Quasinormal modes of a Proca field in Schwarzschild-AdS5 spacetime via the isomonodromy method

A bstract 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-Krtouš-Kubizňák-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 β . 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 β + and β − , 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 Painlevé VI τ function and, using a series expansion of the τ function, we compute the scalar-type and vector-type quasinormal modes (QNMs), focusing on the fundamental modes and the first two overtones as functions of the horizon radius and the mass of the Proca field. We observe the occurrence of a minimum in the real part of the frequencies, which can be explained by the behavior of the potential of the modes. The results using the isomonodromic method are in good agreement with the numerical integration method.