Spectral dimensionality of spacetime around a radiating Schwarzschild black-hole

Abstract

In this work we study the spectral dimensionality of spacetime around a radiating Schwarzschild black hole using a recently introduced formalism of quantum gravity, where the alterations of the gravitational field produced by the radiation are represented on an extended manifold, and describe a non-commutative and non-linear algebra. The ration between classical and quantum perturbations of spacetime can be measured by the parameter $z \geq 0$. When $z=(1+\sqrt{3})/2\simeq 1.3660$, a relativistic observer approaching the Schwarzschild horizon perceives a spectral dimension $N(z)=4\left[\theta(z)-1\right]\simeq 2.8849$. Under these conditions, all studied Schwarzschild black holes with masses ranging from the Planck mass to $10^{46}$ times the Planck mass, present the same stability configuration which suggests the existence of an universal property of these objects under those particular conditions. The difference from the spectral dimension previously obtained at cosmological scales leads to the conclusion that the dimensionality of spacetime is scale-dependent. Another important result presented here, is the fundamental alteration of the effective gravitational potential near the horizon due to Hawking radiation. This quantum phenomenon prevents the potential from diverging to negative infinity as the observable approaches the Schwarzschild horizon.