Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general $\mu$-scheme

Abstract

We investigate black hole shadows and photon sphere properties for two families of covariant quantum-corrected black-hole metrics (hereafter called BH-I and BH-II) formulated within a general $\mu$-scheme, parameterised by a power-law exponent $s$ and an amplitude $\xi$. The extension to general (including non-integer) $s$ is a phenomenological interpolation between the $\mu_0$-scheme ($s=0$) and $\bar\mu$-scheme ($s=1$) and reveals a rich phenomenology masked when $s$ is usually fixed to 1 in previous literature. For BH-I, the photon sphere exists for all parameter values and exhibits an exact cancellation at $s=1$ where the coordinate location $r_{\rm ph}=3M$ is restored for any $\xi$. For BH-II, both the horizon and the photon sphere exhibit critical curves in the $(\xi,s)$ parameter plane; the photon sphere disappears when $\xi$ exceeds a closed-form critical value $\xi_c^{\rm PS}(s)$. We prove analytically that the photon sphere is always unstable ($\lambda_{\rm ph}>0$) throughout the physical parameter space of both metrics, and derive closed-form expressions for the Lyapunov exponent. Systematic parameter scans reveal that for fixed $\xi$ the shadow radius decreases monotonically with $s$ for BH-II and non-monotonically for BH-I. We derive small-parameter analytic expansions for the photon-sphere radius, shadow radius, and Lyapunov exponent, and demonstrate that a single shadow measurement suffers an observational degeneracy in the two-dimensional $(\xi,s)$ space; the degeneracy can be broken by a simultaneous measurement of the Lyapunov exponent. Using Event Horizon Telescope (EHT) measurements, we derive constraints on the $(\xi,s)$ parameter space and find that BH-II is constrained roughly $2$--$4$ times more tightly than BH-I.