An observational signature for extremal black holes

TL;DR

This work tests whether extremal Reissner-Nordström and extremal Kerr black holes imprint a distinct observational signal in scalar perturbations. By deriving an angularly selective signature $s_{\mathcal{I}^{+}}(\vartheta)$ from the radiation field at null infinity and linking its nonzero value to the horizon hair $H[\psi]$, the authors show the signal vanishes for subextremal holes and equals $H[\psi]$ in the extremal case. Using horizon-penetrating hyperboloidal coordinates and a high-order WENO scheme, they numerically verify this relation for RN and Kerr: subextremal cases yield $s_{\mathcal{I}^{+}}(\vartheta) \approx 0$, while extremal cases exhibit a linear dependence on $H[\psi]$ with slope near unity. The key observation is that the horizon charge becomes, in principle, measurable by far-away observers through the late-time behavior of the radiation field, offering a potential falsification of the no-hair hypothesis for extremal black holes within linear perturbation theory.

Abstract

We consider scalar perturbations of the Reissner--Nordström family and the Kerr family. We derive a characteristic expression of the radiation field, at any given unit solid angle of future null infinity, and numerically show that its amplitude gets excited only in the extremal case. Our work, therefore, identifies an observational signature for extremal black holes. Moreover, we show that the source of the excitation is the extremal horizon instability and its magnitude is exactly equal to the conserved horizon charge.

Figures (5)

• Figure 1: Time evolution of $s_{\mathcal{I}^{+}}(\vartheta)$ for three sub-extremal RNs, $Q/M=[0.3,0.4,0.5]$ for any value of $\vartheta$. Note that $s_{\mathcal{I}^{+}}(\vartheta)$ asymptotes to $0$ as we evolve the scalar field at $\mathcal{I}^+$.
• Figure 2: Best-fit line demonstrating a linear relationship between the horizon constant $H[\psi]$ and the asymptotically extracted $s_{\mathcal{I}^{+}}(\vartheta)$ for any $\vartheta$ for extremal RN. The slope of the best fit-line is $1.0\pm0.2$. Each data point is labeled by the initial data used for that computation i.e. the location and width of the Gaussian pulse. The Levenberg–Marquardt fitting errors in each data point are too small to be visible on the scale of the plot.
• Figure 3: Time evolution of $s_{\mathcal{I}^{+}}(\vartheta)$ for three sub-extremal Kerr cases, $a/M=[0.3,0.4,0.5]$ for any value of $\vartheta$. Note that $s^{\color{black}Kerr}_{\mathcal{I}^{+}}(\vartheta)$ asymptotes to $0$ as we evolve the scalar field at $\mathcal{I}^+$.
• Figure 4: Best-fit line demonstrating a linear relationship between the horizon constant $H$ and the asymptotically extracted $s^{\color{black}Kerr}_{\mathcal{I}^{+}}(\vartheta)$ with ${\theta^{1,2}} = \pi/2$ for extremal Kerr black holes. The slope of the best fit-line is $1.0\pm0.1$. Each data point is labelled by the initial data used for that computation i.e. the location and width of the Gaussian pulse. The Levenberg–Marquardt fitting errors are depicted by the vertical bars on each data point.
• Figure 5: Time evolution of $s_{\mathcal{I}^{+}}(\vartheta)$ for ERN and EK for initial data with no support on the horizon. Note that $s_{\mathcal{I}^{+}}(\vartheta)$ asymptotes to $0$ as we evolve the scalar field at $\mathcal{I}^+$.