Ideal photon reservoir and active-area mass balance

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) and use the fixed-reference mathematical emitter model of Exact ideal emitter mass ODE and entropy accounting, with positive constants $G,c$, a positive mass function $M(t_{\mathrm{\infty }})$, and the model horizon radius $r_H(M)=2GM\mathrm{/}{c}^2$. Here $t_{\mathrm{\infty }}$ is measured in seconds, $M$ in kilograms, $r_H$ in metres, $G$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, and $c$ in $\mathrm{m}{\mathrm{s}}^{-1}$. For each $t_{\mathrm{\infty }}\in [0,\tau ]$, including the positive-mass cutoff endpoint supplied by the model, where $M(\tau )=M_{\min }>0$, define the selected crossing-time scale and dimensionless fractional-change diagnostic by

$t_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}(t_{\mathrm{\infty }}):=\frac{r_H(M(t_{\mathrm{\infty }}))}{c},{ϵ}_{\mathrm{q}\mathrm{s}}(t_{\mathrm{\infty }}):=\frac{\mathrm{\mid }{M}^{\mathrm{\prime }}(t_{\mathrm{\infty }})\mathrm{\mid }{t}_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}(t_{\mathrm{\infty }})}{M(t_{\mathrm{\infty }})}\mathrm{.}$

Thus $t_{\mathrm{c}\mathrm{r}\mathrm{o}\mathrm{s}\mathrm{s}}$ has units of seconds and ${ϵ}_{\mathrm{q}\mathrm{s}}$ is dimensionless. This is a diagnostic defined from the adopted mass ODE and the chosen length scale $r_H\mathrm{/}c$; it is not a measured crossing time, an estimate of error against a time-dependent Einstein solution, or a proof that the ideal model approximates such a solution.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Let $G,c,\mathrm{\hslash },\eta ,M_0,M_{\min }>0$ be the relevant constants and initial/cutoff data of Exact ideal emitter mass ODE and entropy accounting, with $0<\eta \le 1$ and $0<M_{\min }<M_0$. Let

$$M(t_{\mathrm{\infty }})=(M_0^3-3\alpha t_{\mathrm{\infty }}{)}^{1\mathrm{/}3},\alpha =\frac{\eta \mathrm{\hslash }{c}^4}{15360\pi G^2},\tau =\frac{M_0^3-M_{\min }^{}}{3\alpha }$$

be the positive solution and cutoff supplied by that proposition. Use the diagnostic of Quasistatic diagnostic for the ideal emitter model. Then

$${ϵ}_{\mathrm{q}\mathrm{s}}(t_{\mathrm{\infty }})=\frac{2G\alpha }{c^{3}M(t_{\mathrm{\infty }}{)}^2}\le \frac{2G\alpha }{c^3{M}_{\min }^{}}\mathrm{.}$$

For any ${ϵ}_0>0$, the condition $M_{\min }^{}\ge 2G\alpha \mathrm{/}(c^3{ϵ}_0)$ therefore implies ${ϵ}_{\mathrm{q}\mathrm{s}}(t_{\mathrm{\infty }})\le {ϵ}_0$ throughout $[0,\tau ]$.

There is also an exact finite-interval bound. If $0\le t_{\mathrm{\infty }}\le t_{\mathrm{\infty }}+\mathrm{\Delta }t\le \tau$ and

$$\delta :=\frac{3\alpha \mathrm{\Delta }t}{M(t_{\mathrm{\infty }}{)}^3}\le \frac{1}{2},$$

then

$$\frac{M(t_{\mathrm{\infty }}+\mathrm{\Delta }t)}{M(t_{\mathrm{\infty }})}=(1-\delta {)}^{1\mathrm{/}3},\mid \frac{M(t_{\mathrm{\infty }}+\mathrm{\Delta }t)}{M(t_{\mathrm{\infty }})}-1\mid \le \frac{2^{2\mathrm{/}3}}{3}\delta \mathrm{.}$$

Both bounds are exact internal consequences of the adopted ODE. They give no error bound between that ODE and an unconstructed Einstein–quantum backreaction solution.

Use SI constants $G_N>0$ in ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $c>0$ in $\mathrm{m}{\mathrm{s}}^{-1}$, $\mathrm{\hslash }>0$ in $\mathrm{J}\mathrm{s}$, and $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$. In Exact ideal emitter mass ODE and entropy accounting, the mathematical constant $G$ is identified with this same $G_N$.

For each fixed mass $M>0$ in kilograms, take the zero-angular-momentum, zero-charge member of the selected Kerr–Newman stationary family Physical KN family work and constancy, with geometric mass length $m=G_{N}M\mathrm{/}{c}^2$ in metres. In the Schwarzschild exterior use coordinates $(x^0,r,\vartheta ,\phi )$, $x^0=c{t}_{\mathrm{\infty }}$ in metres, $r>2m$, and metric

$$d{s}^2=-(1-\frac{2m}{r})(d{x}^0{)}^2+{(1-\frac{2m}{r})}^{-1}d{r}^2+r^{2}d{\mathrm{\Omega }}_2^2,$$

where $d{\mathrm{\Omega }}_2^2$ is the unit-sphere metric. The asymptotic generator is $\xi ={\mathrm{\partial }}_{x^0}$, with $g(\xi ,\xi )\to -1$ at infinity. The selected stationary-family radius, area, and geometric surface gravity are

$$r_H(M)=2m=\frac{2G_{N}M}{c^2},A_H(M)=4\pi r_H(M{)}^2,{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(M)=\frac{1}{4m}=\frac{c^2}{4G_{N}M}\mathrm{.}$$

Thus $r_H$ is in metres, $A_H$ in ${\mathrm{m}}^2$, and ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ in ${\mathrm{m}}^{-1}$, using the length-normalized generator convention of Thermodynamics-GR fields, currents, reference normalization and units.

Adopt the selected reduced-model reference temperature from Restricted reversible horizon entropy identification on this Schwarzschild subfamily:

$$T_{\mathrm{\infty }}(M):=T_K=T_{\mathrm{r}\mathrm{e}\mathrm{d}}=\frac{\mathrm{\hslash }c{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}(M)}{2\pi k_B}=\frac{\mathrm{\hslash }{c}^3}{8\pi G_{N}M{k}_B}>0.$$

This is an adopted model identification for one fixed asymptotic reference. It is not a local proper-temperature field and does not assert a general Hawking state. The reversible entropy state $S_H$ is the value of the stationary-family state function on each Schwarzschild member; evaluating that family function along a changing parameter $M(t_{\mathrm{\infty }})$ is bookkeeping in this model, not a claim that the changing family is an exact spacetime solution.

Adopt the zero-mode-excluded, zero-point-subtracted, two-polarization photon Gibbs model and isotropic freely streaming ray convention of Actual photon box trace and Planck energy/entropy flux, under ${\mathrm{A}\mathrm{C}}_{\omega }$ from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Its energy density $u$, coarse diagonal-mode entropy density $s$, and one-sided fluxes are those of that theorem. This identification is an ideal reservoir assumption; no photon-wall equilibration theorem or physical preparation is asserted.

Fix $0<\eta \le 1$, a constant dimensionless active-area fraction, an initial mass $M_0>0$, and a positive cutoff $0<M_{\min }<M_0$. Let $M(t_{\mathrm{\infty }})>0$ be $C^1$ on an interval containing its first cutoff time $\tau$, with $M(0)=M_0$, and let $t_{\mathrm{\infty }}$ be measured in seconds by the fixed asymptotic time parameter. Define $E_{\mathrm{\infty }}(t_{\mathrm{\infty }})=M(t_{\mathrm{\infty }})c^2$ in joules and the effective far-field emitting area

$$A_{\mathrm{e}\mathrm{f}\mathrm{f}}(M):=\eta A_H(M)\mathrm{.}$$

Adopt zero incoming flux, the power law $P_{\mathrm{\infty }}=A_{\mathrm{e}\mathrm{f}\mathrm{f}}\sigma T_{\mathrm{\infty }}^4$ in watts, the balance equation

$$\frac{d{E}_{\mathrm{\infty }}}{d{t}_{\mathrm{\infty }}}=c^2{M}^{\mathrm{\prime }}(t_{\mathrm{\infty }})=-P_{\mathrm{\infty }}(t_{\mathrm{\infty }}),$$

and the coarse exported entropy rate $S_{\mathrm{r}\mathrm{a}\mathrm{d}}^{\mathrm{\prime }}=(4\mathrm{/}3)P_{\mathrm{\infty }}\mathrm{/}{T}_{\mathrm{\infty }}$, with $S_{\mathrm{r}\mathrm{a}\mathrm{d}}(0)=0$. Here $S_{\mathrm{r}\mathrm{a}\mathrm{d}}$ is the entropy carried by the ideal Gibbs/ray model, not an outside von Neumann entropy. The effective area is not a material patch of the null horizon and $\eta$ is not a frequency-independent greybody factor applied separately to entropy.

The model holds only for the fixed normalization $\xi$. Under $\xi \mapsto b\xi$, $b>0$, the model conventions scale $E_{\xi }\mapsto b{E}_{\xi }$, $t_{\xi }\mapsto t_{\xi }\mathrm{/}b$, ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}\mapsto b{\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$, and leave $A_H$ unchanged. The Stefan closure above is therefore not stipulated as a reference-independent luminosity law. No experiment, coupled backreaction solution, physical extinction event, or universal generalized second law is adopted.

Assume the ideal model and fixed asymptotic normalization of Ideal photon reservoir and active-area mass-balance model, and explicitly assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined by The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). 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

$$\mathrm{\Delta }{S}_H:=S_H(M_0)-S_H(M_{\min })\mathrm{.}$$

For the fixed-reference effective-area model,

$$P_{\mathrm{\infty }}(M)=\frac{\eta \mathrm{\hslash }{c}^6}{15360\pi G_N^2{M}^2},\alpha =\frac{\eta \mathrm{\hslash }{c}^4}{15360\pi G_N^2},M(t_{\mathrm{\infty }})=(M_0^3-3\alpha t_{\mathrm{\infty }}{)}^{1\mathrm{/}3},\tau =\frac{M_0^3-M_{\min }^{}}{3\alpha }\mathrm{.}$$

The model exports energy $c^2(M_0-M_{\min })$ joules, and its coarse photon entropy satisfies

$$S_H(M)=\frac{4\pi k_B{G}_N{M}^2}{\mathrm{\hslash }c}+s_{\ast },s_{\ast }\in \mathbb{R}\text{ constant on the stationary family},$$ $$S_{\mathrm{r}\mathrm{a}\mathrm{d}}(\tau )=\frac{4}{3}\mathrm{\Delta }{S}_H>0,\mathrm{\Delta }(S_H+S_{\mathrm{r}\mathrm{a}\mathrm{d}})=\frac{1}{3}\mathrm{\Delta }{S}_H>0.$$

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,

$${ϵ}_{\mathrm{q}\mathrm{s}}(t_{\mathrm{\infty }})=\frac{2G_N\alpha }{c^{3}M(t_{\mathrm{\infty }}{)}^2},\underset{0\le t_{\mathrm{\infty }}\le \tau }{\sup }{ϵ}_{\mathrm{q}\mathrm{s}}=\frac{2G_N\alpha }{c^3{M}_{\min }^{}}\mathrm{.}$$

Thus $M_{\min }^{}\ge 2G_N\alpha \mathrm{/}(c^3{ϵ}_0)$ implies ${ϵ}_{\mathrm{q}\mathrm{s}}\le {ϵ}_0$ for any ${ϵ}_0>0$. The finite-interval bound in Exact quasistatic and finite-interval mass-change bounds also applies whenever its stated $\delta \le 1\mathrm{/}2$ condition holds.

The Stefan closure is tied to the fixed reference: under $\xi \mapsto b\xi$, power scales as $b^2$, while $T_{\xi }^4$ 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.