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Restricted smooth horizon Einstein/NEC adoption — Examples

1 · Prerequisites

2 · Summary

This companion constructs the zero-spin, zero-charge member of the selected stationary family and verifies its exact SI area-work differential. It records a conditional model calculation, with no measurement or physical-process claim.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

Thought experimentStatement: AI-adaptedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

Classical Schwarzschild SI family work

Physical scope

A conditional exact calculation on the zero-spin, zero-charge Schwarzschild slice of the selected stationary family. It is not a process, observation, evaporation model, or entropy assignment.

Statement

Assume ${A C}_{ω}$ as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Fix $G_{N}, c > 0$ and choose a selected family member with SI mass parameter $m_{*} > 0$ , angular-momentum parameter $J = 0$ , electric-charge parameter $Q_{e} = 0$ , and zero magnetic flux. Let $E_{p a r} : = m_{*} c^{2}$ be the energy-valued parameter from Physical KN family work and constancy. On this Schwarzschild slice the selected outer horizon has radius $r_{H} = 2 G_{N} m_{*} / c^{2}$ , area $A_{H} = \frac{16 π G_{N}^{2} m_{*}^{2}}{c^{4}},$ and acceleration-unit surface gravity $κ_{a c c} = \frac{c^{4}}{4 G_{N} m_{*}} .$ As $m_{*}$ varies over $(0, \infty)$ with the other parameters fixed, the family differentials satisfy $d E_{p a r} = \frac{κ_{a c c} c^{2}}{8 π G_{N}} d A_{H} .$ This is a conditional identity along stationary model parameters; it does not describe an accretion process, report an observation, or assign temperature or entropy.

Facts & Assumptions

Given: The selected KN family theorem and its zero-spin, zero-charge Schwarzschild slice, fixed constants $G_{N}, c > 0$ , and $m_{*} > 0$ .

[F1]

Physical KN family work and constancy: with geometric mass parameter $m = G_{N} m_{*} / c^{2}$ , $J = Q_{e} = 0$ gives $r_{H} = 2 m$ , $A_{H} = 16 π m^{2}$ , $κ_{g e o m} = 1 / (4 m)$ , $κ_{a c c} = c^{2} κ_{g e o m}$ , and $E_{p a r} = m_{*} c^{2}$ . The physical horizon theorem's nonzero-spin-or-charge domain is completed at this point by the exact Schwarzschild supplier Exact Schwarzschild horizon data at the zero-rotation KN point.

[F2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means every countable family of nonempty sets has a choice function; the KN family supplier carries this convention from its geometric flux parametrization.

[A1]

$G_{N}, c$ are fixed positive constants and $m_{*}$ ranges over $(0, \infty)$ ; no measured input or physical process is assumed.

Proof

1.1givenF1F2A1algebra

The inherited ${A C}_{ω}$ premise [F2] is retained; the finite parameter calculation below makes no stronger choice. Set $m : = G_{N} m_{*} / c^{2}$ . By [F1], the geometric horizon radius is $2 m$ , so $A_{H} = 16 π m^{2} = 16 π G_{N}^{2} m_{*}^{2} / c^{4}$ and $κ_{a c c} = c^{2} / (4 m) = c^{4} / (4 G_{N} m_{*})$ . Also $E_{p a r} = m_{*} c^{2}$ . The special case is defined for every $m_{*} > 0$ ; it is not a limiting inference from a charged or rotating horizon.

2.1A1step 1.1algebra∎

Differentiating the explicit area function on the open interval $m_{*} > 0$ gives $d A_{H} = \frac{32 π G_{N}^{2} m_{*}}{c^{4}} d m_{*} .$ Consequently $\frac{κ_{a c c} c^{2}}{8 π G_{N}} d A_{H} = \frac{c^{6}}{32 π G_{N}^{2} m_{*}} \frac{32 π G_{N}^{2} m_{*}}{c^{4}} d m_{*} = c^{2} d m_{*} = d E_{p a r} .$ This verifies the stated family differential identity exactly.