Asymptotically Flat Initial Data

An asymptotically flat initial-data set has one or more noncompact ends on which the spatial metric approaches a Euclidean metric and the extrinsic curvature decays. Falloff controls the size of ADM surface integrals, while integrability and constraint assumptions establish their limits. Suitable coordinate regularity and, for angular momentum or center of mass, parity conditions control their invariance. An end is a geometric region of the physical slice, not the finite coordinate point used to compactify it in a calculation.

Required background. Hypersurface Normals and Tangential Projectors supplies the induced metric and extrinsic curvature, while PDEs and Green Functions for Gravitation supplies weighted falloff, elliptic boundary data, and asymptotic estimates.

Helpful background. The Einstein Equation supplies the constraint system; Formulations of General Relativity locates asymptotic data among the formulations in this volume; and Presymplectic Potentials and Currents prepares the relation between asymptotic generators and charges.

Geometry of an asymptotically Euclidean end

Let $(\Sigma,\gamma_{ij},K_{ij})$ solve the constraints. An end $E$ is a connected component outside a compact set that admits coordinates $x^i$ with $r=(\delta_{ij}x^ix^j)^{1/2}$ and

$$\gamma_{ij}=\delta_{ij}+O(r^{-1}), \qquad \partial_k\gamma_{ij}=O(r^{-2}),$$ $$K_{ij}=O(r^{-2}), \qquad \partial_kK_{ij}=O(r^{-3}).$$

These classical pointwise rates are a readable baseline. Precise theorems usually use weighted Hölder or Sobolev spaces and add differentiability, integrability, and constraint-decay assumptions. Gourgoulhon states the coordinate falloff and discusses the residual asymptotic transformations in Gourgoulhon 2007, §7.2, eqs. (7.1)–(7.7), pp. 103–105, PDF.

The leading Euclidean metric is not a physical background throughout $\Sigma$. It is an asymptotic comparison structure on $E$. A slice may have several ends, and the limiting charges must be evaluated separately on a large sphere in each one.

ADM energy and linear momentum

Let $S_r$ be a coordinate sphere, $s^i$ its outward Euclidean unit normal, and $dS$ its Euclidean area element. Restoring Newton’s constant, the ADM energy of the end is

$$E_{\mathrm{ADM}} =\frac{1}{16\pi G}\lim_{r\to\infty} \int_{S_r} (\partial_j\gamma_{ij}-\partial_i\gamma_{jj})s^i\,dS.$$

The ADM linear momentum is

$$P_i^{\mathrm{ADM}} =\frac{1}{8\pi G}\lim_{r\to\infty} \int_{S_r}(K_{ij}-K\gamma_{ij})s^j\,dS.$$

The sign in the second formula is tied to the convention $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$. Both integrands scale as $r^{-2}$, compensating the $r^2$ area. This proves boundedness at large radius, not convergence by itself. In the usual smooth asymptotically Euclidean class, integrability of the scalar curvature and of the momentum-constraint source, together with the derivative falloff, supplies convergence through volume-divergence identities. Vacuum constraints with $\Lambda=0$ give $R=K_{ij}K^{ij}-K^2=O(r^{-4})$, an integrable decay rate in three dimensions. Gourgoulhon derives the mass and momentum surface terms in Gourgoulhon 2007, §§7.3.1 and 7.4.1, eqs. (7.14)–(7.15) and (7.56), pp. 105–112, PDF.

Coordinate invariance is an additional requirement. Admissible changes of asymptotically Cartesian coordinates on the same slice leave the energy unchanged and rotate the momentum components with the asymptotic spatial frame. Lorentz transformation of the full energy-momentum four-vector also involves the asymptotic observer or slicing. General angle-dependent translations create ambiguities in quantities involving an extra factor of $x^i$, notably angular momentum and center of mass. Regge–Teitelboim conditions remove the offending leading pieces by requiring the $r^{-1}$ part of $\gamma_{ij}-\delta_{ij}$ to be even and the $r^{-2}$ part of $K_{ij}$ to be odd under $x^i\mapsto-x^i$, with corresponding decay for the opposite-parity remainders; see Regge and Teitelboim 1974, §§4–5, pp. 303–311. Stronger gauges can serve the same purpose, but they restrict coordinates rather than define isolatedness itself.

Physical infinity and compactified coordinates

Replacing $r$ by a coordinate such as

$$\rho=\frac{r}{L+r},\qquad L\gt0,$$

places $r=\infty$ at the dimensionless coordinate $\rho=1$; $L$ is a fixed length scale. This is computational compactification: the physical metric still has infinite proper extent in an asymptotically flat end. A conformal compactification instead introduces a rescaled metric and a conformal boundary. Neither operation turns spatial infinity into an ordinary interior point of the physical initial-data manifold.

Puncture data illustrate the distinction. In isotropic Schwarzschild data, $r=0$ appears as a coordinate puncture, but after inversion it represents a second asymptotically flat end. Whether a coordinate endpoint is an end, a regular center, a boundary, or a removable chart singularity must be decided from the metric and topology.

Independent Schwarzschild check

For the time-symmetric conformally flat metric

$$\gamma_{ij}=\psi^4\delta_{ij}, \qquad \psi=1+\frac{GM}{2r}, \qquad K_{ij}=0,$$

one has

$$(\partial_j\gamma_{ij}-\partial_i\gamma_{jj})s^i =-2\partial_r(\psi^4) =\frac{4GM}{r^2}+O(r^{-3}).$$

Integration over $S_r$ gives $16\pi GM+O(r^{-1})$; the ADM formula therefore returns $E_{\mathrm{ADM}}=M$. The momentum integral vanishes because $K_{ij}=0$. This simultaneously checks the normalization and the sign of the mass integrand.

Common pitfalls

Falloff is not the whole asymptotic structure. Function spaces, coordinate freedom, topology, and parity determine which charges exist and are invariant.

Every end has its own outward direction. On a multi-ended slice, “outward” is defined toward infinity within the end under consideration.

Coordinate compactification is not conformal completion. The former changes a coordinate range; the latter changes the metric used to represent infinity.

Exercise: why angular momentum needs more control

Use the falloff above to compare the leading radial orders of the ADM momentum integrand and the analogous rotation integrand, whose asymptotic generator is $\phi^i=O(r)$.

Solution

Because $K_{ij}-K\gamma_{ij}=O(r^{-2})$, contraction with a translation of order one gives an $O(r^{-2})$ integrand. Its sphere integral can have a finite limit. A rotation grows as $O(r)$, so the naive integrand is only $O(r^{-1})$; its integral could grow like $r$. Antisymmetry and parity cancel the dangerous terms for suitable data, but the basic falloff alone does not display that cancellation. Additional parity or asymptotic gauge conditions are therefore needed to define angular momentum without a supertranslation ambiguity.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.
• Regge, Tullio, and Claudio Teitelboim. “Role of Surface Integrals in the Hamiltonian Formulation of General Relativity.” Annals of Physics 88, no. 1 (1974): 286–318. DOI.