Simple Lorentz Boosts

summary:

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.

description:

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.

command:

q_graph -out boost -dir vp -loop 0 -box 4 -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000' -color yellow -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000 |q_boost -vx .5' -color blue -command 'q_add_n -5 -5 -5 -5 0.010 0.010 0.010 0.010 1000 | q_boost -vx .3 -vy .4' -color red

iPod downloads:

boost.povray.100.1000.animation.scan_.m4v

math

equation:

$(t, \vec{R}) \rightarrow (t', R') = (\frac{t}{\sqrt{1 - \beta^2}} - \frac{\vec{\beta} \cdot \vec{R}}{\sqrt{1 - \beta^2}},\vec{R} \times \frac{\vec{V}}{|V|} + \frac{1}{\sqrt{1 - \beta^2}}(\vec{R} - \vec{R} \times \frac{\vec{V}}{|V|} - \vec{\beta} t))$