What is the kinematic transformation resulting from a Lorentz boost in light front coordinates?

Question

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!

Answer 1

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).

Answer 2

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^{-}) $