Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general μ-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 μ-scheme, parameterised by a power-law exponent s and an amplitude ξ. The extension to general (including non-integer) s is a phenomenological interpolation between the μ0-scheme (s=0) and μ̅-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 rph=3M is restored for any ξ. For BH-II, both the horizon and the photon sphere exhibit critical curves in the (ξ,s) parameter plane; the photon sphere disappears when ξ exceeds a closed-form critical value ξcPS(s). We prove analytically that the photon sphere is always unstable (λph>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 ξ 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 (ξ,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 (ξ,s) parameter space and find that BH-II is constrained roughly $2−−4$ times more tightly than BH-I.
Related
- Naked Singularities and Black Hole Thermodynamics from Shadows and Photon Rings in the Quantum Oppenheimer-Snyder Model
- Photon rings and shadows of black holes with non-minimal couplings between curvature and electromagnetic field
- Photon Motion and Shadows of Rotating Black Holes with Nonlinear Electromagnetic and Anisotropic Matter Fields
- Covariant linear response theory for a photon gas in curved spacetime
- Photon spheres, elliptic surfaces, and Seiberg-Witten geometry
BibTeX
@misc{han2026photon,
title = {{Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general $μ$-scheme}},
author = {Yu Han and Meng Liu and Yongzhuang Li},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.28186},
eprint = {2608.28186},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.28186},
}
RIS
TY - UNPB
TI - Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general μ-scheme
AU - Yu Han
AU - Meng Liu
AU - Yongzhuang Li
PY - 2026
DA - 2026/08/28
PB - arXiv
DO - 10.48550/arxiv.2608.28186
UR - https://arxiv.org/abs/2608.28186
AB - 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 μ-scheme, parameterised by a power-law exponent s and an amplitude ξ. The extension to general (including non-integer) s is a phenomenological interpolation between the μ_0-scheme (s=0) and μ̅-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_ph=3M is restored for any ξ. For BH-II, both the horizon and the photon sphere exhibit critical curves in the (ξ,s) parameter plane; the photon sphere disappears when ξ exceeds a closed-form critical value ξ_c^PS(s). We prove analytically that the photon sphere is always unstable (λ_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 ξ 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 (ξ,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 (ξ,s) parameter space and find that BH-II is constrained roughly $2−−4$ times more tightly than BH-I.
ER -