Scaling Limit of the Noncommutative Black Hole

We show that the recent quantum' black hole wave operator in the $\kappa$-Minkowski or bicrossproduct model quantum spacetime has a natural scaling limit $\lambda_p\to 0$ at the event horizon. Here $\lambda_p$ is the Planck time and the geometry at the event horizon in Planck length is maintained at the same time as the limit is taken, resulting in a classical theory with quantum gravity remnants. Among the features is a frequency-dependent skin’ of some $\frac{\omega }{\nu }$ Planck lengths just inside the event horizon for $\omega >0$ and just outside for $\omega <0$, where $\nu$ is the frequency associated to the Schwarzschild radius. We use bessel and hypergeometric functions to analyse propagation through the event horizon and skin in both directions. The analysis confirms a finite redshift at the horizon for positive frequency modes in the exterior.