Photon Spheres and shadow of Schwarzschild black hole on the EUP framework

An explicit correspondence is established between the Extended Uncertainty Principle (EUP) and the metric function by directly relating the radiation temperature function modified by EUP to the modified spacetime metric. Utilising this modified metric, we subsequently derive the corresponding thermodynamic quantities of the black hole, and calculate the photon sphere radius and the size of the black hole shadow. The results of the study indicate that, in comparison with Schwarzschild black holes, the position of the event horizon remains constant under EUP modifications. However, the photon sphere radius increases with growing EUP parameters, while the shadow size decreases with increasing parameters, demonstrating that EUP induces optical shift phenomena. By comparing with observations of the galactic centre black hole $\text{Sgr}{\text{A}^{*}}$ from the Event Horizon Telescope, new constraints are established on EUP parameters.