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Conditional ideal Planck emission and entropy accounting

Physical scope

Conditional exact conclusion in the zero-mode-excluded Planck/Gibbs ray model and fixed-asymptotic-reference positive-cutoff Schwarzschild effective-area ODE. The area entropy input is restricted to the stationary family interface, and the explicit AC_omega assumption is carried. The quasistatic ratio is an internal diagnostic, not an error bound against a dynamic Einstein solution. No actual Hawking flux, coupled backreaction, empirical observation, reference-independent Stefan law, or universal generalized second law is asserted.

Statement

Assume the ideal model and fixed asymptotic normalization of Ideal photon reservoir and active-area mass-balance model, and explicitly assume ${A C}_{ω}$ as defined by The Axiom of Countable Choice ( ${A C}_{ω}$ ). Use $G = G_{N}$ between the mathematical ODE input and the physical Schwarzschild family. Also assume the conditional family-entropy interface stated by Conditional area entropy coefficient with free offset with its actual hypotheses. Let $S_{H} (M)$ denote that state-function value at the stationary Schwarzschild member of mass $M$ , and put

Δ S_{H} : = S_{H} (M_{0}) - S_{H} (M_{\min}) .

For the fixed-reference effective-area model,

P_{\infty} (M) = \frac{η ℏ c^{6}}{15360 π G_{N}^{2} M^{2}}, α = \frac{η ℏ c^{4}}{15360 π G_{N}^{2}}, M (t_{\infty}) = (M_{0}^{3} - 3 α t_{\infty})^{1 / 3}, τ = \frac{M_{0}^{3} - M_{\min}^{}}{3 α} .

The model exports energy $c^{2} (M_{0} - M_{\min})$ joules, and its coarse photon entropy satisfies

S_{H} (M) = \frac{4 π k_{B} G_{N} M^{2}}{ℏ c} + s_{*}, s_{*} \in R constant on the stationary family,

S_{r a d} (τ) = \frac{4}{3} Δ S_{H} > 0, Δ (S_{H} + S_{r a d}) = \frac{1}{3} Δ S_{H} > 0.

The absolute entropy offsets in the mathematical ODE model and the stationary-family state function need not be identified; their differences agree because both have the same area coefficient on this branch.

For the diagnostic of Quasistatic diagnostic for the ideal emitter model,

ϵ_{q s} (t_{\infty}) = \frac{2 G_{N} α}{c^{3} M (t_{\infty})^{2}}, \sup_{0 \leq t_{\infty} \leq τ} ϵ_{q s} = \frac{2 G_{N} α}{c^{3} M_{\min}^{}} .

Thus $M_{\min}^{} \geq 2 G_{N} α / (c^{3} ϵ_{0})$ implies $ϵ_{q s} \leq ϵ_{0}$ for any $ϵ_{0} > 0$ . The finite-interval bound in Exact quasistatic and finite-interval mass-change bounds also applies whenever its stated $δ \leq 1 / 2$ condition holds.

The Stefan closure is tied to the fixed reference: under $ξ \mapsto b ξ$ , power scales as $b^{2}$ , while $T_{ξ}^{4}$ scales as $b^{4}$ and area is unchanged. This is an exact conditional model calculation, not a Hawking-radiation theorem, a time-dependent Einstein solution, a backreaction error estimate, an observed evaporation history, or a universal generalized second law.

Facts & Assumptions

Given: The positive constants, fixed asymptotic reference, zero-mode-excluded photon/ray model, adopted effective area and mass balance, and positive cutoff of Ideal photon reservoir and active-area mass-balance model, together with the explicit ${A C}_{ω}$ assumption.

[F1]

Ideal photon reservoir and active-area mass-balance model: on the selected Schwarzschild stationary family $r_{H} = 2 G_{N} M / c^{2}$ , $A_{H} = 4 π r_{H}^{2}$ , and $κ_{g e o m} = c^{2} / (4 G_{N} M)$ ; it adopts $T_{\infty} = ℏ c κ_{g e o m} / (2 π k_{B})$ , $A_{e f f} = η A_{H}$ , $P_{\infty} = A_{e f f} σ T_{\infty}^{4}$ , $c^{2} M^{'} = - P_{\infty}$ , and coarse entropy rate $S_{r a d}^{'} = (4 / 3) P_{\infty} / T_{\infty}$ until $M_{\min} > 0$ . These are model assumptions.

[F2]

Actual photon box trace and Planck energy/entropy flux: under its zero-mode-excluded two-polarization Gibbs trace and isotropic ray model, $F = σ T^{4}$ , $F_{S} = (4 / 3) F / T$ , and $σ = π^{2} k_{B}^{4} / (60 ℏ^{3} c^{2})$ ; its trace and Fourier/Riemann arguments use ${A C}_{ω}$ .

[F3]

Exact ideal emitter mass ODE and entropy accounting: for the same $G, c, ℏ, k_{B}, η, M_{0}, M_{\min}$ , fixed reference, and adopted ODE, the exact positive solution, cutoff time, energy export, and coarse entropy formula with $S_{B} (M) = 4 π k_{B} G M^{2} / (ℏ c) + S_{0}$ hold.

[F4]

Quasistatic diagnostic for the ideal emitter model: $t_{c r o s s} = r_{H} / c$ and $ϵ_{q s} = ∣ M^{'} ∣ t_{c r o s s} / M$ , as internal ODE definitions only.

[F5]

Exact quasistatic and finite-interval mass-change bounds: the exact $ϵ_{q s}$ formula, positive-cutoff upper bound, threshold criterion, and finite-interval mass-ratio bound stated above follow from the ODE.

[L1]

Conditional area entropy coefficient with free offset currently states that on the connected, strictly subextremal, zero-magnetic-flux stationary Kerr–Newman family, conditional on the reversible-heat identification and selected reduced temperature, $S_{H} = k_{B} c^{3} A_{H} / (4 G_{N} ℏ) + s_{*}$ . Its explicit Countable Choice premise, reversible-heat adoption and same-generator reduced-temperature identification are retained here. The current reduced-temperature Statement also explicitly carries Countable Choice. This use is only the selected stationary-family interface, not an inference of a dynamical spacetime or microscopic entropy.

[A1]

The geometric mass length for the Schwarzschild specialization is $m = G_{N} M / c^{2} > 0$ , $a = q = 0$ ; hence it lies in the strict subextremal family. The model mass schedule is not asserted to solve the Einstein equations.

[A2]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): ${A C}_{ω}$ means that every countable family of nonempty sets has a choice function. This is the exact set-theoretic hypothesis of the cited photon-spectrum chain; no full Axiom of Choice is assumed.

Proof

Given: ${A C}_{ω}$ , the model assumptions of [F1], the mathematical flux relation of [F2], the exact ODE result [F3], and the conditional family interface [L1].

1.1givenF1F2A2algebra

The outgoing one-sided flux in [F2] and the adopted area/temperature in [F1] give $P_{\infty} = η A_{H} σ T_{\infty}^{4}$ . Substituting $A_{H} = 16 π G_{N}^{2} M^{2} / c^{4}$ , $T_{\infty} = ℏ c^{3} / (8 π G_{N} M k_{B})$ , and $σ = π^{2} k_{B}^{4} / (60 ℏ^{3} c^{2})$ yields $P_{\infty} = η ℏ c^{6} / (15360 π G_{N}^{2} M^{2})$ . The adopted balance equation is therefore $M^{'} = - α M^{- 2}$ with $α = η ℏ c^{4} / (15360 π G_{N}^{2}) > 0$ .

1.2givenF1F3A2

The hypotheses of [F3] match the fixed-reference area, temperature, outgoing flux, mass-balance law, and strictly positive cutoff in [F1]. Its exact solution gives the displayed $M (t_{\infty})$ , $τ$ , and energy export. No continuation to $M = 0$ is used.

1.3givenF1F3L1A1A2algebra

Put $m = G_{N} M / c^{2}$ , $a = q = 0$ . Since $m > 0$ , this point lies in the strict subextremal domain of [L1]; the selected Schwarzschild horizon area is the same $A_{H} = 16 π G_{N}^{2} M^{2} / c^{4}$ used by [F1] and [F3]. Therefore [L1] gives $S_{H} (M) = k_{B} c^{3} A_{H} (M) / (4 G_{N} ℏ) + s_{*}$ , while [F3] gives $S_{B} (M) = k_{B} c^{3} A_{H} (M) / (4 G_{N} ℏ) + S_{0}$ . Their constant offsets may differ, but subtracting the values at $M_{0}$ and $M_{\min}$ cancels each offset and gives $S_{H} (M_{0}) - S_{H} (M_{\min}) = S_{B} (M_{0}) - S_{B} (M_{\min}) > 0$ , because $M_{0} > M_{\min} > 0$ . The ${A C}_{ω}$ assumption [A2] is carried for the reduced-spectrum input in [L1].

1.4F3F4F5A2algebra

The definition [F4] and ODE of [F3] give $ϵ_{q s} = ∣ M^{'} ∣ (2 G_{N} M / c^{3}) / M = 2 G_{N} α / (c^{3} M^{2})$ . Since $M \geq M_{\min}$ , the supremum is the displayed positive-cutoff value; the threshold follows by multiplying by $M_{\min}^{} > 0$ . The exact finite-interval ratio and bound are [F5]. These are internal properties of the stipulated ODE.

2.1F3step 1.3A2algebra

By [F3], $S_{r a d} (τ) = (4 / 3) [S_{B} (M_{0}) - S_{B} (M_{\min})]$ . Step 1.3 replaces this difference by $Δ S_{H}$ . Thus $S_{r a d} (τ) = (4 / 3) Δ S_{H}$ , and $S_{H} (M_{\min}) + S_{r a d} (τ) - S_{H} (M_{0}) = - (Δ S_{H}) + (4 / 3) Δ S_{H} = (1 / 3) Δ S_{H} > 0$ . This is the stated coarse model entropy accounting only.

3.1F1algebra∎

Under the rescaling $ξ_{b} = b ξ$ , the postulate's model conventions give $E_{b} (t_{b}) = b E_{\infty} (t_{\infty})$ , $t_{b} = t_{\infty} / b$ , and $κ_{b} = b κ_{g e o m}$ , with $b > 0$ and the same mass history. Define the positive emitted power in the rescaled variables by $P_{b} : = - d E_{b} / d t_{b}$ . Differentiating the rescaled energy with respect to the rescaled time and using the stipulated balance equation $d E_{\infty} / d t_{\infty} = - P_{\infty}$ gives $P_{b} = b^{2} P_{\infty}$ . The temperature formula in [F1] gives $T_{b} = ℏ c κ_{b} / (2 π k_{B}) = b T_{\infty}$ , so the Stefan expression would instead scale as $A_{e f f} σ T_{b}^{4} = b^{4} A_{e f f} σ T_{\infty}^{4}$ , with unchanged area and coefficient. Since $P_{\infty} > 0$ , these scalings agree at the fixed normalization $b = 1$ but not for arbitrary $b$ ; the closure in [F1] is therefore not a reference-independent luminosity law.

Conditional ideal Planck emission and entropy accounting

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources