Formation of extremal Reissner-Nordström black holes: insights from numerics

TL;DR

The paper numerically investigates the finite-time formation of extremal Reissner-Nordström black holes via the characteristic gluing construction, focusing on how horizon-data Ansätze, scalar mass, and cosmological constant affect the required mass-to-charge product $\mathfrak e M$. It implements a shooting-based framework within the Einstein-Maxwell-charged scalar field system under spherical symmetry, exploring $C^0$, $C^1$, and $C^2$ gluing with several Ansätze for the horizon scalar profile and evaluating whether gluing yields a valid extremal RN end-state. The results show strong dependence on the chosen Ansatz: minimal $\mathfrak e M$ values to reach extremality vary across basis choices, and introducing scalar mass $\frak m$ imposes upper bounds on the allowable mass-to-charge ratio, with pronounced differences between regularity classes and cosmological constant values. Overall, the work provides concrete quantitative bounds and demonstrates that, for several physically reasonable Ansätze, extremality can be achieved in finite time for sub-unity mass-to-charge ratios, while indicating clear limits set by $\frak m/\frak e$ and by the regularity of the gluing data, offering insights into the feasibility of third-law violations in this gravitational setting.

Abstract

An extremal Reissner-Nordström black hole can form in finite time in the gravitational collapse of a massless charged scalar field. The proof of this is based on the method of characteristic gluing, which involves making an Ansatz for the scalar field at the horizon. We perform a numerical investigation of the characteristic gluing procedure for several different Ansätze. In each case, gluing is possible only if the final black hole mass is large enough. We find that the minimum required mass varies significantly for different Ansätze. We also consider the effect of including a mass term for the scalar field. In this case, for each Ansatz we determine the maximum mass-to-charge ratio for the scalar field such that gluing is possible. Analogous results are obtained for a non-zero cosmological constant.

Figures (12)

• Figure 1: Setup for characteristic gluing with zero cosmological constant along the outgoing null cone $\mathcal{C}\equiv\{0\}\times[0,1]$. The metric and matter content in regions $\mathfrak{R}_1$ and $\mathfrak{R}_2$ are known and are given by $\mathcal{Q}_1$ and $\mathcal{Q}_2$, respectively. Data must be imposed on $\mathcal{C}$ such that the Einstein equations are satisfied. In the spherically-symmetric situation considered in this paper, $\mathfrak{R}_1$ can be extended to $r=0$, the centre of symmetry, to obtain the dark triangular region. The rectangle $\mathfrak{R}_2$ can be extended to get a complete future null infinity $\mathcal{I}^+$, as well as a segment of past null infinity $\mathcal{I}^-$, and spacelike ($i^0$) and future timelike ($i^+$) infinity. Once a valid solution to the characteristic gluing problem is found, the well-posedness of the characteristic initial value problem Luk:2011vf can be exploited to obtain a local extension of spacetime. To extend this local solution to $r=0$, one can use Cauchy stability Kehle:2022uvc. A Cauchy surface is shown in red.
• Figure 2: Left: Characteristic gluing with a positive cosmological constant. The Cauchy slice (red) is complete and extends to past null infinity. Right: Characteristic gluing with a negative cosmological constant. The Cauchy slice extends to the timelike boundary.
• Figure 3: Left: The function $I$ for the even (solid black), odd (dashed blue) and polynomial (dash-dotted red) Ansätze. Right:$\mathfrak{q}_{\rm max}$ against $\mathfrak{e} M I_{\rm max}$ for different values of $\Lambda M^2$: from thickest to thinnest, $\Lambda M^2=0.17$, $0.12$, $0.1$, $0$, $-0.5$, $-2$. Notice that for $\Lambda M^2>1/9$, there is a lower bound on the charge of the final black hole.
• Figure 4: Left:$\sup\partial_Ur$ (in units $M=1$) against $\Lambda M^2$ for the gluing solution achieved with the minimal value of $\mathfrak{e} M$, with $\mathfrak{m}=0$. The different curves represent different dimensionless charges: $\mathfrak{q}=1$ (solid), $\mathfrak{q}=0.9$ (dashed) and $\mathfrak{q}=0.4$ (dotted). The dot denotes the maximum value of $\Lambda M^2$ (which depends on $\mathfrak{q}$). In all cases we have $\sup \partial_U r<0$ so the candidate solution is a valid solution. Right: Profiles $\rho$ for $\alpha_1=\alpha_{\rm max}$, for the even (solid black), odd (dashed blue), polynomial (dash-dotted red) and modified even (with $\gamma=0.36$; dotted black) Ansätze.
• Figure 5: Left: Maximum charge $\mathfrak{q}_{\rm max}$ as a function of $\mathfrak{e} M$ for $\mathfrak{m}/\mathfrak{e}=0$ (solid blue), $0.2$ (dashed red) and $0.5$ (dash-dotted green), with the even Ansatz. The blue curve attains extremality at $\mathfrak{e} M\approx15.96$, denoted by a star. The latter two curves exhibit a kink when the limiting condition $\sup \partial_U r<0$ becomes important. The kink is more prominent for larger $\mathfrak{m}/\mathfrak{e}$; the insets show that of the red and green curves. After the kink, $\mathfrak{q}_{\rm max}$ rises very slowly with $\mathfrak{e} M$. The green curve reaches a maximum at $\mathfrak{q}<1$, and then decreases slowly. Right, top:$\mathfrak{q}_{\rm max}$ versus $\mathfrak{e} M$ with $\mathfrak{m}/\mathfrak{e}=0.2$, showing the very slow increase towards extremality, which is eventually reached at $\mathfrak{e} M\approx47.46$, at the red star. Note the logarithmic scale on the horizontal axis. Right, bottom: The slow rise and decrease in $\mathfrak{q}_{\rm max}$ following the kink in the green curve ($\mathfrak{m}/\mathfrak{e}=0.5$). Note that all of this plot has much larger $\mathfrak{e} M$ than the kink mentioned above.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2