Bound States Inside the Horizon and Black Hole Entropy
Abstract
We study the bound states perturbations in the interior of the Schwarzschild black hole and examine whether or not their microstates spectrum can furnish a statistical description of the black hole entropy. These states contain a discrete set of modes characterized by the labels (s, p,ℓ,m), where s represents the spin of the species, p denotes the radial excitation number while (ℓ,m) are the usual angular momentum labels. Requiring that the shortest wavelength resolved by the highly excited bound states remains above the quantum gravity scale introduces an angular cutoff ℓmax∼ rH/LPl. The resulting angular degeneracy therefore scales with the horizon area. We count the corresponding configurations in the microcanonical ensemble and find an entropy of the form Sbs=αA/ℏ G. Demanding that the bound state sector to be in thermal equilibrium with the black hole at the Hawking temperature fixes the fraction of the bound state energy in total black hole energy. For the gravitational perturbations sector, the equilibrium bound state contribution in entropy and the energy fraction are typically small. We also examine the conditions under which the bound states could account for the full black hole entropy. Although such saturation is possible at the level of state counting, it requires an order unity fraction in the black hole energy and therefore the backreactions on the background geometry cannot be ignored.
Related
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BibTeX
@misc{firouzjahi2026bound,
title = {{Bound States Inside the Horizon and Black Hole Entropy}},
author = {Hassan Firouzjahi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21426},
eprint = {2610.21426},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21426},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bound States Inside the Horizon and Black Hole Entropy
AU - Hassan Firouzjahi
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21426
UR - https://arxiv.org/abs/2610.21426
AB - We study the bound states perturbations in the interior of the Schwarzschild black hole and examine whether or not their microstates spectrum can furnish a statistical description of the black hole entropy. These states contain a discrete set of modes characterized by the labels (s, p,ℓ,m), where s represents the spin of the species, p denotes the radial excitation number while (ℓ,m) are the usual angular momentum labels. Requiring that the shortest wavelength resolved by the highly excited bound states remains above the quantum gravity scale introduces an angular cutoff ℓ_max∼ r_H/L_Pl. The resulting angular degeneracy therefore scales with the horizon area. We count the corresponding configurations in the microcanonical ensemble and find an entropy of the form S_bs=αA/ℏ G. Demanding that the bound state sector to be in thermal equilibrium with the black hole at the Hawking temperature fixes the fraction of the bound state energy in total black hole energy. For the gravitational perturbations sector, the equilibrium bound state contribution in entropy and the energy fraction are typically small. We also examine the conditions under which the bound states could account for the full black hole entropy. Although such saturation is possible at the level of state counting, it requires an order unity fraction in the black hole energy and therefore the backreactions on the background geometry cannot be ignored.
ER -