Radial quantization of the Schwarzschild geometry: relational observables, evaporation, and remnant state
Abstract
We consider the canonical quantization of the Schwarzschild geometry in a minisuperspace description based on a foliation with leaves labeled by the radial variable. This formulation remains regular across the Killing horizon. In this way, the resulting quantum theory admits a physical Hilbert space of states that solve the radial constraint equation and simultaneously describe the exterior and interior regions of the black hole. We discuss the construction of quantum relational observables, which act as gauge-invariant symmetric operators on the physical Hilbert space. In order to analyze quantum effects on the geometry, we construct a family of semiclassical intelligent states that saturate the Robertson-Schrödinger uncertainty relation for both the black-hole mass and the asymptotic Killing norm, while exhibiting small relative fluctuations. The expectation values of relational observables in these states admit a clear geometric interpretation, and they lead to an effective, quantum-corrected spacetime. In particular, we find that the position of the horizon may be shifted, depending on the correlation of the state. In addition, both the proper acceleration of a static observer and the horizon surface gravity are enhanced with respect to a classical Schwarzschild black hole with the same horizon area. Using the relation between surface gravity and temperature, we then obtain corrections to the thermodynamic properties of the black hole, leading to a logarithmically corrected Bekenstein-Hawking entropy. Finally, we discuss the definition of a state of minimal energy that could be interpreted as a stable remnant characterized by a finite horizon radius, suggesting a possible mechanism for preventing the complete evaporation of the black hole.
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BibTeX
@misc{brizuela2026radial,
title = {{Radial quantization of the Schwarzschild geometry: relational observables, evaporation, and remnant state}},
author = {David Brizuela and Leonardo Chataignier and Mikel López-Escondrillas},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26575},
eprint = {2609.26575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26575},
}
RIS
TY - UNPB
TI - Radial quantization of the Schwarzschild geometry: relational observables, evaporation, and remnant state
AU - David Brizuela
AU - Leonardo Chataignier
AU - Mikel López-Escondrillas
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.26575
UR - https://arxiv.org/abs/2609.26575
AB - We consider the canonical quantization of the Schwarzschild geometry in a minisuperspace description based on a foliation with leaves labeled by the radial variable. This formulation remains regular across the Killing horizon. In this way, the resulting quantum theory admits a physical Hilbert space of states that solve the radial constraint equation and simultaneously describe the exterior and interior regions of the black hole. We discuss the construction of quantum relational observables, which act as gauge-invariant symmetric operators on the physical Hilbert space. In order to analyze quantum effects on the geometry, we construct a family of semiclassical intelligent states that saturate the Robertson-Schrödinger uncertainty relation for both the black-hole mass and the asymptotic Killing norm, while exhibiting small relative fluctuations. The expectation values of relational observables in these states admit a clear geometric interpretation, and they lead to an effective, quantum-corrected spacetime. In particular, we find that the position of the horizon may be shifted, depending on the correlation of the state. In addition, both the proper acceleration of a static observer and the horizon surface gravity are enhanced with respect to a classical Schwarzschild black hole with the same horizon area. Using the relation between surface gravity and temperature, we then obtain corrections to the thermodynamic properties of the black hole, leading to a logarithmically corrected Bekenstein-Hawking entropy. Finally, we discuss the definition of a state of minimal energy that could be interpreted as a stable remnant characterized by a finite horizon radius, suggesting a possible mechanism for preventing the complete evaporation of the black hole.
ER -