Asymptotic safety, asymptotic darkness, and the hoop conjecture in the extreme UV

Assuming the hoop conjecture in classical general relativity and quantum mechanics, any observer who attempts to perform a localized experiment in an arbitrarily small region will be stymied by the formation of a trapped surface within the spatial domain of the experiment. This thought experiment is frequently invoked in arguments for a fundamental minimum length in physics, which in turn is usually considered to be fairly independent of observer or experimental setup. We examine this conclusion in asymptotically safe gravity by modifying a proof of the hoop conjecture for spherically symmetric systems in general relativity to include higher curvature terms in the effective action as well as running couplings. We show that the modified proof fails, and so the argument for the mandatory formation of a trapped surface within the domain of an experiment also falls apart in this context. However, neither is there any contrary proof that local trapped surfaces do not form. Instead, in this approach whether or not an observer can perform local measurements in arbitrarily small regions depends on the specific numerical values of the couplings near the UV fixed point. In this sense, there is no longer any purely local version of the minimum length argument. However, when an experiment is localized to be much smaller than the Planck length we argue that at least one trapped surface must still form outside the experiment. This enshrouding horizon precludes any local information from reaching infinity and so there is still an effective minimum length for observers at infinity.