Instability of the Noncommutative Geometry Inspired Black Hole

Non-commutative geometries have been proposed as an approach to quantum gravity and have led to the construction of non-commutative black holes, whose interior singularities are purportedly eliminated due to quantum effects. Here we find evidence that these black holes are in fact unstable, with infalling matter near the Cauchy (inner) horizon being subject to an infinite blueshift of the type that has been repeatedly demonstrated for the Reissner-Nordstr\(\ddot{\text{o}}\)m black hole. This instability is present even when an ultraviolet cutoff (induced by anticipated non-commutative geometric effects) to a field propagating in that spacetime is included. We demonstrate this by following an analogous argument made for Reissner-Nordstr\(\ddot{\text{o}}\)m black holes, and conclude that stability is dependent on the surface gravities \(\kappa_-\) and \(\kappa_+\) of the inner and outer horizons respectively. In general if \(\kappa_- > \kappa_+\), as we show to be the case here, then the stability of the Cauchy horizon becomes highly questionable, contrary to recent claims.