Quantum Electron Clouds near Black Holes: Black Atoms and Molecules

TL;DR

This work applies DeWitt's curved-space Schrödinger formalism to atomic-scale quantum states in strong gravitational fields near Schwarzschild and Reissner-Nordström black holes. It demonstrates that quantum wavefunctions are strongly attracted toward BH horizons, with bound-state spectra and localization patterns that differ from flat-space atoms, including horizon-shell localization in extremal limits and novel features near inner horizons in RN backgrounds. The study extends to a two-center BH system (black-hole molecule), showing where analytic progress is possible and where numerical methods are required due to curvature-induced coupling. Overall, the paper reveals how gravity can fundamentally modify atomic-scale quantum structures, suggesting possible primordial BHs as hosts for BH-atom-like states and outlining directions for future work on Kerr BHs and more complex geometries.

Abstract

We study quantum mechanical wavefunctions near highly curved spaces, i.e., black holes. By utilizing the formalism developed by DeWitt, we derive the Schrödinger equations in the vicinity of the Schwarzschild and the Reissner-Nordström black hole geometries. The quantum electron cloud for the "black hydrogen atom" - an electron trapped by black holes - is particularly studied. We solve the equations and find that black holes generally attract the wavefunctions, localizing them near the horizon where the electrons are most likely to be trapped. These results imply that not only classical objects but also the quantum material and even the chemical properties of the atoms are affected by strong gravity. We also discuss black hydrogen molecules composed of multi-centered Majumdar-Papapetrou black holes.

Figures (11)

• Figure 1: The probability density $\sqrt{g} |R (r)|^2 r^2$ in the radial direction. Comparison between the flat space and the black hole background cases for $a=1, l = 1$ with $k=1$ (left), $k=5$ (middle) and $k=10$ (right). The normalization factors are appropriately chosen. The red dotted line corresponds to the event horizon $r = a$.
• Figure 2: The 3D plots for the probability density $|\psi (r, \theta, \phi)|^2$ for free particle. The cases of flat space (left) and the black hole background (right). The parameters are fixed to $a =1$, $l = 1$ and $k =1$. The event horizon is shown in the black sphere. The wavefunction is affected by the black hole.
• Figure 3: The radial probability densities of the wavefunction $\sqrt{g} |R (r)|^2 r^2$. Comparison between the hydrogen atom in the flat space and in the black hole background cases for $a=1$. The figure corresponds to the $3p$ orbital ($n = 3$, $l=1$) of the hydrogen atom. We will always use the parameters $\mu = e = \hbar = 1$ in the following.
• Figure 4: The 3D plot of the probability density $|\psi|^2$ for the particle under the attractive force. The flat space (left) and the black hole background (right) cases. The figures correspond to the $3d$ orbital ($n = 3$, $l=2$) of the hydrogen atom. The black hole horizon $a =5$ is shown in the black color.
• Figure 5: Comparison of the radial probability density $\sqrt{g} |R(r)|^2 r^2$ for different $a$ values in the Schwarzschild background with the potential $V$. Each figure corresponds to the $3p$ orbital ($n = 3$, $l = 1$) of the hydrogen atom. The red dotted line denotes the event horizon.