Quantum Schwarzschild Black Hole Optical Aspects

We investigate the optical behavior of a quantum Schwarzschild black hole with a space-time solution including a parameter \(\lambda\) that encodes its discretization. Specifically, we derive the effective potential of this solution. In particular, we study circular orbits around the quantum black hole. Indeed, we find that the effective potential is characterized by a minimum and a maximum yielding double photon spheres denoted by \(r_{p_{1}}\) and \(r_{p_{2}}\). Then, we analyze the double shadow behavior as a function of the parameter \(\lambda\), where we show that it controls the shadow circular size. An inspection of the Innermost Stable Circular Orbits (ISCO) shows that the radius \(r_{\textrm{ISCO}}\) increases as a function of \(\lambda\). Besides, we find that this radius is equal to \(6M\) for an angular momentum \(L=2\sqrt{3}\) independently of \(\lambda\). A numerical analysis shows that the photon sphere of radius \(r_{p_{1}}\) generates a shadow with a radius larger than \(r_{\textrm{ISCO}}\). Thus, a truncation of the effective potential is imposed to exclude such behavior. Finally, the \(\lambda\)-effect is inspected depending on the deflection angle of such a black hole, showing that it increases when higher values of the parameter \(\lambda\) are considered. However, such an increase is limited by an upper bound given by \({6M}/{b}\).