A Nonlinear Endpoint of Charged Horizon Instabilities

TL;DR

This work studies the nonlinear endpoint of the charged Aretakis instability by evolving the full Einstein–Maxwell–Klein–Gordon system in spherical symmetry, starting from a super-extremal RN background and finely tuned ingoing charged wave packets. It demonstrates that dynamical extremal black holes form at a threshold $Q_*$ with near-extremal horizon properties and Aretakis-like gradient growth on the horizon, while curvature gradients can blow up in the interior; the threshold dynamics obey universal scaling with exponent $1/2$, robust across multiple initial-data families, though horizon hair introduces family-dependent constants. The analysis extends to the dispersive side of the threshold, where a blueshift focusing instability drives interior curvature growth even in the absence of a horizon, suggesting possible curvature singularities observable from future null infinity. Overall, the results support a universal near-threshold structure for dynamical extremal spacetimes in the charged-scalar setting, while highlighting subtle family dependence through horizon hair and interior dynamics, and point to important open questions about universality regions and extensions beyond spherical symmetry.

Abstract

We numerically construct asymptotically extremal black holes through the nonlinear evolution of a charged scalar field. Our procedure -- which extends the work of Murata-Reall-Tanahashi to include charged scalar dynamics -- involves the fine-tuned scattering of wave packets within an initially super-extremal Reissner-Nordstrom spacetime. The resulting extremal solution develops an event horizon along which the energy density diverges and the charge density approaches a constant (i.e., the horizon forms with "hair"). We investigate this behavior from the perspective of critical phenomena in gravitational collapse, giving evidence that dynamical extremal black holes act as universal threshold solutions modulo this family-dependent hair. As in the linear instability of fixed extremal backgrounds, the scalar field decays outside the dynamical extremal horizon. But just inside the horizon, the scalar curvature appears to develop unbounded growth. This implies that near-threshold solutions without a black hole could develop correspondingly large curvatures visible from future null infinity.

Figures (17)

• Figure 1: Sample Penrose diagram of dynamical spacetime emanating from initial data pasted along the future null cone of the point $(U_0,V_0)$. When the matter perturbation is purely ingoing, the metric on $\mathcal{N}_A$ can be taken to match the exact RN solution, as depicted here. We evolve from $U_0$ to $U_{\rm max}$ along hypersurfaces of constant $U$, crossing any potential event/apparent horizons along the way.
• Figure 2: Construction of a dynamical extremal black hole. Unlike Figure \ref{['fig:initdatafig']}, here we show only the null diamond within which we evolve our initial data. What we call the background spacetime ($V<V_0$) is super-extremal Reissner-Nordström with parameters $Q_0>M_0$. Given our ingoing pulse of charged scalar field (the red curve), a black hole forms for sufficiently small $Q_0$, and the trapped region exists for $V>V_{\rm trap}$. As the background charge $Q_0$ is increased to $Q_*$, the time $V_{\rm trap}$ at which a trapped region first appears is ostensibly pushed to $\infty$ (right panel); this fine-tuned case is the dynamical extremal black hole. For $Q_0>Q_*$ no trapped regions form in our evolution domain.
• Figure 3: Fine tuning of $Q_0$ to push $V_{\rm trap}$ --- the time at which a black hole forms --- to infinity. We find that $V_{\rm trap}\sim |Q_0-Q_*|^{-1/2}$ in this regime.
• Figure 4: Black hole properties as $Q_0\to Q_*$. Both the charge of the black hole and radius of its event horizon approach the black hole's mass, indicating that the threshold black hole is extremal. The charge of the black hole approaches extremality linearly, while the radius of the event horizon approaches extremality with the critical exponent of $1/2$.
• Figure 5: Initial conditions for the scalar amplitude along $\mathcal{N}_B$. The "single-pulse" --- employed in the previous subsection and shown as a black dashed line in this figure --- is presented in contrast to the "double-pulse" in blue.