IndisputableMonolith.Relativity.Compact.BlackHoleEntropy
This module defines the horizon area for Schwarzschild black holes and derives Bekenstein-Hawking entropy from ledger capacity in the Recognition Science setting. Quantum gravity and black hole thermodynamics researchers cite it to connect classical geometry to information bounds. It supplies targeted definitions and positivity lemmas that rest on the imported metric and constants modules.
claimThe horizon area of a Schwarzschild black hole is $A = 4π r_s^2$. Entropy follows from the ledger capacity limit as $S_{BH} = A/4$ (in RS-native units), with supporting results for positivity and saturation.
background
The module sits inside the static spherically symmetric solutions of the parent Compact module, where the line element takes the form $ds^2 = -f(r)dt^2 + g(r)dr^2 + r^2 dΩ^2$. It imports the RS time quantum $τ_0 = 1$ tick from Constants and the metric geometry from Relativity.Geometry.Metric. Core objects are HorizonArea (event-horizon surface area), LedgerCapacityLimit, and the map bh_entropy_from_ledger that converts geometric area into an information-theoretic bound.
proof idea
This is a definition module, no proofs. It introduces HorizonArea directly from the Schwarzschild radius, then supplies short lemmas (horizon_area_pos, ledger_capacity_pos_of_area_pos, sbh_saturation_uniqueness) that establish non-negativity and uniqueness of the entropy expression under the ledger-capacity hypothesis.
why it matters in Recognition Science
The module supplies the geometric input required by the parent Relativity.Compact module for its treatment of static solutions and entropy. It closes the derivation of Bekenstein-Hawking entropy from ledger capacity, linking the Recognition Composition Law to the area law without additional dynamical assumptions. Downstream work in Compact uses these results to place black-hole thermodynamics inside the phi-ladder and eight-tick structure.
scope and limits
- Does not treat rotating or charged black holes.
- Does not derive Hawking radiation or dynamical evaporation.
- Does not assign numerical entropy values without external mass input.
- Does not prove the area law outside the ledger-capacity hypothesis.