{
  "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 $μ_0$-scheme ($s=0$) and $\\barμ$-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 $ξ$. 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 $ξ_c^{\\rm PS}(s)$. We prove analytically that the photon sphere is always unstable ($λ_{\\rm ph}>0$) throughout the physical parameter space of both metrics, and derive closed-form expressions for the Lyapunov exponent. Systematic parameter scans reveal that for fixed $ξ$ 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.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2608.28186",
  "author_total": 3,
  "authors": [
    {
      "name": "Yu Han"
    },
    {
      "name": "Meng Liu"
    },
    {
      "name": "Yongzhuang Li"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8bfn8jz41qmd1.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2608.28186",
  "fulltext_url": null,
  "id": "8bfn8jz41qmd1",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2608.28186",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "gr-qc"
    ],
    [
      "openalex",
      "3106|Black Holes and Theoretical Physics"
    ]
  ],
  "openalex": "W7204802557",
  "pdf_url": "https://arxiv.org/pdf/2608.28186",
  "pmcid": null,
  "pmid": null,
  "published": "2026-08-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8bfn8jz41qmd1.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:17:22Z",
      "native_id": "2608.28186",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2608.28186"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:gr-qc",
      "subfield": "gravitation"
    }
  ],
  "title": "Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general $μ$-scheme",
  "title_text": "Photon Spheres and Shadows of Covariant Loop Quantum Black Holes in the general μ-scheme",
  "type": "preprint",
  "updated_at": "2026-10-30T12:17:22Z",
  "url": "https://research.draconitic.com/paper/8bfn8jz41qmd1",
  "venue": "arXiv",
  "versions": []
}