# Show that a matrix is a Lorentz transformation

## Homework Statement

Given the matrix
$$\Omega = \begin{pmatrix} 0 & -\psi & 0 & 0 \\ -\psi & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$
show that $e^{\Omega}$ is a Lorentz transformation along the x axis with $\beta = tanh(\psi)$

## Homework Equations

During the lesson we derived from the standard Lorentz transformation matrix the following matrix, where $\psi$ is the rapidity:

$$\Lambda = \begin{pmatrix} cosh(\psi) & -sinh(\psi) & 0 & 0 \\ -sinh(\psi) & cosh(\psi) & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}$$

Other equations:
$cosh(\psi)=\gamma$
$sinh(\psi)=\gamma \beta$

## The Attempt at a Solution

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From $\beta = tanh(\psi)$:
$\psi=arctg(\beta) = \ln\sqrt{\frac{1+\beta}{1-\beta}}$
$e^{-\psi} = \sqrt{\frac{1-\beta}{1+\beta}}$
I think i have to show that the two matrices ($\Lambda$ and $e^{-\Omega}$) are the same, but i can't understand why there are zeros on the diagonal. For the two first zeros on the diagonal $cosh(\psi)=0$, so $\psi = \frac \pi 2$.
I think there's an error somewhere, because with the previous formulas it turns out that $\beta = 1$ and $v=c$.

To be honest, i can't find the right way to solve the problem, maybe it's just algebra? Can you please give me a hint? Thank you!

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PeroK
Homework Helper
Gold Member
What is the definition of $e^{\Omega}$?

Can you calculate $\Omega^n$?

$\Omega^n$ can be obtained multiplying the matrix n times. I underestimated the definition of $e^{\Omega}$, i guess i can't obtain it taking the exponential of each element... now i'm reading the definition on wikipedia.

nrqed
$\Omega^n$ can be obtained multiplying the matrix n times. I underestimated the definition of $e^{\Omega}$, i guess i can't obtain it taking the exponential of each element... now i'm reading the definition on wikipedia.
The exponential of a matrix is defined as in quantum mechanics, through its Taylor expansion (here around $\psi=0$).