Lorentz Transformation from Maxwell's Equations

Question

I've spent the last couple of hours trying to derive the Lorentz Transformation from Maxwell's Equations. What I ended up with is $$L_{nu}^{-1}=left(begin{array}{ll}
frac{1}{sqrt{1-v^{2}}} & frac{-v}{sqrt{1-v^{2}}} 
frac{-v}{sqrt{1-v^{2}}} & frac{1}{sqrt{1-v^{2}}}
end{array}right)$$
Which matches exactly with the Transformation as described in my textbook. And yet, when I search up Lorentz Transformations online, I find no matrix of the above form. What have I actually derived? Am I anywhere close to the Lorentz Transformations? Please advise.

Philip · Answer

As @DanDan0101 points out in their comment to your question, if you define
$$gamma = frac{1}{sqrt{1-v^2}},$$
where $v$ is the velocity measured in units of $c$ (otherwise this wouldn't make any sense dimensionally), then your matrix is just
$$L = begin{pmatrix}gamma & -gamma v  -gamma v & gammaend{pmatrix},$$
the "forward" Lorentz Transformation matrix, see this answer to What is a Lorentz boost and how to calculate it?

Dr jh · Answer

This is the Lorentz transformation for an object in a universe with one spatial dimension and a time dimension. Notice that in most texts $gamma = frac{1}{sqrt{1-frac{v^2}{c^2}}}$, and it seems like in your particulat textbook the authors have opted to go with units in which the speed of light, $c=1$. Your answer is therefore correct, it's just that some authors prefer using these units to simplify calculations.

Answered by Dr jh on September 21, 2020

robphy · Answer

As @Philip points out in their answer to your question, if you define, $v=tanhtheta$ (so that $ gamma=coshtheta$), then
$$ 
L = begin{pmatrix}coshtheta & -sinhtheta  -sinhtheta& coshthetaend{pmatrix}
$$
which is the rapidity form of a Lorentz boost

Lorentz Transformation from Maxwell's Equations

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