Charged scalar field instability between the event and cosmological horizons

Recently, a new interesting instability of a charged scalar field in the Reissner-Nordström-de Sitter background has been found [Z. Zhu et al. Phys. Rev. D 90, 044042 (2014)] through the time-domain integration of the perturbation equation. We investigate further properties of this instability, confirm its existence by concordant frequency-domain and time-domain calculations, and show that it occurs at however small value of the coupling $eQ$, where $e$ and $Q$ are charges of a scalar field and black hole, respectively. We also investigate the parametric region of instability and show that the critical values of $eQ$ at which the stabilization happens strongly depends on the value of cosmological constant $\mathrm{\Lambda }$ and softly on $Q$. We show that all the unstable modes are superradiant, but not all the superradiant modes are unstable. We analytically prove that superradiance is a necessary (but not sufficient) condition for the instability in this case and, thereby, demonstrate the superradiant origin of the instability.

Figure 1

Real and imaginary part (oscillating profiles) together with the absolute value (nonoscillating profile) of the signal for $eQ=0.4$, $r_{-}=0.07r_{+}$, $r_{+}=0.186r_c$ ($Q=0.5M$, $\mathrm{\Lambda }{M}^2=0.023$). The dominant frequency is $\omega r_{+}=0.089160+0.000754i$.

Figure 2

Time-domain profile for $eQ=0.001$, $\mu =0$, $r_{-}=0$, $r_{+}=0.2r_c$ ($Q\to 0$, $\mathrm{\Lambda }{M}^2\approx 0.023$).

Figure 3

Time-domain profiles for $eQ=0.43$ (red, growing), $eQ=0.44$ (green), $eQ=0.45$ (blue, decaying), $\mu =0$, $r_{-}=0$, $r_{+}=0.2r_c$ ($Q\to 0$, $\mathrm{\Lambda }{M}^2\approx 0.023$).

Figure 4

Time-domain profiles for $eQ=0.51$ (red, growing), $eQ=0.52$ (green), $eQ=0.53$ (blue, decaying), $\mu =0$, $r_{-}=0.837r_{+}$, $r_{+}=0.106r_c$ ($Q\approx M$, $\mathrm{\Lambda }{M}^2\approx 0.023$).

Figure 5

Time-domain profiles for charged $eQ=0.4$ massless (black, upper) and massive: $\mu r_{+}=0.01$ (blue), $\mu r_{+}=0.02$ (green), and $\mu r_{+}=0.03$ (red, lower, stable) scalar fields in Reissner-Nordström-de Sitter background: $r_{-}=0.07r_{+}$, $r_{+}=0.186r_c$ ($Q=0.5M$, $\mathrm{\Lambda }{M}^2=0.023$).

Figure 6

The instability region for a weakly charged Reissner-Nordström-de Sitter black hole ($Q\to 0$, $eQ\ne 0$).