Quasistatic diagnostic for the ideal emitter model

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.