Schwarzschild scalar Killing-flux balance

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $s>0$ and let $g_{\mathrm{e}\mathrm{x}\mathrm{t}}$ be the regular advanced-chart extension of the Schwarzschild exterior, with $f(r)=1-s\mathrm{/}r$ and stationary field $T={\mathrm{\partial }}_v$. Give its domain $M_{\mathrm{e}\mathrm{x}\mathrm{t}}={\mathbb{R}}_v\times (0,\mathrm{\infty }{)}_r\times S^2$ the product orientation determined by $dv\wedge dr\wedge {\omega }_{S^2}$, where ${\omega }_{S^2}$ is the positive area form of the unit round sphere in its outward orientation; write $d\mathrm{\Omega }$ for its associated positive area measure. In this orientation, ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g_{\mathrm{e}\mathrm{x}\mathrm{t}}}=r^{2}dv\wedge dr\wedge {\omega }_{S^2}$. Let $C$ be a compact finite smooth singular four-chain in the exterior closure $r\ge s$, with oriented boundary $\mathrm{\partial }C={\mathrm{\Sigma }}_2-{\mathrm{\Sigma }}_1+{\mathcal{H}}^{+}+B_{\mathrm{o}\mathrm{u}\mathrm{t}}$, where ${\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2$ are spacelike faces with future unit normals $n_1,n_2$, ${\mathcal{H}}^{+}$ is a future-horizon face, and $B_{\mathrm{o}\mathrm{u}\mathrm{t}}$ is the remaining signed outer boundary. Let $\psi$ be smooth on a neighborhood of $C$ and solve ${\mathrm{\square }}_{g_{\mathrm{e}\mathrm{x}\mathrm{t}}}\psi =0$. Define $j_T^a:=-Q^a{}_b[\psi ]T^b$. Orient each spacelike face ${\mathrm{\Sigma }}_j$ so that its adapted unit normal is the future unit normal $n_j$, and define

$E_j:={\mathrm{\Phi }}_{j_T}({\mathrm{\Sigma }}_j)={\int }_{{\mathrm{\Sigma }}_j}{Q}_{ab}[\psi ]n_j^a{T}^{b}d{\mu }_{{\mathrm{\Sigma }}_j}\mathrm{.}$

Define $F_{\mathrm{o}\mathrm{u}\mathrm{t}}$ as the signed flux of $j_T$ through $B_{\mathrm{o}\mathrm{u}\mathrm{t}}$ and $F_H:={\int }_{{\mathcal{H}}^{+}}(T\psi {)}^{2}d{\mu }_H$, where $d{\mu }_H=s^{2}dvd\mathrm{\Omega }$ is the positive horizon measure in the advanced chart. Then $E_1,E_2,F_H\ge 0$ and

$E_2+F_H+F_{\mathrm{o}\mathrm{u}\mathrm{t}}=E_1\mathrm{.}$

The signed outer flux need not be nonnegative. On the exterior static slices $x^0=\text{constant}$, $n=f^{-1\mathrm{/}2}T$ and $Q(n,T)=\sqrt{f}Q(n,n)$; this stationary slice energy degenerates as $r\downarrow s$, where $T$ becomes null in the regular extension.

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The calculation proves conservation with the stated side flux and the degeneracy of the stationary Schwarzschild energy at the horizon. A redshift estimate remains deferred: its proof route requires a multiplier vector field that is timelike across the horizon and has positive deformation tensor near the nonextremal horizon. A Morawetz estimate also remains deferred: its proof route requires a radial multiplier whose bulk term degenerates at the photon sphere $r=3s\mathrm{/}2$ (here $s=2M$), together with a controlled far-region $r$-weighted estimate. This finite-region stationary balance proves none of those estimates. The item does not supply a Kerr scalar-decay theorem or any gravitational perturbation stability result.