Pseudospectrum of Reissner-Nordström black holes: Quasinormal mode instability and universality

Black hole spectroscopy is a powerful tool to probe the Kerr nature of astrophysical compact objects and their environment. The observation of multiple ringdown modes in gravitational waveforms could soon lead to high-precision gravitational spectroscopy, so it is critical to understand if the quasinormal mode spectrum is stable against perturbations. It was recently shown that the pseudospectrum can shed light on the spectral stability of black hole quasinormal modes. We study the pseudospectrum of Reissner-Nordström spacetimes and we find a spectral instability of scalar and gravitoelectric quasinormal modes in subextremal and extremal black holes, extending similar findings for the Schwarzschild spacetime. The asymptotic structure of pseudospectral contour levels is the same for scalar and gravitoelectric perturbations. By making different gauge choices in the hyperboloidal slicing of the spacetime, we find that the broad features of the pseudospectra are remarkably gauge independent. The gravitational-led and electromagnetic-led quasinormal modes of extremal Reissner-Nordström black holes exhibit “strong” isospectrality: not only their spectrum coincides, but the whole pseudospectrum is the same for both classes of perturbations. We observe that a conformal duality between the extremal horizon and spacetime boundaries at infinity is responsible for such strong isospectrality property.

Figure 1

Pseudospectra of a BH with charge $Q/M=0.5$. Top: $\ell =0$ scalar QNMs (red dots) and $\epsilon$-pseudospectra boundaries (white lines). Middle: $\ell =1$ electromagnetic-led QNMs and $\epsilon$-pseudospectra boundaries. Bottom: the same for $\ell =2$ gravitational-led QNMs. The contour levels ${\log }_{10}(\epsilon )$ range from $-55$ (top level) to $-5$ (bottom level) in steps of 5. The blue dots designate branch-cut (nonconvergent) modes.

Figure 2

Comparison of pseudospectral levels for gravitational-led $\ell =2$ QNMs with varying BH charge. Here, $Q/M=0$, 0.5 and 1 correspond to red, green and blue contours, respectively. The red and green contour levels ${\log }_{10}(\epsilon )$ range from $-55$ (top level) to $-5$ (bottom level) in steps of 5, while the blue ones from $-50$ (top level) to $-5$ (bottom level) in steps of 5.

Figure 3

Comparison of the $\ell =2$ gravitational-led $\epsilon$ pseudospectra for a RN BH with $Q/M=0.5$. The pseudospectra in red were computed in the areal radius fixing gauge, while those in blue where computed in the Cauchy horizon fixing gauge. In both cases, the contour levels ${\log }_{10}(\epsilon )$ range from $-55$ (top level) to $-5$ (bottom level) in increments of 5.

Figure 4

Top: $\ell =0$ scalar QNMs (red dots) and $\epsilon$-pseudospectra boundaries (white lines) of a RN BH with $Q/M=1$. The contour levels ${\log }_{10}(\epsilon )$ range from $-45$ (top level) to $-5$ (bottom level) in steps of 5. Bottom: enlarged region around the first few QNMs of the top panel. The contour levels ${\log }_{10}(\epsilon )$ range from $-23$ (top level) to $-3$ (bottom level) in steps of 2. The blue dots designate branch-cut (nonconvergent) modes.

Figure 5

Left: enlargement of the region around the real axis (shown as a dashed black line) of $\ell =0$ scalar $\epsilon$-pseudospectral boundaries (white lines) for a RN BH with $Q/M=0.5$. Right: the same, but for $Q/M=1$. A single zero-frequency QNM (shown as a red dot) exists in this case. The contour levels ${\log }_{10}(\epsilon )$ range from $-6$ (top level) to $-2$ (bottom level) in steps of 1. The blue dots designate branch-cut (nonconvergent) modes.

Figure 6

Overplotted contours from Fig. 5, with the same parameters and contour levels.

Figure 7

Left: $\ell =2$ gravitational-led QNMs (red dots) and $\epsilon$-pseudospectra boundaries (white lines) of a RN BH with $Q/M=1$. Right: superimposed $\ell =1$ electromagnetic-led (black lines) and $\ell =2$ gravitational-led (green dashed lines) $\epsilon$-pseudospectral contours of a RN BH with $Q/M=1$. In both cases, the contour levels ${\log }_{10}(\epsilon )$ range from $-50$ (top level) to $-5$ (bottom level) in steps of 5. The blue dots designate branch-cut (nonconvergent) modes.

Figure 8

Superimposed scalar (red), electromagnetic-led (green), and gravitational-led (blue) pseudospectral levels for a RN BH with $Q/M=0.5$. All perturbations share the same angular index $\ell =2$. The contour levels ${\log }_{10}(\epsilon )$ for the left plot range from $-50$ (top level) to $-5$ (bottom level) in steps of 5; for the right plot, they range from $-20$ (top level) to $-2$ (bottom level) in steps of 2.

Figure 9

Position on the real axis of the asymptotic kink $\mathrm{Re}(M\omega {)}_{\mathrm{kink}}$ for the $\ell =2$ gravitational-led pseudospectral levels of a RN BH with $Q/M=0.5$ with respect to the increment of the grid interpolation points $N$ (see right panel of Fig. 8).

Figure 10

“Error bounds” ${\mathcal{E}}^{\epsilon }(L)$ around the spectrum of $\ell =1$ scalar QNMs of a RN BH with $Q/M=0.5$. The red dots correspond to RN QNMs, while the $\epsilon$-contours illustrate the proximity (to a distance $\epsilon$) of pseudospectra contours to the actual QNM spectrum. The contour levels ${\log }_{10}(\epsilon )$ range from $-0.04$ (outer) to $-1.94$ (inner) in steps of 0.1. The blue dots designate branch-cut (nonconvergent) modes. This plot shows the typical structure of the pseudospectrum of a spectrally stable (normal) operator.