Self-complete and GUP-Modified Charged and Spinning Black Holes

We explore some implications of our previous proposal, motivated in part by the Generalised Uncertainty Principle (GUP) and the possibility that black holes have quantum mechanical hair that the ADM mass of a system has the form $M + βM_\mathrm{Pl}^2/(2M)$, where $M$ is the bare mass, $M_\mathrm{Pl}$ is the Planck mass and $β$ is a positive constant. This also suggests some connection between black holes and elementary particles and supports the suggestion that gravity is self-complete. We extend our model to charged and rotating black holes, since this is clearly relevant to elementary particles. The standard Reissner-Nordström and Kerr solutions include zero-temperature states, representing the smallest possible black holes, and already exhibit features of the GUP-modified Schwarzschild solution. However, interesting new features arise if the charged and rotating solutions are themselves GUP-modified. In particular, there is an interesting transition below some value of $β$ from the GUP solutions (spanning both super-Planckian and sub-Planckian regimes) to separated super-Planckian and sub-Planckian solutions. Equivalently, for a given value of $β$, there is a critical value of the charge and spin above which the solutions bifurcate into sub-Planckian and super-Planckian phases, separated by a mass gap in which no black holes can form.