无变量分离的克尔准正规模式：带物理信息神经网络的二维双曲面Teukolsky求解器
Kerr Quasinormal Modes without Variable Separation: A Two-Dimensional Hyperboloidal Teukolsky Solver with Physics-Informed Neural Networks
AI总结：
研究用物理信息神经网络（PINNs）构建无变量分离的二维双曲面Teukolsky求解器，求解克尔准正规模式，精度良好，为非克尔黑洞等的扰动本征问题提供了灵活的数值方法。
AI中文摘要：
我们利用物理信息神经网络（PINNs），直接在Teukolsky方程的二维双曲面表述中求解克尔时空的引力准正规模式（QNM）本征值问题。该表述无需变量分离，因此保留了耦合的径向-角向结构。这类方案为计算克尔黑洞之外的黑洞QNM提供了原型，这类黑洞的扰动方程是不可分离的。我们构建了角动量递增的序列，接近极值极限。我们重点关注基模（ℓ,m,n）=(2,0,0)、(2,1,0)、(2,2,0)、(3,3,0)和(4,4,0)，以及第一泛音（2,2,1）。独立基准评估表明，所有报告的实频和虚频分量的误差均低于0.5%，中位数偏差为0.1%。在近极值区域，阻尼率变小、模式寿命最长，该精度仍得以保持。这些结果为多维黑洞扰动本征问题建立了非谱数值路径，虽未达到专用克尔求解器的更高精度，但提供了更大灵活性且需要更少分析预处理。不可分离的旋转背景和耦合系统，如引力-电磁克尔-纽曼扰动，是该构造的自然扩展。
英文摘要：
We use physics-informed neural networks (PINNs) to solve the gravitational quasinormal-mode (QNM) eigenvalue problem for Kerr spacetime directly in the two-dimensional hyperboloidal formulation of the Teukolsky equation. This formulation does not require separation of variables and thus retains the coupled radial--angular structure. Such a scheme provides a prototype for calculating the QNMs of beyond-Kerr black holes for which the perturbation equations are non-separable. Sequences with increasing angular momentum are constructed, reaching close to the extremal limit. We focus on the fundamental modes $(\ell,m,n)=(2,0,0)$, $(2,1,0)$, $(2,2,0)$, $(3,3,0)$ and $(4,4,0)$, together with the first overtone $(2,2,1)$. Independent benchmark evaluation shows that every reported real and imaginary frequency component remains below $0.5\%$ error, with a median deviation of $0.1\%$. This accuracy is maintained in the near-extremal regime, where the damping rate becomes small and the modes are longest-lived. The results establish a non-spectral numerical route to multidimensional black-hole perturbation eigenproblems which does not match the substantially higher precision of dedicated Kerr solvers but offers greater flexibility and requires less analytical pre-processing. Non-separable rotating backgrounds and coupled systems, such as gravitational--electromagnetic Kerr--Newman perturbations, are natural extensions of the same construction.
