Inertial system S moves at constant velocity $\mathbf{v}\mathbf{=}\mathbf{\beta c}\left(\mathrm{cos\varphi }\hat{x}+\mathrm{sin\varphi }\hat{y}\right)$with respect to S. Their axes are parallel to one other, and their origins coincide at data-custom-editor="chemistry" $\mathbf{t}\mathbf{=}\mathbf{t}\mathbf{=}\mathbf{0}$, as usual. Find the Lorentz transformation matrix A.

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:

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

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" $$\begin{gathered} \mathrm{X}=\cos \mathrm{\varphi x}+\sin \mathrm{\varphi y}.....\left(\mathrm{i}\right) \\ \mathrm{Y}=-\sin \mathrm{\varphi x}+\cos \mathrm{\varphi y}...\left(\mathrm{ii}\right) \end{gathered}$$

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

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

$$\begin{gathered} \overline{\mathrm{X}}=\mathrm{\gamma }\left(\mathrm{X}-\mathrm{vt}\right) \\ =\mathrm{\gamma }\left(\cos \mathrm{\varphi x}+\sin \mathrm{\varphi y}-\mathrm{\beta ct}\right)....\left(\mathrm{vii}\right) \end{gathered}$$

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

$\overline{\mathrm{Y}}=\mathrm{Y}=-\sin \mathrm{\varphi x}+\cos \mathrm{\varphi y}$

By further calculation of equation (vi):

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

Multiplying both sides with, c:

role="math" localid="1658829796222" $\begin{array}{rcl}\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{c\gamma }\left(\mathrm{t}-\frac{\mathrm{v}}{{\mathrm{c}}^2}\mathrm{X}\right)\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{c\gamma t}-\frac{\mathrm{v}}{\mathrm{c}}\mathrm{\gamma X}\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{c\gamma t}-\mathrm{\beta \gamma X}\\ \mathrm{or},\mathrm{c}\overline{\mathrm{t}}& =& \mathrm{\gamma }(\mathrm{ct}-\mathrm{\beta X})....\left(\mathrm{viii}\right)\end{array}$

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

$$\begin{gathered} \mathrm{c}\overline{\mathrm{t}}=\mathrm{\gamma }(\mathrm{ct}-\mathrm{\beta X}) \\ \mathrm{c}\overline{\mathrm{t}}=\mathrm{\gamma }\left[\mathrm{ct}-\mathrm{\beta }(\cos \mathrm{\varphi x}+\sin \mathrm{\varphi y})\right]....\left(\mathrm{ix}\right) \end{gathered}$$

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,

$\begin{array}{rcl}\overline{\mathrm{x}}& =& \mathrm{cos\varphi }\overline{\mathrm{X}}-\mathrm{sin\varphi }\overline{\mathrm{Y}}\\ & =& \mathrm{\gamma cos\varphi }\left(\cos \mathrm{\varphi x}+\sin \mathrm{\varphi y}-\mathrm{\beta ct}\right)-\mathrm{sin\varphi }\left(-\sin \mathrm{\varphi x}+\cos \mathrm{\varphi y}\right)\\ & =& \left({\mathrm{\gamma cos}}^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)\mathrm{x}+(\mathrm{\gamma }-1)\mathrm{sin\varphi cos\varphi y}-\mathrm{\gamma \beta cos\varphi }\left(\mathrm{ct}\right)....\left(\mathrm{x}\right)\end{array}$

And,

$$\begin{gathered} \overline{\mathrm{y}}=\mathrm{sin\varphi }\overline{\mathrm{X}}+\mathrm{cos\varphi }\overline{\mathrm{Y}} \\ =\mathrm{\gamma sin\varphi }[\cos \mathrm{\varphi x}+\sin \mathrm{\varphi y}-\mathrm{\beta ct}]+\mathrm{cos\varphi }[-\sin \mathrm{\varphi x}+\cos \mathrm{\varphi y}] \\ =({\mathrm{\gamma sin}}^2\mathrm{\varphi }+{\cos }^2\mathrm{\varphi })\mathrm{y}+(\mathrm{\gamma }-1)\mathrm{sin\varphi cos\varphi x}-\mathrm{\gamma \beta sin\varphi }\left(\mathrm{ct}\right)........\left(\mathrm{xi}\right) \end{gathered}$$

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

$\left(\begin{array}{c}\mathrm{c}\overline{\mathrm{t}}\\ \overline{\mathrm{x}}\\ \overline{\mathrm{y}}\\ \overline{\mathrm{z}}\end{array}\right)=\left[\begin{array}{cccc}\mathrm{\gamma }& -\mathrm{\gamma \beta cos}\mathrm{\varphi }& -\mathrm{\gamma \beta sin}\mathrm{\varphi }& 0\\ -\mathrm{\gamma \beta cos}\mathrm{\varphi }& \left({\mathrm{\gamma cos}}^2\mathrm{\varphi }+{\sin }^2\mathrm{\varphi }\right)& \left(\mathrm{\gamma }-1\right)\mathrm{sin\varphi cos\varphi }& 0\\ -\mathrm{\gamma \beta sin}\mathrm{\varphi }& \left(\mathrm{\gamma }-1\right)\mathrm{sin\varphi cos\varphi }& \left({\mathrm{\gamma sin}}^2\mathrm{\varphi }+{\cos }^2\mathrm{\varphi }\right)& 0\\ 0& 0& 0& 1\end{array}\right]\left(\begin{array}{c}\mathrm{ct}\\ \mathrm{x}\\ \mathrm{y}\\ \mathrm{z}\end{array}\right)$