Mass-one convention and Harrison denominator

PDF p. 7: §2.1; (2.1); PDF p. 8: §2.1; (2.1)

Let $(M,g,N,T)$ be stationary Lorentzian data, and let ${\mathrm{\Sigma }}_1$ be a supplied smooth spacelike maximal hypersurface for $g$: its induced metric $h$ is positive definite and its mean curvature vanishes. The subscript $1$ labels the original section. Let an asymptotic end of this section have Cartesian coordinates $x^i$, with $r=\mathrm{\mid }x\mathrm{\mid }$, and suppose the following ADM mass limit exists and equals one:

$M_{\mathrm{A}\mathrm{D}\mathrm{M}}=\frac{1}{16\pi }{\lim }_{R\to \mathrm{\infty }}{\int }_{\mathrm{\mid }x\mathrm{\mid }=R}({\mathrm{\partial }}_j{h}_{ij}-{\mathrm{\partial }}_i{h}_{jj})\frac{x^i}{R}d{S}_{\mathrm{e}\mathrm{u}\mathrm{c}\mathrm{l}}=1.$

Here repeated indices run from $1$ to $3$, $h_{ij}$ are the components of the induced metric in the chosen chart, and $d{S}_{\mathrm{e}\mathrm{u}\mathrm{c}\mathrm{l}}$ is Euclidean surface area on the coordinate sphere. This is the mass-one convention for the specified end; convergence of the limit and existence of such a section are assumptions.

On an open set $U$ with the designated asymptotic end, let $\sigma =f+i\psi$ be a supplied normalized Ernst potential: $f=-g(T,T)$, $\psi$ is real, $d\sigma =2T^b{\mathcal{F}}_{ba}d{x}^a$, and $\sigma \to 1$ at the end, with $\mathcal{F}$ as in Killing two-form and normalized Ernst notation. For each real $q\in [0,1]$, define the complex function $B_q=q(1+\sigma )+(1-\sigma )$ and the real function $D_q=\mathrm{\mid }{B}_q{\mathrm{\mid }}^2=B_q\overline{B_q}$ on $U$.

If a supplied bifurcation surface $S$ meets $U$ and a point $p\in S\cap U$ satisfies $T_p=0$, define $d_q=D_q(p)$. Calling this chosen point a pole is terminology for this construction; neither its existence nor its uniqueness is asserted.

Given: The supplied data and the existence assumptions in the Definition.

Goal: Justify the named quantities and conventions.

1.1 The induced metric and the exterior chart give the components and derivatives appearing in the displayed surface integrals. For each $R>R_0>0$ in the exterior chart, the coordinate sphere $S_R=\{x:\mathrm{\mid }x\mathrm{\mid }=R\}\subset {\mathbb{R}}^3$ is nonempty, since $(R,0,0)\in S_R$, and compact by [F8]. To check smooth embeddedness, put $P(x)=(x^1{)}^2+(x^2{)}^2+(x^3{)}^2$. This is a smooth map ${\mathbb{R}}^3\to \mathbb{R}$ and $S_R=P^{-1}(R^2)$. At $x\in S_R$, its differential satisfies $d{P}_x(x)=2R^2\mathrm{\ne }0$, so it is surjective onto $\mathbb{R}$. Thus $R^2$ is a regular value and [F10] makes $S_R$ a smooth embedded hypersurface. [given, F1, F2, F3, F8, F10, algebra]

2.1 The real function $u_R(x)=({\mathrm{\partial }}_j{h}_{ij}-{\mathrm{\partial }}_i{h}_{jj})x^i\mathrm{/}R$ is smooth on a neighborhood of $S_R$, hence continuous on $S_R$. Applying [F9] to $u_R$ on the nonempty compact sphere with its restricted Euclidean metric shows that $u_R$ is bounded above and below, so ${\sup }_{S_R}\mathrm{\mid }{u}_R\mathrm{\mid }<\mathrm{\infty }$. [F2, F3, F9, step 1.1]

3.1 The finite surface measure in [F6] therefore gives ${\int }_{S_R}\mathrm{\mid }{u}_R\mathrm{\mid }d{S}_{\mathrm{e}\mathrm{u}\mathrm{c}\mathrm{l}}\le ({\sup }_{S_R}\mathrm{\mid }{u}_R\mathrm{\mid }){\int }_{S_R}1d{S}_{\mathrm{e}\mathrm{u}\mathrm{c}\mathrm{l}}<\mathrm{\infty }$, so each signed surface integral is defined and finite. Their limit is explicitly assumed to exist and equal one; the notation ${\mathrm{\Sigma }}_1$ is a label and does not construct a section or prove convergence. [given, F6, step 1.1, step 2.1]

4.1 The supplied $\sigma =f+i\psi$ is smooth. For each specified real $q$, set $a_q=(1+q)-(1-q)f$ and $b_q=-(1-q)\psi$, so $B_q=a_q+i{b}_q$ is a uniquely specified smooth complex function. By [F7], $B_q\overline{B_q}=(a_q+i{b}_q)(a_q-i{b}_q)=a_q^2+b_q^2=\mathrm{\mid }{B}_q{\mathrm{\mid }}^2$. This is a unique smooth nonnegative real function $D_q$. If the separately supplied point $p$ satisfies the stated conditions, evaluation gives the unique real number $d_q$. No strict positivity or existence of a zero of $T$ follows from these definitions. [given, F4, F5, F7, algebra] ∎

The cited article supplies attribution for the mass-one convention and its notation. The justification above uses the supplied data and pointwise operations; it does not use the article as a proof of existence or positivity.