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Refocusing of Wheeler-DeWitt wave functions at inner horizons

Takeshi Chiba1,*, Keiju Murata1,†, and Daiki Saito2,‡

  • *Contact author: chibatak@gmail.com
  • †Contact author: murata.keiju@nihon-u.ac.jp
  • ‡Contact author: dsaito@ewha.ac.kr
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Phys. Rev. D 114, 064025 – Published 8 September, 2026

DOI: https://doi.org/10.1103/8cvq-2fnq

Abstract

We study quantum gravitational effects inside hyperbolic black holes with both outer and inner horizons by solving the Wheeler-DeWitt (WDW) equation in the minisuperspace approximation. The WDW equation contains a tachyonic region where the effective potential becomes negative. We develop a numerical method that consistently evolves the wave function across this region and obtain stable solutions throughout the entire minisuperspace. For small values of the parameter κ, which controls the strength of quantum gravitational effects, the wave packet propagates along the classical trajectory with only mild quantum spreading. As κ increases, enhanced quantum effects lead to significant spreading of the wave packet during its propagation. Nevertheless, when the initial state is localized near the outer horizon, the wave packet becomes localized again in the vicinity of the inner horizon. We refer to this recovery of localization as a refocusing phenomenon. This result suggests that if the geometry is classical near the outer horizon, it becomes classical again near the inner horizon. Within the minisuperspace approximation, inner-horizon formation is not obstructed by quantum gravitational effects.

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Phys. Rev. D 114, 064025– Published 8 September, 2026
  • Received 22 June 2026
  • Accepted 12 August 2026
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