Asymptotic ends and ADM surface integrals

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.

Let $E\subset {\mathbb{R}}^3$ be a nonempty open set with Cartesian coordinates $x^1,x^2,x^3$, standard Euclidean metric $\delta ={\delta }_{ij}d{x}^{i}d{x}^j$ with ${\delta }_{ij}=1$ for $i=j$ and $0$ otherwise, inverse components ${\delta }^{ij}$, and Laplacian ${\mathrm{\Delta }}_{\delta }={\sum }_{i=1}^3{\mathrm{\partial }}_i^2$. Let $u\in C^{\mathrm{\infty }}(E,(0,\mathrm{\infty }))$ and set $h=u^4\delta$. Form the product Lorentzian manifold $\widehat{M}={\mathbb{R}}_{\tau }\times E$ with metric $g=-d{\tau }^2+h$ and future orientation ${\mathrm{\partial }}_{\tau }$; its slice $\mathrm{\Sigma }=\{0\}\times E$ is a smooth spacelike hypersurface with induced metric $h$. Use the induced connection $D$ and curvature convention supplied for this slice by the earlier normal-decomposition theorem. In Cartesian coordinates define $R^{\mathrm{\ell }}{}_{bcd}$ by $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b=R^{\mathrm{\ell }}{}_{bcd}{\mathrm{\partial }}_{\mathrm{\ell }}$, ${\mathrm{Ric}}_{ij}^h:=R^k{}_{ikj}$, and $S_h:=h^{ij}{\mathrm{Ric}}_{ij}^h$. For $f=2\log u$, write $\mathrm{\mid }df{\mathrm{\mid }}_{\delta }^2:={\delta }^{ij}({\mathrm{\partial }}_{i}f)({\mathrm{\partial }}_{j}f)$. Then $S_h=-8u^{-5}{\mathrm{\Delta }}_{\delta }u\mathrm{.}$

This statement concerns a smooth positive conformal metric and its induced scalar curvature. The product construction only fixes the geometric convention; no Einstein equation, energy condition, or completeness hypothesis is imposed.

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

The identity is proved here from the library's curvature sign convention. It is used only for explicit smooth conformal metrics; no scalar-curvature sign condition or completeness assertion is assumed.

On a fixed-chart end $E=\{x\in {\mathbb{R}}^3:r=\mathrm{\mid }x\mathrm{\mid }>R_0\}$ with the ADM convention stated below, let $u\in C^{\mathrm{\infty }}(E,(0,\mathrm{\infty }))$ satisfy $u=1+m\mathrm{/}(2r)+\rho$, where $m\in \mathbb{R}$ has units of metres and $\rho =O_2(r^{-2})$. Set $h=u^4\delta$ and choose any $C^1$ symmetric tensor $K=O_1(r^{-2})$. Then the ADM limit exists in the specified chart and $E_{\mathrm{A}\mathrm{D}\mathrm{M}}(h;x)=\frac{c^{4}m}{G_N}\mathrm{.}$

A specified asymptotically Euclidean ADM end

The proof uses the fixed chart and the exact stated falloff. It proves neither coordinate invariance for broader asymptotic classes nor a positivity theorem.

Let $h$ and $h^{\mathrm{\prime }}$ be two $C^2$ Riemannian metrics on the same end $E=\{x\in {\mathbb{R}}^3:r=\mathrm{\mid }x\mathrm{\mid }>R_0\}$, with the same fixed Cartesian chart and asymptotic time normalization stated below. Suppose $h=h^{\mathrm{\prime }}$ outside a compact set $C\subset E$ and both fixed-chart ADM limits exist. Then $E_{\mathrm{A}\mathrm{D}\mathrm{M}}(h;x)=E_{\mathrm{A}\mathrm{D}\mathrm{M}}(h^{\mathrm{\prime }};x)\mathrm{.}$ The symmetric tensors $K$ associated with the two data sets may differ; each need only satisfy the decay in the definition.

A specified asymptotically Euclidean ADM end

This is a fixed-chart statement. It makes no claim of diffeomorphism invariance, constraint preservation, completeness, or scalar-curvature equality.