Quasinormal modes of Proca fields in a Schwarzschild-AdS spacetime

We present new results concerning the Proca massive vector field in a Schwarzschild-anti-de Sitter (Schwarzschild-AdS) black hole geometry. We provide a first principles analysis of Proca vector fields in this geometry using both the vector spherical harmonic (VSH) decomposition and separation method and the Frolov-Krtous-Kubiznak-Santos (FKKS) method that separates the relevant equations in spinning geometries. The analysis in the VSH method shows, on one hand, that it is arduous to separate the scalartype from the vector-type polarizations of the electric sector of the Proca field, and on the other hand, it displays clearly the electric and the magnetic mode sectors. The analysis in the FKKS method is performed by taking the nonrotating limit of the Kerr-AdS spacetime, and shows that the Ansatz decouples the polarizations in the electric mode sector even in the nonrotating limit. On the other hand, it captures only two of the three possible polarizations; indeed, the magnetic mode sector, which is of vector-type, is missing. The reason for the absence of the remaining polarization is related to the degeneracy of the principal tensor in static spherical symmetric spacetimes. The degrees of freedom and quasinormal modes in both separation methods of the Proca field are found. The frequencies of the quasinormal modes are also carefully computed. For the electric mode sector in the VSH method the frequencies are found through an extension, which substitutes number coefficients by matrix coefficients, of the Horowitz-Hubeny numerical procedure, whereas for the magnetic mode sector in the VSH method and the electric sector of the FKKS method it is shown that a direct use of the procedure can be made. The values of the quasinormal mode frequencies obtained for each method are compared and shown to be in good agreement with each other. This further supports the analytical approaches presented here for the behavior of the Proca field in a Schwarzschild-AdS black hole background.

Automated summary

We present new results on the behavior of Proca massive vector fields in a Schwarzschild-anti-de Sitter black hole geometry. The analysis is performed using both vector spherical harmonic decomposition and the Frolov-Krtous-Kubiznak-Santos method. The results show that it is difficult to separate the scalar-type and vector-type polarizations of the electric sector in the vector spherical harmonic method. On the other hand, the Frolov-Krtous-Kubiznak-Santos method captures only two of the three possible polarizations, missing the vector-type magnetic mode sector. The frequencies of the quasinormal modes are carefully computed and compared between the two methods, showing good agreement.