Lorentz boost

Lorentz Boosts with Quaternions

Posted by dougsweetser

QMN:1009.1026

Douglas Sweetser

uploads

latex pdf:

lorentz_w_quats_tex.pdf

nb.pdf:

lorentz_w_quats_master.pdf

Mathematica Notebook:

lorentz_w_quats_master.nb

youtube link:

Boosts w/Quaternions

abstract:

A method for using hyperbolic sines and cosines in a real-valued quaternion to generate a Lorentz boost along an axis is shown. Do precisely what one does for a 3D spacial rotation as a first step, substituting the hyperbolic for regular trig functions, B' = H B H*. That creates four terms that are need, adding two extra terms and containing two omissions. A difference between the same three matrices does the job: B' = H B H* + ( (H H B)* - (H* H* B)* )/2. The inverse transform is created by changing the conjugates on the hyperbolic quaternions. Boosts represented with quaternions must form a group, but it is not compact because the operator uses both addition and multiplication.

Start by understanding rotations around the x axis with quaternions.

$(c t',x',y',z')=(\cos (\alpha ), \sin (\alpha ) ,0,0)(c t,x,y,z)(\cos (\alpha ), -\sin (\alpha ) ,0,0) \quad eq. 1$

$=\left(\left(\sin^2(\alpha )+\cos^2(\alpha )\right)c t,\left(\sin^2(\alpha )+\cos^2(\alpha )\right)x,$
$\left(\cos^2(\alpha )-\sin^2(\alpha )\right)y-2 \sin (\alpha )\cos (\alpha )z,\left(\cos^2(\alpha )-\sin^2(\alpha )\right)z+2 \sin (\alpha )\cos (\alpha )y )$

$=(c t, x, cos(2 \alpha ) y - sin(2\alpha ) z , cos(2\alpha ) z + sin(2\alpha ) y)}$

Document Description

# of pages:

# of figures:

Change Log:

2010 Oct 7: Added reverse Lorentz transformation, comment about non-compactness, and deleted a reference to hypercomplex numbers which are not relevant.

Simple Lorentz Boosts

Posted by doug

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:

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