Overdamped quasibound states inside a Schwarzschild black hole

TL;DR

The paper discovers purely imaginary, overdamped quasibound states for axial gravitational perturbations inside a Schwarzschild black hole, localized in the interior between the horizon and the singularity. Using the Regge-Wheeler equation and Kruskal–Szekeres coordinates, it establishes interior boundary conditions that yield a discrete spectrum whose modes decay before reaching the singularity while remaining regular at the future horizon. The work demonstrates that interior perturbations can hover transiently within the black-hole interior, challenging the simplistic view that all disturbances inexorably plunge into the singularity. These interior resonances have potential implications for semiclassical stress-energy behavior, interior boundary conditions in quantum gravity, and holographic or microstructure-inspired models, though they do not couple to exterior observables.

Abstract

Schwarzschild black-hole interiors, bounded by event horizons and terminated by spacelike singularities, are regions where all physical observers are inevitably destroyed. In the geometric optics approximation, waves follow null geodesics to the singularity. However, outside the geometric optics regime, the behavior of wave propagation can be rich and nuanced, even in such extreme habitats. In this work, we show that axial gravitational perturbations in the interior of a Schwarzschild black hole can form overdamped (non-oscillatory) quasibound states that decay before reaching the singularity. Using Kruskal-Szekeres coordinates to avoid coordinate ambiguities, we identify these modes and analyze their eigenfunctions. Contrary to earlier claims, we find that the Regge-Wheeler master function of these modes have non-zero amplitude at the future event horizon but decay before interacting with the singularity. We consider observations of the modes along timelike geodesics. This work suggests that certain gravitational fluctuations can hover transiently within the black-hole interior, challenging common assumptions about wave behavior in uncharted and extreme regions of spacetime.

Figures (4)

• Figure 1: The continuation of the Regge-Wheeler effective potential $V_\mathrm{RW}$ inside the Schwarzschild black hole. The dashed, solid and dash-dotted curves correspond to $\ell = 2,\, 3$ and $4$, respectively. (a) $V_\mathrm{RW}$ as a function of $r$. (b) $V_\mathrm{RW}$ as a function of $r_*$. The horizon is at $r_*=-\infty$. The horizontal "geometric optics" line indicates a large value of $\omega$ (not to scale).
• Figure 2: Top row The overdamped gravitational modes for $\ell \leq 5$. (a), (c), (e) and (g) with respect to the effective potentials for $\ell = 2,\, 3,\, 4$ and $5$, respectively. The horizontal colored lines indicate the effective energy $-\omega_\mathrm{I}^2$ of each energy mode $n$. Insets are enlargements of the region close to zero effective potential. Bottom row (b), (d), (f) and (h) show the wavefunctions for $\ell = 2,\, 3,\, 4$ and $5$, respectively. The color of each wavefunction matches the color of the energy state in the panel above. The number of nodes is equal to the number of the energy level $n$. The dashed curves corresponds to the smallest value of $|\omega_\mathrm{I}|$. Insets show the long range behavior of the dashed curves. The solid curves in the insets show the corresponding exponential dependence $\mathrm{exp}(\omega_\mathrm{I} r_* )$ of the smallest $|\omega_\mathrm{I}|$.
• Figure 3: An OQBS in Kruskal-Szekeres coordinates. The absolute value of $\Psi(t,r)$ for $\ell=5$, $n=2$ is shown. The diagram corresponds to quadrant II in a Kruskal-Szekeres diagram. The $U=0$ ($V=0$) line is the future (past) horizon. The dashed $UV=1$ curve is the singularity. The red and green curves are examples of timelike geodesic observers. Their world lines happen to cross at the white circle.
• Figure 4: An OQBS as seen by geodesic observers. The red and green curves are the values from Fig. \ref{['fig:UVdiagram']} along the red and green world lines, as functions of each observer's proper time. The two curves are equal at the black circle, which corresponds to the white circle in Fig. \ref{['fig:UVdiagram']}.