Perturbations of slowly rotating black holes: massive vector fields in the Kerr metric

TL;DR

We address the challenge of studying linear perturbations of slowly rotating black holes when the perturbation equations are nonseparable. By developing a general slow-rotation framework, we apply it to massive vector (Proca) perturbations in Kerr, deriving equations up to second order in rotation and uncovering Zeeman-like mode splittings and axial–polar couplings. We demonstrate, for the first time, a Proca superradiant instability in Kerr, which yields stringent astrophysical bounds on the vector field mass, translating into mv ≲ 4×10^{-20} eV under conservative assumptions and potentially mv ≲ 10^{-22} eV with optimistic spin measurements. The results establish a robust, broadly applicable perturbative method for nonseparable perturbations and show that SMBH spin observations offer powerful probes of light vector fields and related beyond-GR physics.

Abstract

We discuss a general method to study linear perturbations of slowly rotating black holes which is valid for any perturbation field, and particularly advantageous when the field equations are not separable. As an illustration of the method we investigate massive vector (Proca) perturbations in the Kerr metric, which do not appear to be separable in the standard Teukolsky formalism. Working in a perturbative scheme, we discuss two important effects induced by rotation: a Zeeman-like shift of nonaxisymmetric quasinormal modes and bound states with different azimuthal number m, and the coupling between axial and polar modes with different multipolar index l. We explicitly compute the perturbation equations up to second order in rotation, but in principle the method can be extended to any order. Working at first order in rotation we show that polar and axial Proca modes can be computed by solving two decoupled sets of equations, and we derive a single master equation describing axial perturbations of spin s=0 and s=+-1. By extending the calculation to second order we can study the superradiant regime of Proca perturbations in a self-consistent way. For the first time we show that Proca fields around Kerr black holes exhibit a superradiant instability, which is significantly stronger than for massive scalar fields. Because of this instability, astrophysical observations of spinning black holes provide the tightest upper limit on the mass of the photon: mv<4x10^-20 eV under our most conservative assumptions. Spin measurements for the largest black holes could reduce this bound to mv<10^-22 eV or lower.

Figures (9)

• Figure 1: (color online) Contour plots in the BH Regge plane Arvanitaki:2010sy corresponding to an instability timescale shorter than a typical accretion timescale, $\tau_{\rm Salpeter}=4.5\times 10^7$ yr, for different values of the vector field mass $m_v={{\mu}}\hbar$ (from left to right: $m_v=10^{-18}{\rm eV}$, $10^{-19}{\rm eV}$, $10^{-20}{\rm eV}$, $2\times10^{-21}{\rm eV}$). For polar modes we consider the $S=-1$ polarization, which provides the strongest instability, and we use both Eq. \ref{['fitII']} (fit II, top panel) and Eq. \ref{['fit']} (fit I, middle panel), and for axial modes we use Eq. \ref{['fit']} (bottom panel). Dashed lines bracket our estimated numerical errors. The experimental points (with error bars) refer to the mass and spin estimates of supermassive BHs listed in Table 2 of Brenneman:2011wz; the rightmost point corresponds to the supermassive BH in Fairall 9 Schmoll:2009gq. Supermassive BHs lying above each of these curves would be unstable on an observable timescale, and therefore they exclude a whole range of Proca field masses.
• Figure 2: (color online) Percentage difference between QNMs of massless vector perturbations in the slow-rotation limit at first order and the "full" numerical solution of the Teukolsky equation in the Kerr metric Berti:2009kk. The deviation scales like $\tilde{a}^2$ as $\tilde{a}\to0$.
• Figure 3: (color online) Comparison between the exact, Teukolsky-based result and the results obtained by our slowly rotating approximation at first and second order for the imaginary part of the scalar fundamental bound-state mode with $\ell=m=1$ and $\mu M=0.1$. For comparison, we have also computed the same mode as obtained by expanding the Teukolsky equation at third and fourth order.
• Figure 4: (color online) Bound state modes obtained with a first-order Breit-Wigner method applied to the full system. We show the determinant $|\det\mathbf{S}_m|$ as a function of the real part of the frequency for $M\mu=0.1$ and $\tilde{a}=0.1$. According to Eq. \ref{['fit_wR']}, the real part of modes with the same $\ell+n+S$ is approximately degenerate for $M\mu\ll 1$.
• Figure 5: (color online) Absolute value of the imaginary part of the axial (left panel) and of the polar $S=-1$ (right panel) vector modes as a function of the BH rotation rate $\tilde{a}$ for $\ell=1$, $M\mu=0.05$ and different values of $m$, computed at first order. For $m=1$, the modes cross the axis and become unstable when the superradiance condition \ref{['superradiance']} is met. In the left panel, the red dot-dashed line denotes the analytical result \ref{['ana_l=1']}.