A black hole cast on a non-commutative background

Abstract

In this work we describe a black hole, set on a non-commutative background. The model, which is relatively simple, is an exact solution of the Einstein Field Equations. Based on a proposition we put forward, we argue that introducing a matter density field on a non-commutative background sets up a mechanism that deforms the field into two distinct fields, one residing dominantly on the lattice tops (hereafter, on-cell) and the other residing dominantly in the inter-lattice regions (hereafter, off-cell). The two fields have different physical and themodynamic characterics which we describe, and some of which play a role in halting collpse to a singularity. For example, not surprisingly the on-cell (off-cell) fields manifest standard on-shell (off-shell) characteristics, respectively. Both the density and the net mass-energy are unchanged by the deformation mechanism. In our treatment the mass of a black hole defines its own size scale L of the interior region it occupies. Moreover, such a length is quantized, L=2N\sqrt{\theta}, in terms of a minimum length scale \sqrt{\theta}. The approach has the advantage that there is no degeneracy in mass-confinement since, here, the black hole density is not a function of the mass (as is the case in some recent treatments). The density is, instead a fixed quantity. As such, the approach puts an upper bound on black hole density, making it a universal parameter. The picture that emerges is that a black hole defined on a non-commutative background is both non-singular, holographic and quantized. Yet we also find, interestingly, that when taken over L the average value of the associated energy-stress tensor of the fields satisfies all classical energy conditions of GR.