The planck length is defined as $l_P = \sqrt{\frac{\hbar G}{c^3}}$. So it is a combination of the constants $c, h, G$ which I believe are all Lorentz invariants. So I think the Planck length should also be a Lorentz invariant!
But there seem to be some confusion about that, see e.g. the following paper
Magueijo 2001: Lorentz invariance with an invariant energy scale:
The combination of gravity $G$, the
quantum $h$ and relativity $c$ gives rise
to the Planck length, $l_p$ or its
inverse, the Planck energy $E_p$ . These
scales mark thresholds beyond which
the old description of spacetime
breaks down and qualitatively new
phenomena are expected to appear. …
This gives rise immediately to a
simple question: in whose reference
frame are $l_P$ and $E_P$ the thresholds for
new phenomena?
But if $l_P$ is a Lorentz invariant their is no question about that. $l_P$ is the same in all reference frames! Another confusing issue is that the Planck mass (from which the Planck length is derived) is often derived by setting equal the Compton length $\lambda_C = \frac{h}{m_0 c}$ ( a Lorentz invariant 4-length) and the Schwarzschild length $r_{s} = \frac{2Gm}{c^2}$ (which I believe is not a Lorentz invariant, since in the derivation of the Schwarzschild metric it is assumed to be a 3-length, measuring a space distance). But since Compton wavelength and Schwarzschild radius are not lengths of the same kind I think such a derivation is not valid.
So my question is:
Is the Planck length a Lorentz invariant and if so, how to derive it then without using the Compton wave length and the Schwarzschild radius ?
Best Answer
A possible answer to the last part of the question: the article Six Easy Roads to the Planck Scale, Adler, Am. J. Phys., 78, 925 (2010) contains multiple "derivations" that you might (or might not) find more satisfactory than the one you mention.
As far as the rest of the question is concerned, others have made the most relevant points. I think a fair summary of what Magueijo is getting at is something like the following:
One frequently hears that "interesting new physics" happens when some length $l$ is less than the Planck length. The Planck length is manifestly Lorentz invariant. The other length $l$, if it is the physical length of some object, is manifestly not Lorentz invariant. What meaning, then, can one assign to such statements?
It seems to me that reasonable people can differ over whether this is an interesting question. I don't find it manifestly insane, myself.