Quasinormal modes of massive fermions in Kerr spacetime: Long-lived modes and the fine structure

Quasinormal modes of a massive Dirac field were calculated for various static black hole backgrounds with the help of the WKB formula. These estimations, however, are rough and valid only for very small values of $\mu M$, where $M$ and $\mu$ are mass of the black hole and field respectively. Thus, no accurate calculations of massive Dirac modes are known even for the Schwarzschild black hole and this is all the more so for the Kerr solution. Here we calculate quasinormal modes of a massive Dirac field in the Kerr background. We have shown that the infinitely long-lived quasinormal modes (quasiresonances), which exist for boson fields, appear also in the fermions’ quasinormal spectrum. Two chiralities of massive fermions lead to an additional “fine structure” in the spectrum. We discuss the effect of this fine structure on the behavior of quasiresonances and the stability. The analysis is also extended to a charged massive field in the Kerr-Newman background.

Figure 1

$\mathrm{Re}(\omega )$ (left) and $\mathrm{Im}(\omega )$ (right) for the fundamental ($n=0$, $\ell =1/2$) quasinormal mode of the neutral, massive Dirac field in the Schwarzschild background: $\sigma =1/2$ (blue) and $\sigma =-1/2$ (cyan). Positive value of the chirality corresponds to higher oscillation frequency and slower decay.

Figure 2

$\mathrm{Re}(\omega )$ (left) and $\mathrm{Im}(\omega )$ (right) for the fundamental quasinormal mode ($\ell =1/2$, $n=0$) of the neutral Dirac field ($\sigma =1/2$) as a function of the field mass ($\mu r_{+}$) in the Reissner-Nordström black hole background: $Q=0$ (top, blue), $Q=0.5r_{+}$ (green), $Q=0.95r_{+}$ (bottom, red).

Figure 3

$\mathrm{Re}(\omega )$ (left) and $\mathrm{Im}(\omega )$ (right) for the fundamental quasinormal mode ($\ell =1/2$, $n=0$) of the neutral ($e=0$), massive ($\mu r_{+}=0.1$) Dirac field in the Kerr ($Q=0$) background: $\sigma =1/2$, $m=1/2$ (red); $\sigma =1/2$, $m=-1/2$ (blue); $\sigma =-1/2$, $m=1/2$ (magenta); $\sigma =-1/2$, $m=-1/2$ (cyan). The positive value of the chirality corresponds to higher oscillation frequency and slower decay in the nonrotating limit. The positive azimuthal value leads to higher oscillation frequency and slower decay for the rotating black hole. Yet, for the near-extremal rotation $\sigma =-1/2$, $m=1/2$ has the smallest, almost vanishing, decay rate.

Figure 4

$\mathrm{Re}(\omega )$ (left) and $\mathrm{Im}(\omega )$ (right) for the fundamental quasinormal mode ($\ell =1/2$, $n=0$) of the massive ($\mu r_{+}=0.2$) Dirac field as a function of the field charge ($eQ$) for the Reissner-Nordström black hole ($Q=0.95r_{+}$): $\sigma =1/2$ (blue) and $\sigma =-1/2$ (cyan). The horizontal axis corresponds to the asymptotic value ($\omega r_{+}=eQ-0.024375i$) given by formula (20).