The nonrotating bifurcation gives positive normalized mass before staticity

Work in the connected spacetime component containing the connected section $C\cup S$ and the selected end; $T$ is nontrivial there since it is timelike at infinity. At $q\in S$, let $n$ be the future unit normal to the original section and $e$ its unit normal to $S$ pointing into $C$. The boost interface in [F1] gives $A_{q}n={\lambda }_{q}e$ and $A_{q}e={\lambda }_{q}n$, with ${\lambda }_q\mathrm{\ne }0$; this follows also by writing its skew map in the orthonormal basis $(n,e)$. Choose a smooth section curve $c(r)$ into $C$ with $c(0)=q$ and $c^{\mathrm{\prime }}(0)=e$. In a smooth ambient chart, $T(c(r))=r{\lambda }_{q}n+O(r^2)$ because $T(q)=0$ and the connection correction to its first derivative vanishes there. Consequently $g(T,T)(c(r))=-{\lambda }_q^2{r}^2+O(r^3)<0$ for small $r>0$. These points lie in $E$; the orientation supplier makes $T$ future timelike there. Taking $T(c(r))\mathrm{/}r\to {\lambda }_{q}n$ forces ${\lambda }_q>0$. This fixes the sign of the already normalized $T$, without rescaling it or choosing a sign for a horizon parameter.

Identify the future boundary sheet using $P=I^{-}(M^{\mathrm{e}\mathrm{n}\mathrm{d}})$, a past set, and ${\mathcal{H}}^{+}=\mathrm{\partial }P$ near $S$. Take a defining function $u$ from [F4], with $du(n)>0$ at $q$. The differential annihilates $T_{q}S$ and is null, so $du(e)=\pm du(n)\mathrm{\ne }0$. The past-set closure property gives points immediately to the timelike past of $q$ in $P$, while a timelike-future point of $q$ cannot belong to $P$, since that would put $q$ in the open past set. In a sufficiently small hypersurface chart, membership in $P$ is constant on each connected side of $u=0$, so its local side is $u<0$. Since $c(r)\in C\subset P$, differentiation at $r=0$ and $du(e)\mathrm{\ne }0$ imply $du(e)<0$, hence $du(e)=-du(n)$. Thus the future normal generator of this sheet at $q$ is $\mathrm{\ell }=n+e$, and $A_q\mathrm{\ell }={\lambda }_q\mathrm{\ell }$. On the future branch $s>0$ of its affine generator, the supplier's exact law is $T={\lambda }_{q}s{\dot{\gamma }}_q$. Affineness gives ${\mathrm{\nabla }}_{T}T={\lambda }_{q}T$, so its signed surface gravity is ${\lambda }_q>0$, with the same limit at $q$. The vacuum zeroth law of Reall §9.3 makes this one constant $\kappa >0$ on the connected local future Killing sheet and its bifurcation limit. The induced metric on the nonempty compact sphere $S$ has positive finite area $A$.

Apply Sudarsky–Wald equation (23), the nonmaximal Komar identity, to the oriented section $C\cup S$ supplied in [F1]. It has compact core, only the bifurcation boundary $S$, and the one normalized asymptotic end. The field is Killing and the geometry is vacuum by [F3]; $T$ itself generates the local horizon, so the rotational term is absent ($\mathrm{\Omega }=0$), and the matter integral is zero. The identity therefore gives $M_{\mathrm{K}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{r}}=\kappa A\mathrm{/}(4\pi )>0$. Its derivation uses the section and its boundary fluxes, requiring neither an all-time ambient extension nor a maximal section.

In the specified end the same section is $t=0$, $T={\mathrm{\partial }}_t$, $g_{0i}=O_6(r^{-2})$ and $g_{ij}={\delta }_{ij}+O_6(r^{-1})$, so it is asymptotically orthogonal to $T$. Reall §8.4 therefore gives $E_{\mathrm{A}\mathrm{D}\mathrm{M}}=M_{\mathrm{K}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{r}}$. Use the Hodge and flux conventions of [F2]: orient the end by $ϵ=\sqrt{-\det g}dt\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, the section by its future normal and the end spheres outward. This orientation extends along the section using its radial chart and smooth future normal. Here $T^{\mathrm{♭}}=g_{00}dt+g_{0i}d{x}^i$, and stationarity gives $(d{T}^{\mathrm{♭}}{)}_{0i}=-{\mathrm{\partial }}_i{g}_{00}=2M_0{x}^i\mathrm{/}{r}^3+O(r^{-3})$ and $(d{T}^{\mathrm{♭}}{)}_{ij}=O(r^{-3})$. In the limiting Minkowski metric the supplied Hodge convention gives $\star (dt\wedge dr)=-d{A}_0$, where $d{A}_0$ is the positively oriented Euclidean sphere area form. The metric and inverse-metric corrections are $O(r^{-1})$, and the differentiated shift terms are $O(r^{-3})$, so on $S_r=\{t=0,\mathrm{\mid }x\mathrm{\mid }=r\}$ one has ${\star d{T}^{\mathrm{♭}}\mid }_{S_r}=(-2M_0\mathrm{/}{r}^2+O(r^{-3}))d{A}_0$, uniformly in angle. Thus (8.24) gives $M_{\mathrm{K}\mathrm{o}\mathrm{m}\mathrm{a}\mathrm{r}}=-\frac{1}{8\pi }{\lim }_{r\to \mathrm{\infty }}{\int }_{S_r}\star d{T}^{\mathrm{♭}}=M_0$, since ${\int }_{S_r}d{A}_0=4\pi r^2$ and the integrated error is $O(r^{-1})$. For completeness, $N=1+O(r^{-1})$ and the covariant end shift is $X_i=g_{0i}=O_6(r^{-2})$. The tangential Killing equation and the convention of [F3] give $2N{k}_{ij}=D_i{X}_j+D_j{X}_i$, hence $k_{ij}=O(r^{-3})$. Reall's momentum formula (8.44) therefore has limit zero, independently of its sign convention for $k$. By (8.45), $M_{\mathrm{A}\mathrm{D}\mathrm{M}}=\sqrt{E_{\mathrm{A}\mathrm{D}\mathrm{M}}^2-\mathrm{\mid }{P}_{\mathrm{A}\mathrm{D}\mathrm{M}}{\mathrm{\mid }}^2}=E_{\mathrm{A}\mathrm{D}\mathrm{M}}$, since step 3.1 made this energy positive. Combining the identities proves the unchanged Statement.