Massive Vector Fields in Rotating Black-Hole Spacetimes: Separability and Quasinormal Modes

We demonstrate the separability of the massive vector (Proca) field equation in general Kerr-NUT-AdS black-hole spacetimes in any number of dimensions, filling a long-standing gap in the literature. The obtained separated equations are studied in more detail for the four-dimensional Kerr geometry and the corresponding quasinormal modes are calculated. Two of the three independent polarizations of the Proca field are shown to emerge from the separation ansatz and the results are found in an excellent agreement with those of the recent numerical study where the full coupled partial differential equations were tackled without using the separability property.

Figure 1

Quasinormal modes. We display the frequency $\omega M$ as a function of $mM$ computed for quasinormal modes with $m_{\varphi }=1$ and $a/M=0.998$. The real (left) and imaginary (right) part of $\omega M$ is displayed for the two polarizations $S=-1$, $+1$, with the top (online blue) curve corresponding to $S=-1$, and the lower (online green) one to $S=+1$. The polarization $S=-1$ corresponds to the most unstable mode. For this case, the (online orange) squares represent the data from [9], and the dashed line gives the analytic approximation (30) derived in [8]. Solving separated ODEs allowed us to extend the numerics below $mM\sim {10}^{-2}$, compared to the range $mM\gtrsim {10}^{-1}$ of [9]. The agreement of the analytic approximation with our numerics for small $mM$ is excellent.