Scalar Quasinormal modes in Reissner--Nordström black holes: implications for Weak Gravity Conjecture

Abstract

Microscopic charged black holes can provide possibilities to test the consistency of the effective field theory (EFT) corrections to Einstein-Maxwell theory. A particularly interesting result is fixing the sign of a certain combination of EFT couplings from the requirement that all charged black holes should be able to evaporate (Weak Gravity Conjecture). In our work, we analysed the EFT corrections to a set of zero-damping quasinormal modes (QNMs) of the scalar wave probe in a nearly extremal Reissner-Nordström black hole. We review the duality of this setup to the problem of the quantum Seiberg-Witten curve of $N=2$ Super-Yang-Mills theory with three flavors. We provide an analytic result for the EFT corrections to the QNMs obtained from the quantization condition imposed on the Seiberg-Witten cycle. Our main result is that the causality requirement of the gravitational theory formulated for the QNMs translates to the same condition on EFT couplings as the one appearing in the Weak Gravity Conjecture.

Figures (2)

• Figure 1: Exact and approximate effective potentials. In all plots we show $Q_W^{num}$, $Q_W^{Heun}$, and $Q_W^{EFT}$ obtained from equations \ref{['QWnum']}, \ref{['Qheun']} and \ref{['EFTQW']}, respectively. The values of $R_+$ and $R_+^{(1)}$ correspond to the outer horizons obtained as an exact numerical root of $G(r)=0$, and the root $G(r)=0$ obtained approximately to the linear order in $g_4$ and given in \ref{['R1+-']}. We choose $l=1,~\omega=-0.03 i$ because it corresponds to one of the slowly damped modes for the RN black hole, see Table \ref{['tabzero']}. The approximate location of the RN horizon corresponds to $r_+=1.01414$ for $Q=0.9999$, and $r_+=1.43589$ for $Q=0.9$. These values are less than $R_+$, so they are located to the left of the vertical axis. The upper two plots represent the case of small EFT coupling $g_4=10^{-5}$. From the upper left plot, it can be seen that the effective potential $Q_W^{EFT}$ is very close to the exact one for large $r$. The upper right plot shows zooming in on the values of $r$ close to the horizons, where $Q_W^{EFT}$ deviates from $Q_W^{num}$, while $Q_W^{Heun}$ still coincides with the exact potential. The lower left plot represents the case of $Q=0.9$, for which the approximation $Q_W^{EFT}$ works very well. The lower right plot shows the visible breakdown of the approximations for the larger value of $g_4=10^{-3}$ near the exact horizon $R_+$. The maxima of the effective potential are not shown because they correspond to very large values of $Q_W$ and appear to be extremely close to $R_+$ for the chosen parameters.
• Figure 2: Effective potential in the critical regime given as parameters $M=1$, $Q=0.5$, $g_4=0.01$. The horizon is at $r_+=1.86607$, the critical unstable radius of the circular photon sphere is at $r_c=2.8229$, corresponding to a critical impact parameter $b_c=4.96793$.