Self-dual black holes in loop quantum gravity: Theory and phenomenology

Abstract

In this paper we have recalled the semiclassical metric obtained from a classical analysis of the loop quantum black hole (LQBH). We show that the regular Reissner--Nordstr\om-like metric is self-dual in the sense of $T$-duality: the form of the metric obtained in loop quantum gravity is invariant under the exchange $r\ensuremath{\rightarrow}{a}_{0}/r$ where ${a}_{0}$ is proportional to the minimum area in loop quantum gravity and $r$ is the standard Schwarzschild radial coordinate at asymptotic infinity. Of particular interest, the symmetry imposes that if an observer in $r\ensuremath{\rightarrow}+\ensuremath{\infty}$ sees a black hole of mass $m$ an observer in the other asymptotic infinity beyond the horizon (at $r\ensuremath{\approx}0$) sees a dual mass ${m}_{P}/m$. We then show that small LQBH are stable and could be a component of dark matter. Ultralight LQBHs created shortly after the big bang would now have a mass of approximately ${10}^{\ensuremath{-}5}{m}_{P}$ and emit radiation with a typical energy of about ${10}^{13}--{10}^{14}\text{ }\text{ }\mathrm{eV}$ but they would also emit cosmic rays of much higher energies, albeit few of them. If these small LQBHs form a majority of the dark matter of the Milky Way's Halo, the production rate of ultra-high-energy-cosmic-rays (UHECR) by these ultralight black holes would be compatible with the observed rate of the Auger detector.