External ADM–Komar equality on ordinary stationary vacuum ends

For a smooth four-dimensional stationary vacuum end with $T={\mathrm{\partial }}_t$, $g-\eta =O_2(r^{-1})$, $g_{0i}=O_2(r^{-2})$, and the future unit time-translation normalization at infinity, the ordinary ADM energy of the $t=0$ end equals its future/outward normalized Komar mass $m_K$. Here $\eta =-d{t}^2+{\sum }_{i=1}^3(d{x}^i{)}^2$, $r=\mathrm{\mid }x\mathrm{\mid }$ in the end chart, and vacuum means $\mathrm{Ric}(g)=0$. There are no Kaluza–Klein directions. No maximality is assumed.

For sufficiently large $r$, put $h=g_{ij}d{x}^{i}d{x}^j$ on $t=0$, let $n$ be its future unit spacetime normal, let $\nu$ be the outward $h$-unit normal to $S_r=\{t=0,\mathrm{\mid }x\mathrm{\mid }=r\}$, and let $d{A}_h$ be the Riemannian area density on $S_r$. Use the spatial-metric ADM energy convention and set $m_K=\frac{1}{4\pi }{\lim }_{r\to \mathrm{\infty }}{\int }_{S_r}{n}^a{\nu }^b{\mathrm{\nabla }}_a{T}_{b}d{A}_h\mathrm{.}$ Both flux limits exist under the stated hypotheses and $E_{\mathrm{A}\mathrm{D}\mathrm{M}}=m_K$. The conventions are in geometric units $G=c=1$ and the inherited ZFC setting.

Stationary Lorentzian data and tensor conventions

Differentiated asymptotic bounds and parameter uniformity

Asymptotic mass and axial angular-momentum conventions

Riemannian volume density

Integral of a compactly supported smooth density

The Axiom of Choice

The imported result is the $K=0$, $n=3$, flat-background, vacuum specialization of Barzegar–Chruściel–Hörzinger, Section V (V.5), with the ADM identification in (III.18)–(III.19). Its proof is not reconstructed here.

The stated falloff implies $\mathrm{\partial }g=O(r^{-2})$, $\delta \mathrm{\Gamma }=O(r^{-2})$ and $\mathrm{Riem}(g)=O(r^{-3})$. Thus (III.13) holds, and the choices $\alpha =1\mathrm{/}2$, $\beta =1.9$, ${\beta }_R=2.9$, $\gamma =2$, $Z=r{\mathrm{\partial }}_r$ meet (IV.16)–(IV.20) and (IV.29)–(IV.30): the four effective decay exponents are $2.4,2.8,3.9,2.4>2$. The Killing field is also parallel for the flat background; there are no extra-dimension terms and the Ricci term vanishes. These are checks of the imported hypotheses, using only the bounds in the Statement.

For the compatible future/outward orientation, the physical binormal element is $d{S}_{ab}^g=(n_b{\nu }_a-n_a{\nu }_b)d{A}_h$. Killing antisymmetry gives $T^{a;b}d{S}_{ab}^g=2n^a{\nu }^b{\mathrm{\nabla }}_a{T}_{b}d{A}_h$. The surface-element convention (III.34), when taken for the flat background, differs from this physical element by relative $O(r^{-1})$ in the fixed chart. Since $\mathrm{\nabla }T=O(r^{-2})$ and sphere area is $O(r^2)$, this replacement changes the integrated flux by $O(r^{-1})$, which tends to zero. Thus the $1\mathrm{/}(8\pi )$ Komar factor in this specialization of (V.5) yields the $1\mathrm{/}(4\pi )$ scalar-flux convention in the Statement. No mass-coefficient expansion or extrinsic-decay lemma enters this normalization.