Asymptotically Flat and Hyperboloidal Initial Data

Asymptotically flat Cauchy data approach spatial infinity, whereas hyperboloidal data meet future or past null infinity after conformal compactification. Both can describe isolated systems, but they use different ends, decay conditions, charges, and regularity problems. ADM energy belongs to spatial asymptotics; Bondi data require control at null infinity and are not obtained by relabeling an ADM boundary integral.

Required background. Analysis for Mathematical Relativity supplies weighted spaces, The Einstein Equation supplies the constraints, Hypersurface Normals and Tangential Projectors supplies intrinsic and extrinsic data, and Differential Equations for Relativity supplies asymptotic boundary conditions.

Helpful background. Causality and Global Structure explains spatial and null infinity, while PDEs and Green Functions for Gravitation reviews weighted estimates.

Asymptotically Euclidean ends

On an asymptotically Euclidean end with coordinates $x^i$ and $r=|x|$, a standard four-dimensional vacuum class has the indicative falloff

$$h_{ij}=\delta_{ij}+O(r^{-1}), \qquad \partial_kh_{ij}=O(r^{-2}), \qquad K_{ij}=O(r^{-2}),$$

with corresponding higher-derivative or weighted Sobolev control. The ADM energy is

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

The momentum boundary integral uses $K_{ij}-Kh_{ij}$, consistently with $K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu$. Coordinate invariance of these limits requires an admissible asymptotic coordinate class, not merely pointwise convergence $h\to\delta$. Angular momentum and center of mass need stronger decay, parity conditions, or controlled asymptotic expansions to remove divergent odd/even pieces.

Weighted Sobolev spaces encode decay and differentiability simultaneously, but notation for the weight varies. A theorem must define the integral norm, not infer a pointwise estimate solely from the subscript. The worked power-law test explains one explicit convention. Carlotto reviews asymptotically flat constraint constructions and the associated functional setting in Carlotto 2021, §§2–3.

Checking the ADM normalization

For time-symmetric Schwarzschild data on $r\gt0$, write

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

Here $m$ is a length and $M$ is the physical mass. The metric is scalar-flat because $\Delta_\delta\psi=0$ away from $r=0$; the puncture represents another asymptotic end, not a matter delta-function included in this manifold. Since $\partial_jh_{ij}-\partial_ih_{jj}=-2\partial_i\psi^4$ and $\partial_r\psi^4=-2m/r^2+O(r^{-3})$, the surface integral tends to $16\pi m$. Thus $E_{\mathrm{ADM}}=m/G=M$. This checks both the $16\pi G$ normalization and the distinction between physical mass and geometrized length.

Hyperboloidal ends

A hyperboloidal slice is spacelike but asymptotic to a null hypersurface. Its physical metric is complete in the appropriate direction, while a defining function $\Omega$ compactifies it:

$$\overline h_{ij}=\Omega^2h_{ij}, \qquad \Omega=0,\quad d\Omega\ne0 \quad\text{at }\partial\overline\Sigma.$$

The boundary represents a cut of conformal null infinity rather than a point at spatial infinity. Hyperboloidal extrinsic curvature is not asymptotic to zero in the same way as an asymptotically Euclidean slice; a constant-mean-curvature hyperboloid in Minkowski spacetime already has nonzero pure-trace $K_{ij}$.

Smooth conformal completion imposes more than leading falloff. The conformal constraint equations can develop apparently singular coefficients at $\Omega=0$, and regularity requires compatibility conditions. In the pure-trace construction of Andersson, Chruściel, and Friedrich 1992, Theorem 1.1, pp. 589–590, smoothness is equivalent to vanishing of the trace-free second fundamental form of the conformal boundary; for vacuum data with $\Lambda=0$, this is the shear-free condition. For more general seed data, logarithmic terms occur generically, and their removal requires explicit boundary compatibility conditions, as shown in Andersson and Chruściel 1993, pp. 2830–2832.

What the charges measure

ADM four-momentum is a charge at spatial infinity of a complete asymptotically Euclidean slice. Bondi mass and momentum live on cuts of null infinity and evolve through radiation flux. A hyperboloidal initial surface can meet such a cut, but the Bondi quantities depend on the conformal and radiative structure of the development. They are not determined by inserting hyperboloidal falloff into the ADM formula.

The contrast is summarized as follows:

Feature	Asymptotically Euclidean data	Hyperboloidal data
Limiting boundary	Spatial infinity	A cut of null infinity
Typical $K_{ij}$	Decays toward zero	Approaches a nonzero trace part in common CMC constructions
Compactification	Optional analytical tool	Central to representing the boundary
Main regularity issue	Weighted decay, coordinates, parity	Conformal smoothness, shear, logarithmic obstructions
Natural energy	ADM at spatial infinity	Bondi-type quantity after null-infinity structure is fixed

Neither class is intrinsically “more physical.” Asymptotically Euclidean data are adapted to complete Cauchy developments and global charges; hyperboloidal data are adapted to radiation and conformal infinity. Gluing or matching them requires a theorem controlling the transition, not just agreement of leading powers.

Check your understanding

In Minkowski spacetime, compare the $t=0$ plane with the future unit hyperboloid $t=\sqrt{1+r^2}$. Why can the second be hyperboloidal although its extrinsic curvature does not decay to zero?

Solution

The plane approaches spatial infinity and has $K_{ij}=0$. Restoring a radius $a\gt0$, the future hyperboloid $t=\sqrt{r^2+a^2}$ has $K_{ij}=-h_{ij}/a$ and scalar mean curvature $K=-3/a$. Its trace-free extrinsic curvature vanishes. In an orthonormal spatial frame $K_{\hat i\hat j}=-\delta_{\hat i\hat j}/a$ remains nonzero; coordinate components by themselves can grow with $r$. Hyperboloidal regularity concerns the conformally rescaled geometry at its null boundary, not decay of the entire $K_{ij}$ to zero. The explicit compactification and conformal diagram show where the two slices end.

References

• Andersson, Lars, and Piotr T. Chruściel. “On ‘Hyperboloidal’ Cauchy Data for Vacuum Einstein Equations and Obstructions to Smoothness of ‘Null Infinity’.” Physical Review Letters 70 (1993): 2829–2832. DOI.
• Andersson, Lars, Piotr T. Chruściel, and Helmut Friedrich. “On the Regularity of Solutions to the Yamabe Equation and the Existence of Smooth Hyperboloidal Initial Data for Einstein’s Field Equations.” Communications in Mathematical Physics 149 (1992): 587–612. DOI.
• Carlotto, Alessandro. “The General Relativistic Constraint Equations.” Living Reviews in Relativity 24 (2021): 2. DOI.