I am currently looking a bit into special relativity. Consider the matrix(adsbygoogle = window.adsbygoogle || []).push({});

[itex]\Lambda=\left( \begin{array}{cc}

\gamma & -\gamma \beta c \\

-\gamma \beta c & \gamma \end{array} \right)[/itex]

where

[itex]\beta=\frac{v}{c},\quad \gamma=\frac{1}{\sqrt{1-\beta^2}}[/itex]

and c is the speed of light.

Then, an observer moving in the positive x direction will observe

[itex]x^{\mu '}=\Lambda^{\mu '}_\nu x^\nu[/itex]

The eigenvectors of the transformation matrix describes light rays and are

[itex]u^\mu_\pm=\left( \begin{array}{c}

t \\

\pm ct \end{array} \right)[/itex]

That eigenvectors of the Lorentz transformation are light rays simply means that light rays stay light rays under any Lorentz transformation which is the starting point of special relativity.

But the eigenvalue of the Lorentz transformation is

[itex]e_\pm=\frac{1\mp \beta}{\sqrt{1-\beta^2}}[/itex]

And I was wondering what the meaning of this eigenvalue is. Do anyone know?

And, we have the interesting identity (which may be somewhat intuitive when you think about it)

[itex]e_+e_-=1[/itex]

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# What do the eigennumbers of the Lorentz transformation represent?

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