Normalized Komar data on a nonrotating horizon section

Work in ZFC, including choice, on an oriented smooth time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, with its Levi–Civita connection $\mathrm{\nabla }$. Let $C$ be a smooth compact embedded spacelike horizon cross-section with induced positive metric $q$. Supply smooth normal fields $n,s$ along $C$ with $g(n,n)=-1$, $g(s,s)=1$, $g(n,s)=0$, $n$ future-directed and $s$ pointing outward toward the selected exterior. Orient $C$ by declaring $(v_1,v_2)$ positive exactly when $(n,s,v_1,v_2)$ is positive in $M$, and put $dA={\mathrm{vol}}_q$ in that orientation. Thus ${\int }_{C}fdA$ is the intrinsic integral of the smooth top form $f{\mathrm{vol}}_q$ on oriented $C$, equivalently its positive Riemannian area integral. Put $b_{ab}=s_a{n}_b-n_a{s}_b$. For a smooth Killing field $T$, define the oriented horizon Komar functional by $Q_C[T]=-(8\pi {)}^{-1}{\int }_C({\mathrm{\nabla }}^a{T}^b)b_{ab}dA$.

In the nonrotating case require $T=c(n+s)$ on $C$ for a supplied smooth positive function $c$, and require that $T$ generate the smooth Killing horizon with ${\mathrm{\nabla }}_{T}T=\kappa T$. The signed surface-gravity convention is equivalently $d(g(T,T))=-2\kappa T^{\mathrm{♭}}$ on the nonzero-generator part, with $T^{\mathrm{♭}}=g(T,\cdot )$. No bifurcation sphere is assumed.

For the exterior normalization and mass notation, additionally supply an asymptotically flat spacelike end $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ with specified chart $\chi :\{x\in {\mathbb{R}}^3:\mathrm{\mid }x\mathrm{\mid }>R\}\to S_{\mathrm{e}\mathrm{x}\mathrm{t}}$ and induced positive metric $\gamma$, using the end and derivative-decay convention stated in the references below. Set $r=\mathrm{\mid }x\mathrm{\mid }$ and $S_r=\chi (\{\mathrm{\mid }x\mathrm{\mid }=r\})$ for $r>R$. The field $T$ is future timelike at this end and is normalized there by $-g(T,T)\to 1$ as $r\to \mathrm{\infty }$. Let $n_{\mathrm{e}\mathrm{x}\mathrm{t}}$ be the end's future unit normal and $s_r$ its outward unit normal to $S_r$ within $S_{\mathrm{e}\mathrm{x}\mathrm{t}}$, with $dr(s_r)>0$. Orient $S_r$ so that $(v_1,v_2)$ is positive exactly when $(n_{\mathrm{e}\mathrm{x}\mathrm{t}},s_r,v_1,v_2)$ is positive in $M$. The operators $d$ and ${\star }_g$ are the exterior derivative and Lorentzian Hodge star for $g$ and the supplied ambient orientation, with $\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{vol}}_g$. If $j_r:S_r\hookrightarrow M$ is the inclusion, ${\int }_{S_r}{\star }_{g}d{T}^{\mathrm{♭}}$ means the intrinsic integral of $j_r^{\ast }({\star }_{g}d{T}^{\mathrm{♭}})$ on this oriented compact sphere. The notation $M_K$ denotes the exterior Komar mass in the source convention $M_K=-(8\pi {)}^{-1}{\lim }_{r\to \mathrm{\infty }}{\int }_{S_r}{\star }_{g}d{T}^{\mathrm{♭}}$, whenever that limit exists. No identity $M_K=Q_C[T]$ is built into this definition.

Stationary Lorentzian data and tensor conventions

Bifurcate Killing horizons, rotation and normalization

Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

Riemannian volume form on an oriented manifold

The Axiom of Choice

Asymptotically flat spacelike end and derivative decay

Lorentzian exterior calculus and Hodge star

Integration on an oriented embedded submanifold

Reall, Black Holes, §8.2 (8.24) and §9.1 (9.1)–(9.2)

Given: The supplied compact section, normal fields, Killing field and, for the exterior notation, selected end chart and normalization.

Goal: Check that the local functional and conditional exterior mass notation are meaningful.

1.1 The smooth ordered normal pair and ambient orientation give a smooth orientation on $C$ by the stated positive-basis rule. Lowering the normal fields and antisymmetrizing gives a smooth covariant two-tensor $b$. Raising the derivative of $T$ gives the smooth scalar $f=({\mathrm{\nabla }}^a{T}^b)b_{ab}$. Thus $f{\mathrm{vol}}_q$ is a smooth top form with compact support on compact $C$. Apply the integration supplier with the identity embedding $C\to C$: its finite-chart integral is defined and finite. This proves finiteness of $Q_C[T]$ without extending $dA$ to an ambient two-form. [given, F1, F2]

1.2 For each $r>R$, the end chart maps the Euclidean sphere to a compact smooth embedded surface $S_r$. The end metric is positive definite and $dr$ is nowhere zero, so its metric-dual vector normalized to unit length gives the smooth outward normal $s_r$; the positive-basis rule with $n_{\mathrm{e}\mathrm{x}\mathrm{t}}$ orients $S_r$. The smooth ambient two-form ${\star }_{g}d{T}^{\mathrm{♭}}$ has a smooth pullback to $S_r$, whose support is compact. The integration supplier therefore defines each finite sphere integral. Its specified limit is a definite real number if it exists, and the source convention then defines $M_K$. [given, F2, F3, F4]

2.1 The outward and future choices fix the binormal sign convention; the positive area integral has the meaning established in step 1.1. The end normalization specifies the scale of $T$, and the signed equation defining $\kappa$ is meaningful on the nonzero-generator part. The exterior limit and any equality $M_K=Q_C[T]$ remain separate analytic assertions; neither is required to define the local functional. [given, F1, F4, step 1.1, step 1.2] ∎

Reall's lecture notes supply only the exterior Komar normalization in §8.2 (8.24), the coordinate-sphere convention following (8.14), and the signed Killing-horizon surface gravity and normalization in §9.1 (9.1)–(9.2) and its preceding paragraph, under the hypotheses and reading limits recorded in the source metadata. The local justification defines the integrals using the named mathematical suppliers. Global degenerate-end transport and zero-mass exclusion are separate imported applications, not claims justified by this notation.