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

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!

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).

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

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

we have:

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

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

• In the 2nd equation, it's $+ \ beta x^{-} ...$ Nov 10, 2021 at 19:53
• 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. Nov 10, 2021 at 21:04
• 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?
– STDK
Nov 10, 2021 at 23:19
• 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." Nov 11, 2021 at 7:34