 #1
Athenian
 143
 33
 TL;DR Summary

QUESTION:
How can one tell (or figure out) the boost speed(s), direction coordinates, and rotation through matrix multiplication?
Recently, I've been studying about Lorentz boosts and found out that two perpendicular Lorentz boosts equal to a rotation after a boost. Below is an example matrix multiplication of this happening:
$$
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & 0 & \frac{1}{\sqrt{3}} & 0 \\
0 & 1 & 0 & 0 \\
\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{3}} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 \\
\frac{1}{\sqrt{3}} & \frac{2}{\sqrt{3}} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \frac{4 \sqrt{3}}{7} & \frac{1}{7} & 0 \\
0 & \frac{1}{7} & \frac{4 \sqrt{3}}{7} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{cccc}
\frac{4}{3} & \frac{2}{3} & \frac{1}{\sqrt{3}} & 0 \\
\frac{2}{3} & \frac{25}{21} & \frac{2}{7 \sqrt{3}} & 0 \\
\frac{1}{\sqrt{3}} & \frac{2}{7 \sqrt{3}} & \frac{8}{7} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
Source: https://physics.stackexchange.com/q...ingtoarotationafteraboost/515690#515690
According to the expert (on the website above) providing the example matrix multiplication, he stated that, "the lefthandside represents a boost by ##c/2## along the ##x##direction followed by a boost by ##c/2## along the ##y##direction, and that the righthandside represents a boost by ##\sqrt{7}c/4## in the direction ##(2/\sqrt{7},\sqrt{3/7},0)## followed by rotation around the ##z##axis by ##\cos^{1}(4\sqrt{3}/7)## or ##8.21## degrees".
However, my question is how was he able to know the boosts for the ##x## and the ##y## directions are both ##c/2##? Furthermore, how was he able to know what are the magnitude, direction, and rotation about the ##z##axis of the matrices after the matrix multiplication? For some reason, I am having a hard time figuring out how he was able to find these numerical values.
Finally, a last yet notsobright question, how was the author able to get the answer yet split the matrix in two in that specific manner provided in the example above? Does this help with the solution process?
Any help to provide any amount of insight to answering my question will be much appreciated. Thank you very much for your time and assistance!
$$
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & 0 & \frac{1}{\sqrt{3}} & 0 \\
0 & 1 & 0 & 0 \\
\frac{1}{\sqrt{3}} & 0 & \frac{2}{\sqrt{3}} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{cccc}
\frac{2}{\sqrt{3}} & \frac{1}{\sqrt{3}} & 0 & 0 \\
\frac{1}{\sqrt{3}} & \frac{2}{\sqrt{3}} & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1 & 0 & 0 & 0 \\
0 & \frac{4 \sqrt{3}}{7} & \frac{1}{7} & 0 \\
0 & \frac{1}{7} & \frac{4 \sqrt{3}}{7} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{cccc}
\frac{4}{3} & \frac{2}{3} & \frac{1}{\sqrt{3}} & 0 \\
\frac{2}{3} & \frac{25}{21} & \frac{2}{7 \sqrt{3}} & 0 \\
\frac{1}{\sqrt{3}} & \frac{2}{7 \sqrt{3}} & \frac{8}{7} & 0 \\
0 & 0 & 0 & 1 \\
\end{array}
\right)
$$
Source: https://physics.stackexchange.com/q...ingtoarotationafteraboost/515690#515690
According to the expert (on the website above) providing the example matrix multiplication, he stated that, "the lefthandside represents a boost by ##c/2## along the ##x##direction followed by a boost by ##c/2## along the ##y##direction, and that the righthandside represents a boost by ##\sqrt{7}c/4## in the direction ##(2/\sqrt{7},\sqrt{3/7},0)## followed by rotation around the ##z##axis by ##\cos^{1}(4\sqrt{3}/7)## or ##8.21## degrees".
However, my question is how was he able to know the boosts for the ##x## and the ##y## directions are both ##c/2##? Furthermore, how was he able to know what are the magnitude, direction, and rotation about the ##z##axis of the matrices after the matrix multiplication? For some reason, I am having a hard time figuring out how he was able to find these numerical values.
Finally, a last yet notsobright question, how was the author able to get the answer yet split the matrix in two in that specific manner provided in the example above? Does this help with the solution process?
Any help to provide any amount of insight to answering my question will be much appreciated. Thank you very much for your time and assistance!