Physical KN family work and constancy

Fix $G_N,c,{\mu }_0>0$ and ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$. Let $m_{\ast }>0$ be the selected SI mass parameter, $J\in \mathbb{R}$ the selected angular-momentum parameter, and $Q_e\in \mathbb{R}$ the selected electric charge parameter, with zero magnetic flux. Define the geometric length parameters and field normalization by $m:=\frac{G_N{m}_{\ast }}{c^2},a:=\frac{J}{m_{\ast }c},q:=\sqrt{\frac{G_N}{4\pi {\epsilon }_0{c}^4}}{Q}_e,\alpha :=\sqrt{\frac{4\pi G_N}{{\mu }_0{c}^4}}\mathrm{.}$ Assume the strict subextremal condition $m^2>a^2+q^2$. Set $s:=\sqrt{m^2-a^2-q^2},r_{+}:=m+s,D:=r_{+}^2+a^2,A_H:=4\pi D\mathrm{.}$ Assume ${\mathrm{A}\mathrm{C}}_{\omega }$ as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$), the choice convention inherited by the selected geometric flux parametrization; no full Axiom of Choice is used. In the selected smooth ingoing KN model, the outer Killing horizon has generator $\chi :={\mathrm{\partial }}_v+\frac{a}{D}\mathrm{\Psi },$ where $v$ is a length coordinate and $\mathrm{\Psi }=X{\mathrm{\partial }}_Y-Y{\mathrm{\partial }}_X$ is the smooth rotation field of the unit two-sphere. Normalize time by $t=v\mathrm{/}c$, and define ${\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}:=\frac{ca}{D},{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}:=\frac{c^{2}s}{D},{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}:=\frac{cq{r}_{+}}{\alpha D}\mathrm{.}$ Thus $c\chi ={\mathrm{\partial }}_t+{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}\mathrm{\Psi }$; ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}$ is the acceleration-unit surface gravity; and ${\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}$ is the voltage in the displayed gauge, constant on the horizon cross-section and vanishing at spatial infinity. The quantities ${\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}$, ${\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}$, and ${\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}$ are constant on each fixed cross-section because the selected horizon formulas depend only on the family parameters. Let $E_{\mathrm{p}\mathrm{a}\mathrm{r}}:=m_{\ast }{c}^2$ be the energy-valued parameter derived from the selected mass input. For ordinary differentials on the strict physical parameter domain, with $G_N,c,{\mu }_0$ fixed, the exact family identities are $d{E}_{\mathrm{p}\mathrm{a}\mathrm{r}}=\frac{{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}{c}^2}{8\pi G_N}d{A}_H+{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}dJ+{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}d{Q}_e,$ and $E_{\mathrm{p}\mathrm{a}\mathrm{r}}=\frac{{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}{c}^2}{4\pi G_N}{A}_H+2{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}J+{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}{Q}_e\mathrm{.}$ The quantities have units $[E_{\mathrm{p}\mathrm{a}\mathrm{r}}]=\mathrm{J}$, $[A_H]={\mathrm{m}}^2$, $[{\kappa }_{\mathrm{a}\mathrm{c}\mathrm{c}}]=\mathrm{m}{\mathrm{s}}^{-2}$, $[{\mathrm{\Omega }}_{\mathrm{S}\mathrm{I}}]={\mathrm{s}}^{-1}$, and $[{\mathrm{\Phi }}_{\mathrm{S}\mathrm{I}}]=\mathrm{V}$. The joint Schwarzschild point $J=Q_e=0$ is included by the separate exact Schwarzschild horizon-data supplier; the generic physical horizon theorem itself states a nonzero-rotation-or-charge hypothesis.

These identities concern the selected stationary family and its stated normalizations. They are not a dynamical-process first law, and $E_{\mathrm{p}\mathrm{a}\mathrm{r}}$ is not identified here with ADM or Komar gravitational energy. No operational temperature, physical entropy, Hawking calibration, or extremal-boundary claim follows from this calculation.

The explicit zero-spin, zero-charge case uses [F6] because the generic physical horizon interface [L1] assumes $a^2+q^2>0$. The strict family excludes the extremal boundary, where $s=0$; no positive-temperature division or thermodynamic interpretation is made.