标题： Proca场在Schwarzschild-AdS$_5$时空中的准正规模通过等单环方法研究 显示英文标题

标题： Quasinormal modes of a Proca field in Schwarzschild-AdS$_5$ spacetime via the isomonodromy method

摘要： 我们研究了五维Schwarzschild-anti-de Sitter（Schwarzschild-AdS$_{5}$）黑洞几何中的Proca场扰动。利用矢量球谐函数（VSH）方法，我们表明Proca场根据其在三维球面上的张量行为分解为标量类型和向量类型分量。场的两个自由度由标量类型的分量描述，由于质量项这些分量是耦合的，而剩下的两个自由度由一个完全解耦的向量类型的分量描述。受Frolov-Krtouš-Kubizňák-Santos（FKKS）零自旋极限下的假设启发，我们通过场变换来解耦标量类型的分量，代价是引入了一个复分离参数$\beta$。这个参数可以解析确定，并且其值对应于标量类型部分的两种不同极化："电磁"和"非电磁"，分别记为$\beta_{+}$和$\beta_{-}$。在标量类型部分，每种极化的径向微分方程是具有五个奇点的Fuchs型微分方程；而在向量类型部分，径向方程有四个奇点。通过等单值性方法，我们将边值问题重新表述为Painlevé VI$\tau$函数的初始条件问题，并利用$\tau$函数的级数展开计算了小视界极限下的标量类型和向量类型的准正规模（QNMs）。我们的结果与数值积分方法得到的结果总体上非常一致。这表明等单值性方法是一种在小视界极限下以高精度计算准正规模的可靠方法。 显示英文摘要

摘要： We consider Proca field perturbations in a five-dimensional Schwarzschild-anti-de Sitter (Schwarzschild-AdS$_{5}$) black hole geometry. Using the vector spherical harmonic (VSH) method, we show that the Proca field decomposes into scalar-type and vector-type components according to their tensorial behavior on the three-sphere. Two degrees of freedom of the field are described by scalar-type components, which are coupled due to the mass term, while the remaining two degrees of freedom are described by a vector-type component, which decouples completely. Motivated by the Frolov-Krtou\v{s}-Kubiz\v{n}\'{a}k-Santos (FKKS) ansatz in the limit of zero spin, we use a field transformation to decouple the scalar-type components at the expense of introducing a complex separation parameter $\beta$. This parameter can be determined analytically, and its values correspond to two distinct polarizations of the scalar-type sector: "electromagnetic" and "non-electromagnetic", denoted by $\beta_{+}$ and $\beta_{-}$, respectively. In the scalar-type sector, the radial differential equation for each polarization is a Fuchsian differential equation with five singularities, whereas in the vector-type sector, the radial equation has four singularities. By means of the isomonodromy method, we reformulate the boundary value problem in terms of the initial conditions of the Painlev\'{e} VI $\tau$ function and, using a series expansion of the $\tau$ function, we compute the scalar-type and vector-type quasinormal modes (QNMs) in the small horizon limit. Our results are in overall very good agreement with those obtained via the numerical integration method. This shows that the isomonodromy method is a reliable method to compute quasinormal modes in the small horizon limit with high accuracy.