Quasinormal modes of charged fields in Reissner-Nordström backgrounds by Borel-Padé summation of Bender-Wu series

We extend recent work of Hatsuda on the computation of quasinormal mode frequencies via analytic continuation of bound state energies and Borel-Padé resummation of the Bender-Wu perturbation series to the case of charged fields in the background of Reissner-Nordström black holes. We compare the quasinormal mode frequencies obtained in this manner to calculations using Leaver’s method of continued fractions, and find good agreement for damped modes with the imaginary part remaining finite in the extremal limit. We also present numerical evidence that the frequencies of certain zero-damped modes with the imaginary part tending to zero in the extremal limit can be computed when constructing the Bender-Wu expansion about a peak of the potential inside the outer horizon of the black hole.

Figure 1

Imaginary part ${\omega }_{\mathrm{I}}$ of the quasinormal mode frequency for odd-parity gravitational/electromagnetic perturbations of the $D=4$ Reissner-Nordström black hole, as computed for the $l=2$, $\nu =0$ mode found with Hatsuda’s method [15]. The horizontal axis shows the black hole charge measured as a fraction of extremality, whereas the vertical axis shows the frequency measured in units of $r_S^{-1}=(2G_{N}M{)}^{-1}$. The fact that ${\omega }_{\mathrm{I}}$ remains finite as $Q\to Q_{\mathrm{ext}}$ indicates that the algorithm is finding a damped mode, not a zero-damped mode, in the extremal limit.

Figure 2

Dependence of the real and imaginary parts of the lowest-lying $l=0$ quasinormal mode frequency of a charged scalar on the scalar mass parameter $\mu$ (horizontal axis) and its charge-to-mass ratio $f$ (as labeled on each curve). We measure all quantities in units set by the outer horizon radius, $r_{+}$, and have fixed $r_{-}/r_{+}=1/2$. The continuous blue lines interpolate between results obtained with our extension of Hatsuda’s algorithm, while the red dots were obtained with Leaver’s method. They agree within the precision of the calculation.

Figure 3

Dependence of the real and imaginary parts of the lowest-lying $l=0$ quasinormal mode frequency of a charged scalar on the ratio of horizon coordinates $r_{-}/r_{+}$ and the scalar mass parameter $\mu$ (as labeled on each curve). We have fixed the scalar charge fraction $f=-1$. Blue lines are our algorithm and red dots are Leaver’s method. In the right-hand plot, we have suppressed the label $\mu r_{+}=0.06$ on the middle curve for legibility.

Figure 4

Dependence of the imaginary part of the lowest-lying $l=0$ quasinormal mode frequency of a massless scalar, ${\omega }_{\mathrm{I}}$, measured in different units. Left panel: ${\omega }_{\mathrm{I}}{r}_S$ as a function of $Q/Q_{\mathrm{ext}}$ for the same choices of $\mu r_{+}$ shown in Fig. 3. Right panel: ${\omega }_{\mathrm{I}}/(\pi T)$ as a function of $r_{-}/r_{+}$, selecting a particular $\mu$ since the curves of different $\mu$ are difficult to separate when plotted in this manner.

Figure 5

Dependence of the potential $V(r)$ for a massive, neutral scalar field in a $D=4$ Reissner-Nordström background on the scalar mass parameter $\mu$. We have fixed the ratio of inner and outer horizon coordinates, $r_{-}/r_{+}=0.9$. The potential has a maximum at $r_0>r_{+}$ when $\mu r_{+}\lesssim 0.29$, as in the solid blue and orange curves. For larger values of $\mu$, as in the dashed green and red curves, there is no maximum at $r>r_{+}$. For all values of $\mu r_{+}$, there is a minimum and a maximum of $V(r)$ behind the outer horizon.

Figure 6

Dependence of the critical points, $r_0$, of the potential $V(r)$ on the scalar mass parameter $\mu r_{+}$, for the fixed choice $r_{-}/r_{+}=0.9$. Initially, there is a real maximum of $V(r)$ on the left and a real minimum on the right. As $\mu \to {\mu }_{\mathrm{c}}$, these two critical points merge and then move away from the real axis, appearing as two complex conjugate critical points. Some specific choices of $\mu r_{+}$, labeled in the plot, are marked to illustrate the behavior.

Figure 7

Breakdown of the algorithm near $\mu ={\mu }_{\mathrm{c}}$, where the critical points $r_0$ bifurcate as shown in Fig. 6. As in the plots of Sec. 4, the red dots show results we obtained with Leaver’s method, which we believe to be an accurate calculation of the quasinormal mode frequency. This result shows that Hatsuda’s algorithm breaks down near the critical $\mu$ where a bifurcation in the critical points of $V(r)$ occurs. None of the branches of $r_0$ accurately match the true quasinormal modes in this region.

Figure 8

Calculation of the quasinormal mode frequencies for $\mu >{\mu }_{\mathrm{c}}$ using the branch of $r_0$ labeled “top” in Fig. 7 (continuous green curve) as well as Leaver’s method (red dots). Although the method breaks down badly near ${\mu }_{\mathrm{c}}$, we see that it approximately reproduces the quasinormal mode with smallest ${\omega }_{\mathrm{I}}$ above ${\mu }_{\mathrm{c}}$ until this mode disappears at $\mu r_{+}\approx 0.53$.

Figure 9

Quasinormal modes of the massless, neutral scalar field in Reissner-Nordström backgrounds near extremality. Solid blue and orange curves are DMs and ZDMs computed with Hatsuda’s algorithm. The ZDMs are computed by expanding around the maximum of $V(r)$ behind the outer horizon. Dotted green and red curves are the same modes computed with Leaver’s method.

Figure 10

Imaginary part ${\omega }_{\mathrm{I}}$ of the quasinormal modes of the massless, neutral scalar field in Reissner-Nordström backgrounds near extremality, plotted in units of the black hole temperature. This is the same data as the right-hand panel of Fig. 9, but shows more clearly that ${\omega }_{\mathrm{I}}\propto T$ in the extremal limit for the ZDMs.

Figure 11

Imaginary part ${\omega }_{\mathrm{I}}$ of the $l=0$ ZDMs of a massive, charged scalar field in a nearly extremal Reissner-Nordström background, computed with our extension of Hatsuda’s method when expanding around a critical point $r_0$ inside the outer horizon. We have fixed the mass parameter $\mu r_{+}=7$, and vary the scalar charge parameter $f=\sqrt{2}q/(\kappa \mu )$. We plot the six lowest-lying ZDMs, with $\nu$ increasing from 0 for the lowest curve to 5 for the highest curve. These results agree to high precision with analytic results of Hod [34].

Figure 12

The ratio of ${\omega }_{\mathrm{I}}$ to $\pi T$ for $\nu =0$, $l=0$ ZDMs of a massive, charged scalar field in a very near-extremal Reissner-Nordström background, for selected choices of $\mu r_{+}$ and $f$ (labeled curves). We see that to high precision, the ZDMs computed with our method have ${\omega }_{\mathrm{I}}$ approaching a constant ($\mu$- and $q$-dependent) multiple of temperature in the extremal limit.