Ideal photon reservoir and active-area mass-balance model

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.