Can black holes evaporate past extremality?

TL;DR

The work investigates whether charged black holes can evaporate past extremality using a thin-shell model with a flat interior and an external charged Vaidya spacetime. It identifies two interior fates once the horizon disappears: a timelike singularity or an expanding null remnant that re-emerges after evaporation, with the latter carrying correlations to outgoing Hawking quanta. Including backreaction near the inner horizon shows the shell rapidly approaches a near-horizon regime, and during evaporation the internal energy $m$ collapses to zero at a finite time, leaving an outgoing null shell that expands to infinity and leaves behind a low-curvature exterior. The scenario offers a semiclassical avenue to address information paradox questions without quantum gravity, and suggests a qualitatively new interior structure where collapsing matter can ultimately resurface as an expanding remnant.

Abstract

Black holes with sufficiently large initial charge and mass will Hawking-evaporate towards the extremal limit. The emission slows as the temperature approaches zero, but still reaches the point where a single Hawking quantum would make the object superextremal, removing the horizon. We take this semiclassical prediction at face value and ask: When the emission occurs, what is revealed? Using a model of thin-shell collapse with subsequent accretion/evaporation by a null flux of ingoing positive/negative energy (charged Vaidya spacetime glued to a flat interior), we find two possible outcomes: (i) For shells that are initially very tightly bound, a timelike singularity forms and later appears; or (ii) for unbound or modestly bound shells, the matter re-emerges as a null shell that expands to infinity. This expanding remnant has been bathed in the ingoing Hawking quanta during evaporation and presumably carries correlations with the outgoing quanta, offering the attractive possibility of studying information paradox issues in a setup where spacetime curvatures are globally small, so that quantum gravity is not required. Even for ordinary black holes that evaporate down to the Planck size, we propose a radical new scenario for the interior: rather than forming a singularity, the collapsing matter settles onto an \textit{outgoing} null trajectory \textit{inside} the horizon for the entirety of evaporation.

Figures (7)

• Figure 1: Charged spherical collapse without backreaction. Possible paths for the surface of collapsing matter are superimposed on a portion of the RN Penrose diagram; the part to the left of each path should be replaced with the regular interior spacetime. The trajectories are color-coded according to whether they reach the singularity or the "Cauchy horizon" (defined as shown, and not strictly a Cauchy horizon). The two different Cauchy-bound trajectories would emerge in different regions of the maximally extended RN spacetime, but for collapse and evaporation (Fig. \ref{['fig:outcomes']}) their behavior is qualitatively identical. Causal regions are labeled $A$, $B$, and $C$ for future reference.
• Figure 2: Charged spherical collapse with backreaction from Hawking radiation. The collapsing matter is taken to be a thin spherical shell, and Hawking radiation is modeled by regions of ingoing ("in") and outgoing ("out") charged Vaidya spacetime separated by a timelike pair formation front (brown curve) lying somewhat outside the shrinking apparent horizon. If a timelike singularity would form without backreaction (Fig. \ref{['fig:collapse-without-backreaction']} red curve), then a timelike singularity forms and eventually is revealed as naked (left panel). If the Cauchy horizon would be approached without backreaction (Fig. \ref{['fig:collapse-without-backreaction']} lavender curves), then the matter becomes null at a finite time and re-expands to null infinity after evaporation.
• Figure 3: Penrose diagram illustrating the meaning of various definitions and coordinates. The diagram precisely reflects the causal structure of the coordinate patch only for constant subextremal $M$ and $Q$ (RN spacetime), but it still forms good intuition for the slowly-evolving case we consider. We use capital letters for the coordinates on the diagram, since mostly we will think about the history of a shell described by $v=V(\tau)$ and $r=R(\tau)$. The gray equations indicate properties of these functions as $f=0$ horizons are crossed in the direction shown with gray arrows. The regions bewteen the horizons are labeled $A$, $B$, and $C$. The sign $s$ is defined in Eq. \ref{['Vdot']}.
• Figure 4: Parameter space for thin shells with a flat interior and RN exterior. White areas are disallowed, while colored areas indicate the behavior of the shell. Unbound superextremal shells (top left) always bounce, meaning they re-expand to infinity. Modestly bound superextremal shells cannot be constructed (white area), but more significantly bound superextremal shells collapse to naked singularities (top right). Subextremal shells (below dashed line) form black holes, either collapsing to a timelike singlarity (right of gray curve) or approaching the Cauchy horizon (left of gray curve). "Branch" indicates that the horizon is approached from region $B$, while "bounce" indicates the the horizon is approached from region $C$. (See Fig. \ref{['fig:collapse-without-backreaction']} or Fig. \ref{['fig:regions']} for the definitions of these regions.)
• Figure 5: Numerical solution for $R(v)$ and $m(v)$ with initial values $R(0)=10$ and $m(0)=0.1$, expressed in units where the initial mass $M$ is equal to $1$. The black hole mass evolves as $M=1-v/100$. The shell bounces at a radius just below the inner horizon radius and then approaches the stable value \ref{['hug']} set by the evaporation timescale. At $v\approx0.16259$ the mass evolves to zero and the shell becomes null with initial radius $R\approx.13417$, just below the inner horizon at $r_-\approx.13422$. The main plot uses a logarithmic scale, while the inset uses a linear scale showing the indicated horizontal range, together with a vertical range of $5\times 10^{-4}$.