plane, the blue line is compressed in time but overlaps the yellow line. The line in red has been boosted in both the x and y axis. There is no complex plane where the red line is colinear with the yellow because the change is distributed in two planes. The reason neither red nor blue is colinear in either ty or tz planes is that only time is changed by the boost. The red line has a highest boost, so appears on the screen for a smaller time.
