A specified asymptotically Euclidean ADM end

For the geometric initial-data convention, let $(M,g)$ be a smooth time-oriented Lorentzian four-manifold and let $\iota :\mathrm{\Sigma }\hookrightarrow M$ be a smooth embedded three-manifold whose induced metric $h:={\iota }^{\ast }g$ is positive definite. Its future unit normal $n$ satisfies $g(n,n)=-1$ and $g(n,X)=0$ for $X\in T\mathrm{\Sigma }$. The second fundamental form is $K(X,Y):=g({\mathrm{\nabla }}_{X}n,Y)$, with $\mathrm{\nabla }$ the Levi-Civita connection of $g$ and the curvature sign convention stated in the Definition references section. The end data below are specified as a Riemannian metric and a symmetric tensor of these types; no embedding or Einstein constraint equation is required.

Let $E=\{x\in {\mathbb{R}}^3:r>R_0\}$, where $R_0>0$ is a length, $x=(x^1,x^2,x^3)$ are Cartesian coordinates measured in metres, and $r:=\sqrt{(x^1{)}^2+(x^2{)}^2+(x^3{)}^2}$. Write $\delta :={\delta }_{ij}d{x}^{i}d{x}^j$ for the standard Euclidean metric, where ${\delta }_{ij}=1$ if $i=j$ and $0$ otherwise; repeated spatial indices are summed over $1,2,3$. A fixed-chart asymptotically Euclidean end consists of this exterior domain, a $C^2$ symmetric positive-definite covariant $2$-tensor $h=h_{ij}d{x}^{i}d{x}^j$, and a $C^1$ symmetric covariant $2$-tensor $K=K_{ij}d{x}^{i}d{x}^j$ such that $h_{ij}(x)={\delta }_{ij}+\frac{a_{ij}}{r}+{\rho }_{ij}(x),K_{ij}(x)=O_1(r^{-2}),$ where $a_{ij}=a_{ji}$ is a constant real matrix, $ϵ>0$, and $\rho =O_2(r^{-1-ϵ})$. For any tensor components $T_{ij}$, the notation $T=O_k(r^{-q})$ means that for every multi-index $\alpha =({\alpha }_1,{\alpha }_2,{\alpha }_3)$ with $\mathrm{\mid }\alpha \mathrm{\mid }:={\alpha }_1+{\alpha }_2+{\alpha }_3\le k$ there are constants $C_{\alpha },R_{\alpha }$ such that $\mathrm{\mid }{\mathrm{\partial }}^{\alpha }{T}_{ij}(x)\mathrm{\mid }\le C_{\alpha }{r}^{-q-\mathrm{\mid }\alpha \mathrm{\mid }}$ for $r\ge R_{\alpha }$, uniformly in angle. Thus the definition requires two Cartesian derivatives of the metric remainder and one of $K$. The coordinates and $r$ have units of metres; ${\delta }_{ij}$ and $h_{ij}$ are dimensionless, $a_{ij}$ has units of metres, and $K$ has inverse-length units in a spatial orthonormal frame.

For $R>R_0$, the coordinate sphere $S_R:=\{x:r=R\}$ is a smooth embedded surface in $E$. Let ${\mathrm{ȷ}}_R:S_R\hookrightarrow E$ be its inclusion and ${\gamma }_R:={\mathrm{ȷ}}_R^{\ast }\delta$ its induced Riemannian metric. The Euclidean area measure $dS$ is the Riemannian area measure of ${\gamma }_R$: for any smooth rank-two local parametrization $X:U\subseteq {\mathbb{R}}^2\to S_R$ with coordinates $(y^1,y^2)$, its coordinate density is $dS=\sqrt{\det [\delta ({\mathrm{\partial }}_{y^a}X,{\mathrm{\partial }}_{y^b}X){]}_{a,b=1}^2}d{y}^{1}d{y}^2\mathrm{.}$

The change-of-variables formula makes this density independent of the chosen surface coordinates; since $x$ is measured in metres, $dS$ has units of square metres. The sphere's outward Euclidean unit normal has components $n^i:=x^i\mathrm{/}R$. If its limiting integral exists as $R\to \mathrm{\infty }$—the spatial-infinity limit in this chosen chart—the fixed-chart ADM energy and its associated mass are $E_{\mathrm{A}\mathrm{D}\mathrm{M}}(h;x):=\frac{c^4}{16\pi G_N}{\lim }_{R\to \mathrm{\infty }}{\int }_{S_R}({\mathrm{\partial }}_j{h}_{ij}-{\mathrm{\partial }}_i{h}_{jj})n^{i}dS,M_{\mathrm{A}\mathrm{D}\mathrm{M}}:=\frac{E_{\mathrm{A}\mathrm{D}\mathrm{M}}}{c^2}\mathrm{.}$ Here $c>0$ is the SI speed-of-light conversion constant with units metres per second, and $G_N>0$ is the Newtonian gravitational conversion constant with units metres cubed per kilogram per second squared. The limit has units of metres, $E_{\mathrm{A}\mathrm{D}\mathrm{M}}$ has units of joules, and $M_{\mathrm{A}\mathrm{D}\mathrm{M}}$ has units of kilograms. The prefactor fixes the asymptotic time-translation normalization; in SI coordinates use $x^0=ct$. No rescaling of this convention is part of the definition.

Gauss-Codazzi decomposition in Gaussian normal coordinates and the constraint equations in terms of the induced metric and second fundamental form

The formula depends only on $h$, not on $K$; the stated decay of $K$ records the initial-data setting. This definition does not assert that $(h,K)$ satisfies the Einstein constraint equations, that the limit is invariant under changes of chart, or that the energy is positive. It is a limit at spatial infinity, not a finite-radius surface value or a local gravitational energy density. Angular momentum and center of mass require additional decay and parity hypotheses and are outside this definition.

The sketch is a two-dimensional section through the Cartesian end. The shaded disk is excluded from $E$, its dashed boundary is $r=R_0$, and the outer circle is a representative coordinate sphere $S_R$ with $R>R_0$. The arrow at the outer circle points along the outward Euclidean unit normal $n^i=x^i\mathrm{/}R$. The sketch suppresses the other angular directions and plays no role in any proof; the definitions above specify the full three-dimensional objects.