91Ó°ÊÓ

Lorentz Information is an important part of physisc sthat deals with the linear tranformations from a specific co-ordinate frame in space-time to a non-static frame, having a constant velocity with a respect to the former.

Consider the matrix is:

$\left(\begin{array}{c}\overline{{\mathrm{A}}_{\mathrm{y}}}\\ \overline{{\mathrm{A}}_{\mathrm{z}}}\end{array}\right)=\left(\begin{array}{cc}\cos \mathrm{Ï•}& \sin \mathrm{Ï•}\\ -\sin \mathrm{Ï•}& \cos \mathrm{Ï•}\end{array}\right)\left(\begin{array}{c}{\mathrm{A}}_{\mathrm{y}}\\ {\mathrm{A}}_{\mathrm{z}}\end{array}\right)$

If we take $\overline{{\mathrm{A}}_{\mathrm{x}}}$as X and $\overline{{\mathrm{A}}_{\mathrm{y}}}$as Y ${\mathrm{A}}_{\mathrm{x}}$and as x ${\mathrm{A}}_{\mathrm{y}}$and as y:

localid="1658826300610" $$\begin{gathered} \mathrm{X}=\cos \mathrm{Ï•³\ae }+\sin \mathrm{Ï•²â}.....\left(\mathrm{i}\right) \\ \mathrm{Y}=-\sin \mathrm{Ï•³\ae }+\cos \mathrm{Ï•²â}...\left(\mathrm{ii}\right) \end{gathered}$$

$$\begin{gathered} \overline{\mathrm{X}}=\mathrm{γ}\left(\mathrm{X}-\mathrm{vt}\right)...\left(\mathrm{iii}\right) \\ \overline{\mathrm{Y}}=\mathrm{Y}...\left(\mathrm{iv}\right) \\ \overline{\mathrm{Z}}=\mathrm{Z}...\left(\mathrm{v}\right) \\ \overline{\mathrm{t}}=\mathrm{γ}\left(\mathrm{t}-\frac{\mathrm{v}}{{\mathrm{c}}^2}\mathrm{X}\right)...\left(\mathrm{vi}\right) \end{gathered}$$

Now, from putting the value of equation (i) in equation (iii):

$$\begin{gathered} \overline{\mathrm{X}}=\mathrm{γ}\left(\mathrm{X}-\mathrm{vt}\right) \\ =\mathrm{γ}\left(\cos \mathrm{Ï•³\ae }+\sin \mathrm{Ï•²â}-\mathrm{⳦t}\right)....\left(\mathrm{vii}\right) \end{gathered}$$

Equating equation (iv) with (ii) we get:

$\stackrel{¯}{\mathrm{Y}}=\mathrm{Y}=-\sin \mathrm{Ï•³\ae }+\cos \mathrm{Ï•²â}$

By further calculation of equation (vi):

$\overline{\mathrm{t}}=\mathrm{γ}\left(\mathrm{t}-\frac{\mathrm{v}}{{\mathrm{c}}^2}\mathrm{X}\right)$

Multiplying both sides with, c:

role="math" localid="1658829796222" $\begin{array}{rcl}\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{³¦Î³}\left(\mathrm{t}-\frac{\mathrm{v}}{{\mathrm{c}}^2}\mathrm{X}\right)\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{³¦Î³t}-\frac{\mathrm{v}}{\mathrm{c}}\mathrm{γ³Ý}\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{³¦Î³t}-\mathrm{βγ³Ý}\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{γ}(\mathrm{ct}-\mathrm{β³Ý})....\left(\mathrm{viii}\right)\end{array}$

Putting the value from equation (i) to (viii):

$$\begin{gathered} \mathrm{c}\overline{\mathrm{t}}=\mathrm{γ}(\mathrm{ct}-\mathrm{β³Ý}) \\ \mathrm{c}\overline{\mathrm{t}}=\mathrm{γ}\left[\mathrm{ct}-\mathrm{β}(\cos \mathrm{Ï•³\ae }+\sin \mathrm{Ï•²â})\right]....\left(\mathrm{ix}\right) \end{gathered}$$

To get the Lorenz matrix, we have to rotate from to by using the equation (1.29) with negative and putting the respectives values from the above equations. We get

Therefore,

$\begin{array}{rcl}\overline{\mathrm{x}}& =& \mathrm{³¦´Ç²õÏ•}\overline{\mathrm{X}}-\mathrm{²õ¾\pm ²ÔÏ•}\overline{\mathrm{Y}}\\ & =& \mathrm{㳦´Ç²õÏ•}\left(\cos \mathrm{Ï•³\ae }+\sin \mathrm{Ï•²â}-\mathrm{⳦t}\right)-\mathrm{²õ¾\pm ²ÔÏ•}\left(-\sin \mathrm{Ï•³\ae }+\cos \mathrm{Ï•²â}\right)\\ & =& \left({\mathrm{㳦´Ç²õ}}^2\mathrm{Ï•}+{\sin }^2\mathrm{Ï•}\right)\mathrm{x}+(\mathrm{γ}-1)\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•²â}-\mathrm{γ⳦osÏ•}\left(\mathrm{ct}\right)....\left(\mathrm{x}\right)\end{array}$

And,

$$\begin{gathered} \overline{\mathrm{y}}=\mathrm{²õ¾\pm ²ÔÏ•}\overline{\mathrm{X}}+\mathrm{³¦´Ç²õÏ•}\overline{\mathrm{Y}} \\ =\mathrm{γ²õ¾\pm ²ÔÏ•}[\cos \mathrm{Ï•³\ae }+\sin \mathrm{Ï•²â}-\mathrm{⳦t}]+\mathrm{³¦´Ç²õÏ•}[-\sin \mathrm{Ï•³\ae }+\cos \mathrm{Ï•²â}] \\ =({\mathrm{γ²õ¾\pm ²Ô}}^2\mathrm{Ï•}+{\cos }^2\mathrm{Ï•})\mathrm{y}+(\mathrm{γ}-1)\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•³\ae }-\mathrm{γβ²õ¾\pm ²ÔÏ•}\left(\mathrm{ct}\right)........\left(\mathrm{xi}\right) \end{gathered}$$

By convention (x) and (xi) into matrix form:

$\left(\begin{array}{c}\mathrm{c}\overline{\mathrm{t}}\\ \overline{\mathrm{x}}\\ \overline{\mathrm{y}}\\ \overline{\mathrm{z}}\end{array}\right)=\left[\begin{array}{cccc}\mathrm{γ}& -\mathrm{γ⳦os}\mathrm{Ï•}& -\mathrm{γβ²õ¾\pm ²Ô}\mathrm{Ï•}& 0\\ -\mathrm{γ⳦os}\mathrm{Ï•}& \left({\mathrm{㳦´Ç²õ}}^2\mathrm{Ï•}+{\sin }^2\mathrm{Ï•}\right)& \left(\mathrm{γ}-1\right)\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•}& 0\\ -\mathrm{γβ²õ¾\pm ²Ô}\mathrm{Ï•}& \left(\mathrm{γ}-1\right)\mathrm{²õ¾\pm ²ÔÏ•³¦´Ç²õÏ•}& \left({\mathrm{γ²õ¾\pm ²Ô}}^2\mathrm{Ï•}+{\cos }^2\mathrm{Ï•}\right)& 0\\ 0& 0& 0& 1\end{array}\right]\left(\begin{array}{c}\mathrm{ct}\\ \mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right)$