Massive spin-2 fields on black hole spacetimes: Instability of the Schwarzschild and Kerr solutions and bounds on the graviton mass

TL;DR

This work investigates massive spin-2 fluctuations in ghost-free bimetric/massive gravity on black hole spacetimes, deriving master equations that describe a massive graviton propagating on Schwarzschild and slowly rotating Kerr backgrounds. It reveals a strong spherically symmetric monopole instability for Schwarzschild BHs and shows Kerr BHs are generically unstable as well due to superradiance, with the polar-dipole channel predicting the fastest growth. By computing quasinormal modes and quasibound states, the authors map out a rich spectrum and identify a hydrogenic pattern for many modes alongside a distinctive polar-dipole state with unusually long binding and rapid decay. These results yield a conservative bound on the graviton mass from spinning BH observations, μ ≲ 5×10^{-23} eV, and highlight BHs as powerful laboratories to test extensions of gravity and constrain ultralight tensor fields, while signaling the need for nonlinear studies to determine the final fate of instabilities.

Abstract

Massive bosonic fields of arbitrary spin are predicted by general extensions of the Standard Model. It has been recently shown that there exists a family of bimetric theories of gravity - including massive gravity - which are free of Boulware-Deser ghosts at the nonlinear level. This opens up the possibility to describe consistently the dynamics of massive spin-2 particles in a gravitational field. Within this context, we develop the study of massive spin-2 fluctuations - including massive gravitons - around Schwarzschild and slowly-rotating Kerr black holes. Our work has two important outcomes. First, we show that the Schwarzschild geometry is linearly unstable for small tensor masses, against a spherically symmetric mode. Second, we provide solid evidence that the Kerr geometry is also generically unstable, both against the spherical mode and against long-lived superradiant modes. In the absence of nonlinear effects, the observation of spinning black holes bounds the graviton mass to be smaller than 5x10^{-23} eV.

Figures (8)

• Figure 1: Details of the instability of Schwarzschild (de Sitter) BHs against spherically symmetric polar modes of a massive spin-2 field. The left panel shows the inverse of the instability timescale $\omega_I=1/\tau$ as a function of the graviton mass $\mu$ for different values of the cosmological constant $\Lambda_g=\Lambda_f$, including the asymptotically flat case $\Lambda_g=0$. Curves are truncated when the Higuchi bound is reached $\mu^2=2\Lambda_g/3$Higuchi1987397. For any value of $\Lambda_g$, unstable modes exist in the range $0<M\mu\lesssim 0.47$, the upper bound being only mildly sensitive to $\Lambda_g$. The right panel shows some eigenfunctions in the asymptotically flat case. The eigenfunctions decay exponentially at spatial infinity and are progressively peaked closer and closer to the BH horizon for masses close to the threshold mass $M\mu\sim 0.43$.
• Figure 2: QNM frequencies for axial $l=1,2$ modes, for a range of field masses $M\mu=0,0.04,\ldots,0.52$. Points with largest $|\omega_I|$ correspond to $\mu\to0$. The fundamental mode ($n=0$, circles) and the first overtones ($n=1$, triangles) are shown. In the massless limit the "vector" modes have the same QNM frequency as the electromagnetic field, and the "tensor" modes have the same QNM frequency as the massless gravity perturbations.
• Figure 3: Axial (Top) and polar (bottom) quasibound state levels of the massive spin-2 field. The left and right panels show the real part, $\omega_R/\mu$, and the imaginary part, $\omega_I/\mu$, of the mode as a function of the mass coupling $M\mu$, respectively. We label the modes by their angular momentum $l$, overtone number $n$ and spin projection $S$. Except for the polar dipole $l=1$, the spectrum is hydrogenic in the massless limit.
• Figure 4: Comparison between the numerical and analytical results for the the axial mode $l=1,\,S=1,\,n=0$ as a function of the mass coupling $M\mu$. The solid line shows the numerical data and the dashed shows the analytical formula \ref{['wI_ana']}.
• Figure 5: Absolute value of the imaginary part of the axial and polar quasibound modes as a function of the BH rotation rate $\tilde{a}$ for different values of $l$ and $m$ and different values of the mass coupling $\mu M$, computed at first order. Left top panel: axial dipole for $l=m=1$. Right top panel: axial mode $S=-1$ for a mass coupling $M\mu=0.15$ and different values of $m$. Left bottom panel: polar dipole mode for $l=m=1$. Right bottom panel: polar mode $l=m=2$, $S=-2$. For any mode with $m\geq 0$, the imaginary part crosses the axis and become unstable when the superradiance condition is met.