Show that a matrix is a Lorentz transformation

AI Thread Summary
The discussion focuses on demonstrating that the matrix Ω represents a Lorentz transformation along the x-axis, with the rapidity ψ related to the velocity β by the equation β = tanh(ψ). Participants explore the relationship between the exponential of the matrix e^Ω and the standard Lorentz transformation matrix Λ, noting that the zeros on the diagonal of Ω may indicate a misunderstanding of the matrix's properties. There is confusion regarding the calculation of e^Ω, with suggestions to use the Taylor expansion and matrix exponentiation techniques. The conversation highlights the need for clarity on the definition of matrix exponentials and the algebra involved in deriving the Lorentz transformation. Understanding these concepts is crucial for solving the problem accurately.
fineTuner
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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


[/B]
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!
 
Physics news on Phys.org
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.
 
fineTuner said:
##\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##).
 
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