χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy
- 1Independent Researcher , retired person Kharkiv , Ukraine
Abstract
The χ‑operator geometry provides a new spectral framework for describing extremal black holes, whose zero‑temperature states exhibit non‑zero entropy and a collapse of surface gravity. In this work, we reconstruct the extremal horizon as a χ‑spectral degeneracy manifold and show that the vanishing of temperature corresponds to a structural reduction of the χ‑spectrum rather than thermodynamic triviality. The χ‑operator encodes the geometric, quantum, and gravitational degrees of freedom in a unified spectral form, allowing extremal black holes to be treated as zero‑temperature χ‑states with finite horizon complexity. This approach reveals a new mechanism for entropy retention in extremal configurations and establishes a purely operator‑geometric origin of extremal horizon structure.
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BibTeX
@misc{mukha2026χoperator,
title = {{χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy}},
author = {Anatolii Mukha},
year = {2026},
month = aug,
doi = {10.33774/coe-2026-qc6jw},
url = {https://doi.org/10.33774/coe-2026-qc6jw},
note = {CC BY},
}
RIS
TY - UNPB
TI - χ‑Operator Geometry of Extremal Black Holes: Spectral Degeneracy, Zero Temperature, and Non‑Zero Entropy
AU - Anatolii Mukha
PY - 2026
DA - 2026/08/14
DO - 10.33774/coe-2026-qc6jw
UR - https://doi.org/10.33774/coe-2026-qc6jw
AB - The χ‑operator geometry provides a new spectral framework for describing extremal black holes, whose zero‑temperature states exhibit non‑zero entropy and a collapse of surface gravity. In this work, we reconstruct the extremal horizon as a χ‑spectral degeneracy manifold and show that the vanishing of temperature corresponds to a structural reduction of the χ‑spectrum rather than thermodynamic triviality. The χ‑operator encodes the geometric, quantum, and gravitational degrees of freedom in a unified spectral form, allowing extremal black holes to be treated as zero‑temperature χ‑states with finite horizon complexity. This approach reveals a new mechanism for entropy retention in extremal configurations and establishes a purely operator‑geometric origin of extremal horizon structure.
ER -