Quantum Suppression of Mass Inflation in Reissner-Nordström Interiors via Wheeler-DeWitt Equation

TL;DR

This work extends a Wheeler-DeWitt quantization to the interior of static black holes in Einstein-Maxwell-$\Lambda$ theory, focusing on Reissner-Nordström. Using a Kantowski-Sachs reduction with $X=\ln a$, $Y=\ln b$, and $Q$, the authors derive a WDW equation that treats $Q$ as a timelike interior coordinate and find horizon-sourced, on-shell wavefunctions that decay away from the classical locus. By separating variables and imposing normalizability, they obtain bounded solutions expressed via modified Bessel functions; with a Gaussian ansatz they localize the wavefunction around the classical trajectory and horizon centers, and identify three qualitatively distinct interior behaviors: monotonic decay, quantum bounce, and annihilation-to-nothing. In the neutral limit ($r_Q\to0$) the Schwarzschild interior emerges as a bounded, monotonic WDW state, providing a singularity-avoiding picture that unifies classical and quantum interiors. The boundary-condition choices at the event and Cauchy horizons lead to potential quantum-gravitational suppression of mass inflation, and the results motivate extensions to Kerr and regular black holes to test the generality of these interior structures.

Abstract

We construct a canonical quantization, the Wheeler-DeWitt equation, of the interior geometry of static and spherically symmetric black holes in Einstein-Maxwell-$Λ$ framework, focusing on Reissner-Nordström. The wave function of the Wheeler-DeWitt equation for the Reissner-Nordström black hole is set to be on-shell and exhibiting exponential damping away from the classical locus. Horizon boundary conditions for the wave function generate two regimes: a single inward mode from event horizon yields monotonic decay, while superpositions produce either a quantum bounce (single time arrow) or interference-driven annihilation-to-nothing (two time arrows). We show that these are generic features of static black hole interiors. Furthermore, the wave function of the Schwarzschild black hole, obtained as the charge-neutral limit of the Reissner-Nordström black hole, is monotonically decaying and no longer unbounded. Moreover, this framework unifies classical and quantum interiors, suggests a quantum gravitational suppression to the mass inflation, and motivates extensions to Kerr and regular black holes.

Figures (6)

• Figure 1: The yellow surface is the constraint surface Eq. (\ref{['eq:XY']}) for the Reissner–Nordström black hole. The colored lines are Eq. (\ref{['eq:pl']}) with $r_Q=\pm0.48$(green), $r_Q=\pm0.2$(red) and $\pm0.1$(blue), where the solid lines are $r_Q$ with $+$ and dotted lines are $-$. The black line is the Schwarzschild black hole, $r_Q=0$.
• Figure 2: Left: The squared modulus of the wave function $\Psi_+$ Eq.(\ref{['eq:gbwf']}) with $\{r_s,\sigma,r_Q\}=\{1,1,0.2\}$ is shown. The red line is the classical trajectory. Right: The planar cross-section shows that the amplitude maximum traces the steepest-descent contour.
• Figure 3: The squared modulus of the wave function Eq.(\ref{['eq:gbwf']}) with $\{r_s,\sigma,r_Q\}=\{1,2.3,0.48\}$ is shown. The green line is the classical trajectory. $\Psi_+$ corresponds to the upper panel and $\Psi_-$ to the lower; the right-hand inset shows a planar cross-section in which the amplitude maximum traces the steepest-descent contour.
• Figure 4: Left: The squared modulus of the wave function $\Psi_e$ Eq.(\ref{['eq:gbwfe']}) with $\{r_s,\sigma,r_Q\}=\{1,1,0\}$ is shown. The black line is the classical trajectory. Right: The planar cross-section shows that the amplitude maximum traces the steepest-descent contour.
• Figure 5: Penrose diagram of Reissner–Nordström black hole is presented. The arrow is the arrow of time. The waves $\Psi_+$ from $r_+$ and $\Psi_-$ from $r_-$ refer to the red region. The yellow region is the empty space without any waves.