AI 中文总结
研究一般μ方案下协变圈量子黑洞BH-I、BH-II的光子球与阴影，解析推导相关量表达式，利用EHT测量约束参数空间，发现BH-II约束更紧。
AI 中文摘要
我们研究了在一般μ方案中构建的两类协变量子修正黑洞度规（以下称为BH-I和BH-II）的黑洞阴影与光子球性质，该方案由幂律指数s和振幅ξ参数化。向一般s（包括非整数）的扩展是μ₀方案（s=0）和μ̄方案（s=1）之间的唯象插值，揭示了现有文献中通常将s固定为1时被掩盖的丰富唯象学。对于BH-I，光子球在所有参数值下均存在，且在s=1时呈现精确抵消，此时坐标位置r_ph=3M对任意ξ均恢复。对于BH-II，视界和光子球在(ξ,s)参数平面中均呈现临界曲线；当ξ超过闭式临界值ξ_c^PS(s)时，光子球消失。我们解析证明，在两类度规的整个物理参数空间中，光子球始终不稳定（λ_ph>0），并推导了李雅普诺夫指数的闭式表达式。系统的参数扫描显示，对于固定ξ，BH-II的阴影半径随s单调减小，而BH-I的阴影半径随s非单调变化。我们推导了光子球半径、阴影半径和李雅普诺夫指数的小参数解析展开式，证明单次阴影测量在二维(ξ,s)空间中存在观测简并；通过同时测量李雅普诺夫指数可打破该简并。利用事件视界望远镜（EHT）的测量结果，我们推导了对(ξ,s)参数空间的约束，发现BH-II的约束比BH-I紧约2至4倍。
英文摘要
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.
Comments31 pages, 7 figures
