Quantum Black Hole as a Harmonic Oscillator from the Perspective of the Minimum Uncertainty Approach

TL;DR

The paper maps the interior Schwarzschild black hole mass eigenvalue problem to a quantum harmonic oscillator via a reparametrization, enabling a direct comparison between standard and minimal-length (GUP) quantizations. Standard quantization yields a discrete, positive area and mass spectrum with oscillatory interior wavefunctions, but the state is not square-integrable, limiting physical interpretation. The minimal-uncertainty approach introduces a deformed commutator and, after imposing convergence/truncation conditions, yields a square-integrable wavefunction and an area spectrum that scales as $A_s(n)\sim 16\pi\beta\ell_{\mathrm{Pl}}^2\,n^2$ for large $n$, with the deformation parameter $\beta$ tied to a discrete quantum number $m$. This framework also reveals tunneling connecting the black hole interior to exterior and to a white-hole region, which vanishes as $\beta\to0$, illustrating how minimal-length effects regularize the wavefunction and reshape the semiclassical structure of the black hole. The results point to a rich interplay between quantum gravity effects and black-hole interiors, suggesting algebraic structures (e.g., $SU(1,1)$) and further connections to LQG-inspired models as promising avenues for future work.

Abstract

Starting from the eigenvalue equation for the mass of a black hole derived by Mäkelä and Repo, we show that, by reparametrizing the radial coordinate and the wave function, it can be rewritten as the eigenvalue equation of a quantum harmonic oscillator. We then study the interior of a Schwarzschild black hole using two quantization approaches. In the standard quantization, the area and mass spectra are discrete, characterized by a quantum number $n$, but the wave function is not square-integrable, limiting its physical interpretation. In contrast, a minimal-uncertainty quantization approach yields an area spectrum that grows as $n^2$, and consequently the mass $M$ also increases. In this framework, the wave function is finite and square-integrable, with convergence requiring that the deformation parameter $β$ be regulated by a discrete quantum number $m$. The wave function exhibits quantum tunneling connecting the black hole interior with both its exterior and a white hole region, effects that disappear in the limit $β\to 0$. These results demonstrate how minimal-length effects both regularize the wave function and modify the semiclassical structure of the black hole.

Figures (7)

• Figure 1: Square of the wave function $\lvert \psi(x) \rvert^2$ plotted for different quantum numbers $n$.
• Figure 2: Squared modulus of the wave function \ref{['eq:Mod_Square_wav_func_Gup_case']}, obtained within the minimal-uncertainty formalism, shown in the semiclassical limit $R_s \gg \ell_{\mathrm{Pl}}$.
• Figure 3: Oscillation amplitude in the exterior region of the black hole, extending from $x \to \infty$ down to the point $x_{+}$.
• Figure 4: Transition (tunneling) region across the event horizon, mediating the connection between the exterior and the interior of the black hole.
• Figure 5: Oscillation amplitude $|\Psi_n^{\text{GUP}}(x)|^2$ within the black hole interior, covering the region from $x_{0_+}$ to $x_{0_-}$, where the amplitude initially grows rapidly, then decreases, and finally rises again as it approaches $x_{0_-}$.