Why the Length of a Quantum String Cannot Be Lorentz Contracted

We propose a quantum gravity-extended form of the classical length contraction law obtained in Special Relativity. More specifically, the framework of our discussion is the UV self-complete theory of quantum gravity. Against this background, we show how our results are consistent with, i) the generalized form of the Uncertainty Principle (GUP), ii) the so called hoop-conjecture which we interpret, presently, as the saturation of a Lorentz boost by the formation of a black hole in a two-body scattering, and iii) the intriguing notion of ``classicalization'' of trans-Planckian physics. Pushing these ideas to their logical conclusion, we argue that there is a physical limit to the Lorentz contraction rule in the form of some minimal universal length determined by quantum gravity, say the Planck Length, or any of its current embodiments such as the string length, or the TeV quantum gravity length scale. In the latter case, we determine the critical boost that separates the ordinary ``particle phase,'' characterized by the Compton wavelength, from the ``black hole phase'', characterized by the effective Schwarzschild radius of the colliding system. Interestingly enough, the geometric mean of these two fundamental lengths turns out to be the Planck Length. Finally, with the ``classicalization'' of quantum gravity in mind, we comment on the remarkable identity, to our knowledge never noticed before, between three seemingly independent universal quantities, namely, a) the ``string tension'', b) the ``linear energy density'' or tension that exists at the core of all Schwarzschild black holes, and c) the ``superforce'' i.e., the Planckian limit of the static electro-gravitational force and, presumably, the unification point of all fundamental forces.