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I was reading that Lorentz boosts in light-front coordinates reduce to simple kinematic transformations, but I fail to see how this is resolved. For example, if I take a Lorentz boost in the $z$-direction for $x^+$, we have

$$ (x^+)' = \gamma \left[ x^+ - \frac{vx^-}{c^2} \right] $$

Where $x^+ = ct + z$ is the light front time coordinate, and $x^- = ct - z$ is the light front space coordinate. In addition, $t$ is the instant form time coordinate, and $z$ is an instant form Cartesian coordinate. After expanding and simplifying, this yields

$$ (x^+)' = \frac{\gamma}{c^2}[ct(c^2 - v)+z(c^2+v)] $$

I've been trying to figure out how to manipulate this to see it as a kinematic transformation of $x^+$, but I fail to see how to do so. I was trying to somehow factor things so that we have $ct + z$ multiplied by some other factor, which could of course then be reduced to $x^+$ times some other factor.

Any hints or additional insight would be very much appreciated!

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2 Answers 2

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Here are some suggestions:

  1. Start in the usual (t,z) coordinates. Find the eigenvectors and eigenvalues of a boost [matrix]. [This leads you to the $x^+$, $x^-$ coordinates... but the boost transformation for them is not the same as it is for the $t,z$ coordinates.]

  2. A related approach. In the usual (t,z) coordinates, How does the vector $\left(\begin{array}{c}1\\1\end{array}\right)$ transform under the boost? [insert factors of c, as needed]. Then, repeat for $\left(\begin{array}{c}1\\-1\end{array}\right)$.

  3. Consult a reference on the "Bondi k-calculus" (e.g. mine).

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See formula (39) : https://www.scielo.br/j/rbef/a/wh9hgMSRTJzvws8ZxzL8NNp/?format=pdf&lang=en

with :$\beta= tanh(\varphi)\;\;,\gamma=cosh(\varphi)\;\;, \beta\gamma = sinh(\varphi)$

https://en.wikipedia.org/wiki/Lorentz_transformation

we have:

$x'^{+}=\gamma(x^{+}-\beta x^{+})$

$ x'^{-}=\gamma(x^{-}+\beta x^{-}) $

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  • $\begingroup$ In the 2nd equation, it's $+ \ beta x^{-} ...$ $\endgroup$
    – The Tiler
    Nov 10, 2021 at 19:53
  • $\begingroup$ You do not need to leave a comment. By clicking on the link 'Edit' below your answer, you can edit your post and modify the 2nd equation by yourself. $\endgroup$
    – Christophe
    Nov 10, 2021 at 21:04
  • $\begingroup$ I still don't quite follow, as our answers don't match. If you expand yours out and compare it to mine, the signs differ. I don't mean to imply that either one of us is wrong, but do you know why my method doesn't lead us to the conclusion in your answer? $\endgroup$
    – STDK
    Nov 10, 2021 at 23:19
  • $\begingroup$ In the link it says: " This result tells us that Lorentz transformations in the light front are very peculiar, behaving just like a kind of “scaling” factor. It is rather peculiar in that it does not mix different coordinates like it happens in the usual Minkowski space-time Lorentz transformation.If the frame in movement is going in the op." $\endgroup$
    – The Tiler
    Nov 11, 2021 at 7:34

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