Reconcile Planck-scale discreteness and the Lorentz-Fitzgerald contraction
Carlo Rovelli, Simone Speziale
TL;DR
The paper resolves the apparent tension between Planck-scale discreteness and Lorentz contraction by analyzing the area operator in loop quantum gravity. It shows that the minimal observable area $A_0$ is a spectral edge, not a fixed geometric constant, so boosts do not contract this value; instead, the boosted observable $A'$ remains noncommuting with $A$, leading to frame-dependent probability distributions over area eigenvalues. A local boost generator $M(\beta)$ is proposed, and its action can be implemented unitarily ($U(\beta)=e^{-iM(\beta)}$) under suitable Hermiticity conditions, ensuring the same spectrum for $A$ and $A'$. The results clarify how Planck-scale discreteness can coexist with local Lorentz invariance in a diffeomorphism-invariant quantum gravity framework, with implications for how geometry is observed in quantum regimes.
Abstract
A Planck-scale minimal observable length appears in many approaches to quantum gravity. It is sometimes argued that this minimal length might conflict with Lorentz invariance, because a boosted observer could see the minimal length further Lorentz contracted. We show that this is not the case within loop quantum gravity. In loop quantum gravity the minimal length (more precisely, minimal area) does not appear as a fixed property of geometry, but rather as the minimal (nonzero) eigenvalue of a quantum observable. The boosted observer can see the same observable spectrum, with the same minimal area. What changes continuously in the boost transformation is not the value of the minimal length: it is the probability distribution of seeing one or the other of the discrete eigenvalues of the area. We discuss several difficulties associated with boosts and area measurement in quantum gravity. We compute the transformation of the area operator under a local boost, propose an explicit expression for the generator of local boosts and give the conditions under which its action is unitary.