Regular Harrison Einstein–Maxwell family

Work in ZFC, including The Axiom of Choice. The stationary, hypersurface, asymptotic, form, electrovacuum and circle-action interfaces are Stationary Lorentzian data and tensor conventions, Bifurcate Killing horizons, rotation and normalization, Spacelike sections, maximality and lapse–shift data, Differentiated asymptotic bounds and parameter uniformity, Complex self-dual forms and metric-dependent contraction, Geometric electrovacuum equations and charge convention and Periodic isometries and invariant normal averaging.

The precise imported GR data are a connected oriented time-oriented smooth vacuum four-spacetime, a spacelike acausal edgeless partial Cauchy hypersurface ${\mathrm{\Sigma }}^0={\mathrm{\Phi }}_0(\{\mathrm{\mid }x\mathrm{\mid }>1\mathrm{/}2\})$, ${\mathrm{\Sigma }}_1={\mathrm{\Phi }}_0(\{\mathrm{\mid }x\mathrm{\mid }>1\})$, and $S_0={\mathrm{\Phi }}_0(\{\mathrm{\mid }x\mathrm{\mid }=1\})$. The stationary end has $T={\mathrm{\partial }}_t$ and $g_{00}=-1+2\mathrm{/}r+O_6(r^{-2})$, $g_{ij}={\delta }_{ij}+O_6(r^{-1})$, $g_{0i}=-2{ϵ}_{ijk}{S}^j{x}^k{r}^{-3}+O_6(r^{-3})$. The spacetime orientation is fixed by $\mathrm{d}t\wedge \mathrm{d}{x}^1\wedge \mathrm{d}{x}^2\wedge \mathrm{d}{x}^3$ in this chart and is used for every ${\star }_g$ and ${\star }_{g_q}$. Here $S_r=\{t=0,\mathrm{\mid }x\mathrm{\mid }=r\}$ has the outward boundary orientation induced by $\mathrm{d}{x}^1\wedge \mathrm{d}{x}^2\wedge \mathrm{d}{x}^3$ on the zero-time end slice. Its exterior $E=I^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})\cap I^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$ is globally hyperbolic; ${\mathrm{\Sigma }}^0\cap I^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}^0\cap I^{+}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})={\mathrm{\Sigma }}_1$; $T$ is nowhere zero on $E$, with complete orbits meeting ${\mathrm{\Sigma }}_1$ transversally. SBS requires the two chronological boundary hypersurfaces $H^{\pm }$ near $S_0$ to be smooth, null, transverse and nonexpanding, intersecting in $S_0$, with $T$ tangent to both and not identically zero on $S_0$. The induced geometry extends smoothly to $S_0$ and ${\mathrm{tr}}_{h}k=0$ on ${\mathrm{\Sigma }}_1$. These are §1.1 GR, SBS and maximality of the independent external source, with mass rescaled to one; they are additional hypotheses rather than consequences of the interface definitions in [A1].

Use the normalized vacuum Ernst data of the source: $f=-g(T,T)$, $F=\frac{1}{2}\mathrm{d}{T}^{\mathrm{♭}}$, $\mathcal{F}=F+i{\star }_{g}F$, $\sigma =f+i\psi$ with $\psi (\mathrm{\infty })=0$, $i_T\mathcal{F}=\frac{1}{2}\mathrm{d}\sigma$, $f<1$ on $E$, and $f\le 0$ on $S_0$. The local source fields obey $[K,Z]=0$, $T=K+\mathrm{\Omega }Z$ with constant $\mathrm{\Omega }\mathrm{\ne }0$, $K{\mathrm{\mid }}_{S_0}=0$, and the periodic field $Z$ has a pole $p$ where $T(p)=0$.

The shared stationary primitive supplier Smooth invariant Harrison primitive near the horizon and at infinity gives $\mathrm{d}{W}_0=4((f-1)F+\psi {\star }_{g}F)$, $W_0(T)=D_0(p)-D_0$, smooth extension across $S_0$, local $K,Z$ invariance and the radial end choice $W_{0i}=O_5(r^{-3})$. Its normalization, decay and extension are existence inputs, not consequences of the following polynomial calculations.

The denominator, primitive lift, metric, electromagnetic tensor and boundary-scope arguments are Positive Harrison denominator and its parameter identity, Lifting a normalized stationary primitive to the Harrison parameter, Regular Lorentzian metric from the normalized rank-one map, Smooth Maxwell tensor formula without division by the stationary norm and Scope of smooth field equations at an exterior horizon boundary.

Radu Proposition 3.1, §3.1–3.3 and Appendix A are imported for the normalized vacuum inputs in [F1], the exterior Einstein–Maxwell equations (including exterior $f=0$), fluxes, and local horizon Killing-field and surface-gravity preservation, under [A2].