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So having worked out beforehand that $Λ(v) = \begin{pmatrix} γ&0&\frac{-γv}{c}\\ 0&1&0\\\frac{-γv}{c}&0&γ\end{pmatrix}$ where $Λ(v) ∈ SO(2,1)$ is a matrix representation of a Lorentz boost to a reference frame $S'$ moving at velocity $v=vy$ relative to an intertial frame. $SO(2,1)$ is the Lorentz Group in 2 spatial dimensions and one time dimension and $γ = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$

The basis elements of the Lie algebra $so(2,1)$ which I worked out are:

$ ζ_1 = \begin{pmatrix} 0&0&0\\ 0&0&-1\\0&-1&0 \end{pmatrix}, ζ_2\begin{pmatrix} 0&0&1\\ 0&0&0\\1&0&0 \end{pmatrix}, ζ_3 =\begin{pmatrix} 0&-1&0\\ 1&0&0\\0&0&0 \end{pmatrix} $

I'm wondering how to write $Λ(v)$ in exponential form in terms of a basis element of $so(2,1)$

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Hint: work out $exp(\eta \zeta_1)$ and figure out how $\eta$ is related to $v, \gamma$

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