Lorentz Transformation

Lorentz transformation is usually expressed as:

$$\{\begin{array}{rl}{t}^{\prime }& =\gamma \left(t-\frac{vx}{c^2}\right)\\ x^{\prime }& =\gamma (x-vt)\\ y^{\prime }& =y\\ z^{\prime }& =z\end{array}\text{\hspace{1em}(1)}$$

where $\gamma =\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

Notes

We admit Constancy of the Speed of Light and Principle of Relativity We could derive this transformation by:

$$\begin{array}{rl}\left(\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right)& =\frac{1}{-2c}\left(\begin{array}{cc}c& c\\ 1& -1\end{array}\right)\left(\begin{array}{cc}{\lambda }_1& \\ & {\lambda }_2\end{array}\right)\left(\begin{array}{cc}-1& -c\\ -1& c\end{array}\right)\left(\begin{array}{c}x\\ t\end{array}\right)\\ \left(\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right)& =\left(\begin{array}{cc}({\lambda }_1+{\lambda }_2)/2& c({\lambda }_1-{\lambda }_2)/2\\ {\lambda }_1-{\lambda }_2/2c& ({\lambda }_1+{\lambda }_2)/2\end{array}\right)\left(\begin{array}{c}x\\ t\end{array}\right)\\ 0& =v({\lambda }_1+{\lambda }_2)/2+c({\lambda }_1-{\lambda }_2)/2\\ \frac{{\lambda }_1}{{\lambda }_2}& =\frac{c-v}{c+v}=\frac{1-\beta }{1+\beta }\\ {\lambda }_1{\lambda }_2& =1\\ {\lambda }_1& =\sqrt{\frac{1-\beta }{1+\beta }}\\ {\lambda }_2& =\sqrt{\frac{1+\beta }{1-\beta }}\\ \left(\begin{array}{c}{x}^{\prime }\\ t^{\prime }\end{array}\right)& =\gamma \left(\begin{array}{cc}1& -v\\ -\frac{v}{c^2}& 1\end{array}\right)\left(\begin{array}{c}x\\ t\end{array}\right)\\ \gamma & =\frac{1}{\sqrt{1-{\beta }^2}}\end{array}$$

In this transformation, what is invariant in this transform?

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