AI 中文总结
该研究证明，当临界曲线段在屏幕上的位置和取向未知时，可从满足$0<a<1$、$0<i\leq\pi/2$的旋转非极端克尔黑洞的一段连通正长度临界曲线，唯一确定其无量纲自旋参数$a$与倾角$i$。
AI 中文摘要
高分辨率观测有望探测黑洞图像的光子环结构。临界曲线是观察者屏幕上的几何定义的极限轨迹，连续高阶光子子环会向其汇聚。对于克尔黑洞，该极限曲线的形状取决于无量纲自旋参数$a$和倾角$i$。我们研究当临界曲线段在屏幕上的位置和取向未知时，能否从一段连通且长度为正的临界曲线唯一确定这些参数。对于满足$0<a<1$、$0<i\leq\pi/2$的旋转非极端克尔黑洞，我们证明了全局唯一性：若两段此类曲线在屏幕的保定向刚体运动后作为点集重合，则它们的克尔参数对重合，且两个刚体运动完全相同。该证明利用不可约隐式多项式，表明一段曲线可确定包含它的完整代数曲线，再从该多项式的刚体运动不变量中恢复参数对$(a,i)$。
英文摘要
High-resolution observations are expected to probe the photon-ring structure of black hole images. The critical curve is the geometrically defined limiting locus on the observer's screen toward which successive higher-order photon subrings accumulate. For a Kerr black hole, the shape of this limiting curve depends on the dimensionless spin parameter $a$ and the inclination angle $i$. We investigate whether these parameters can be determined uniquely from a connected positive-length segment of the critical curve when the segment's position and orientation on the screen are unknown. For rotating non-extremal Kerr black holes with $0<a<1$ and $0<i\leqπ/2$, we prove global uniqueness: if two such segments coincide as point sets after orientation-preserving rigid motions of the screen, then their Kerr parameter pairs coincide and the two rigid motions are identical. The proof uses an irreducible implicit polynomial to show that a segment determines the full algebraic curve containing it and then recovers the parameter pair $(a,i)$ from rigid-motion invariants of that polynomial.
Comments29 pages
