First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson--Visser-Type Regular Black Hole: Geometrothermodynamics, Phase Structure, and Observational Discriminants
Abstract
We construct the first-law-consistent entropy of a geometrically minimal-length-deformed Schwarzschild spacetime, obtained via the areal-radius substitution R(r)=√(r2+ℓ2) on r∈[0,+∞) with f(R)=1−2M/R, whose nonvanishing Einstein tensor sources an effective geometric fluid with no classical matter counterpart. Integrating the first law gives S=π[rhRh+ℓ2ln((rh+Rh)/ℓ)]. This coincides in functional form with the semiclassical term found independently by Joshi and Joshi, but we fix its boundary condition S(rh=0)=0 on independent physical grounds and adopt it, rather than the Bekenstein--Hawking area law, as the complete entropy of the model, building the free energy, geometrothermodynamics, and mode-stability analysis on it. Evaporation, governed by the Helmholtz free energy F(M), terminates at Mmin=ℓ/2 in a previously unrecognised endpoint: a degenerate extremal regular black hole, where the regular centre coincides with a degenerate Killing horizon of quadratic order, f≈ r2/(2ℓ2), at areal radius ℓ, with TH→ 0, S→ 0, C→ 0+, and finite curvature everywhere; we display its Penrose--Carter structure for the first time. The same entropy, with ℓ, defines the equilibrium state space of a Legendre-invariant geometrothermodynamic (GTD) description whose curvature scalar diverges independently at the Davies-type transition M*=ℓ/√2 and at Mmin, a divergence with no counterpart in the minimal-length black hole literature, specific to a genuine horizon at S=0. We embed this remnant within the observational discriminant noted qualitatively by Tsukamoto: exact shadow degeneracy combined with a measurable photon-ring flux enhancement rn=e−2π/a>e−2π, accessible to next-generation very-long-baseline interferometry.
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BibTeX
@misc{toghrai2026firstlaw,
title = {{First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson--Visser-Type Regular Black Hole: Geometrothermodynamics, Phase Structure, and Observational Discriminants}},
author = {T. Toghrai and N. Mansour and A. Daassou and R. Benbrik},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17588},
eprint = {2609.17588},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17588},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - First-Law Entropy and a Degenerate Extremal Remnant in a Minimal-Length Simpson--Visser-Type Regular Black Hole: Geometrothermodynamics, Phase Structure, and Observational Discriminants
AU - T. Toghrai
AU - N. Mansour
AU - A. Daassou
AU - R. Benbrik
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.17588
UR - https://arxiv.org/abs/2609.17588
AB - We construct the first-law-consistent entropy of a geometrically minimal-length-deformed Schwarzschild spacetime, obtained via the areal-radius substitution R(r)=√(r^2+ℓ^2) on r∈[0,+∞) with f(R)=1−2M/R, whose nonvanishing Einstein tensor sources an effective geometric fluid with no classical matter counterpart. Integrating the first law gives S=π[r_hR_h+ℓ^2ln((r_h+R_h)/ℓ)]. This coincides in functional form with the semiclassical term found independently by Joshi and Joshi, but we fix its boundary condition S(r_h=0)=0 on independent physical grounds and adopt it, rather than the Bekenstein--Hawking area law, as the complete entropy of the model, building the free energy, geometrothermodynamics, and mode-stability analysis on it. Evaporation, governed by the Helmholtz free energy F(M), terminates at M_min=ℓ/2 in a previously unrecognised endpoint: a degenerate extremal regular black hole, where the regular centre coincides with a degenerate Killing horizon of quadratic order, f≈ r^2/(2ℓ^2), at areal radius ℓ, with T_H→ 0, S→ 0, C→ 0^+, and finite curvature everywhere; we display its Penrose--Carter structure for the first time. The same entropy, with ℓ, defines the equilibrium state space of a Legendre-invariant geometrothermodynamic (GTD) description whose curvature scalar diverges independently at the Davies-type transition M^*=ℓ/√2 and at M_min, a divergence with no counterpart in the minimal-length black hole literature, specific to a genuine horizon at S=0. We embed this remnant within the observational discriminant noted qualitatively by Tsukamoto: exact shadow degeneracy combined with a measurable photon-ring flux enhancement r_n=e^−2π/a>e^−2π, accessible to next-generation very-long-baseline interferometry.
ER -