Canonical Quantum Statistics and Gravitational Thermodynamics A Pedagogical Treatment of Gravity, Information, and Entropy

This technical report develops the mathematical and conceptual framework connecting general relativity, quantum field theory in curved spacetime, black-hole thermodynamics, and quantum information. Beginning with the differential-geometric and field-theoretic foundations, it systematically establishes the network of relations linking horizon geometry, entanglement, and gravitational dynamics. The core topics treated are: (i) Hawking's semiclassical theory of black-hole radiation, derived through Bogoliubov analysis and the KMS condition on the Hartle–Hawking state; (ii) the Bekenstein–Hawking entropy and the four laws of black-hole mechanics; (iii) Jacobson's derivation of Einstein's equations as an equation of state on local Rindler horizons, with explicit attention to the role of energy–momentum conservation as an independent input and the cosmological constant as an integration constant; (iv) Wald's Noether-charge formulation of black-hole entropy and its higher-derivative generalisations; (v) the Unruh effect and the observer-dependence of the particle concept; (vi) the AdS/CFT correspondence and the Ryu–Takayanagi formula, together with the Lewkowycz–Maldacena replica derivation; (vii) generalised entropy, the quantum extremal surface prescription, and the island formula in JT gravity coupled to matter; and (viii) the Page curve and what it does and does not establish concerning the black-hole information paradox. The presentation distinguishes carefully between rigorously established results — Hawking temperature, the area theorem, the four laws, the leading-order RT formula, and the FGHMV linearised-Einstein theorem — and active research programmes, including the microscopic origin of the area law, the operational mechanism of information return, holography beyond AdS, and the interpretation of gravity as emergent. Three appendices provide a full Bogoliubov calculation, the Wald Noether-charge construction, and a mathematical reference.