Conformal and Hyperboloidal Formulations

Conformal and hyperboloidal methods bring future null infinity into a finite evolution domain without replacing it by an artificial timelike boundary. They combine three logically distinct choices: a conformal metric that represents the physical asymptotic geometry, spacelike slices that approach outgoing null directions, and coordinate or conformal gauges that keep the boundary regular and fixed. Their value comes from making radiation accessible at $\mathscr I^+$; their difficulty is that smoothness at $\mathscr I^+$ imposes real conditions on fields and data.

Helpful background. This chapter draws on The Einstein Equation, Hypersurface Normals and Tangential Projectors, The Riemann Curvature Tensor, Formulations of General Relativity, and Causality and Global Structure.

Enter the conformal picture

Tildes mark physical fields throughout this chapter. The regular unphysical metric is

$$g_{\mu\nu}=\Omega^2\widetilde g_{\mu\nu}, \qquad \widetilde g_{\mu\nu}=\Omega^{-2}g_{\mu\nu}.$$

The physical region has $\Omega\gt 0$. A regular conformal boundary satisfies

$$\Omega=0, \qquad d\Omega\ne0,$$

while $g$ remains smooth and nondegenerate. At future null infinity, vacuum regularity further gives

$$g^{\mu\nu}\nabla_\mu\Omega\nabla_\nu\Omega=0 \qquad\text{on }\mathscr I^+.$$

Thus $d\Omega$ is nonzero even though its metric dual is null. We orient $d\Omega$ toward increasing $\Omega$, into the physical interior. Any spatial outward normal therefore points in the opposite direction at the compact boundary.

The conformal representative is not unique:

$$g\mapsto\omega^2g, \qquad \Omega\mapsto\omega\Omega, \qquad \omega\gt 0.$$

This freedom can control the expansion of scri, but cannot repair incompatible conformal data or turn a polyhomogeneous completion into a smooth one by decree.

Route through the chapter

The six pages follow the construction in dependency order.

1. Conformal Rescaling of Geometric Fields derives the connection, Ricci, scalar-wave, normal, induced-metric, and extrinsic-curvature transformations. A compactified outgoing Minkowski chart fixes every power of $\Omega$.
2. Conformal Einstein Field Equations distinguishes the directly rescaled Einstein equation, which retains formal poles, from an enlarged regular system using the Schouten tensor, Friedrich scalar, and rescaled Weyl curvature. Reconstruction recovers physical vacuum wherever $\Omega\gt 0$.
3. Hyperboloidal Slices embeds a CMC hyperboloid in Minkowski spacetime. It remains physically spacelike, has $\widetilde K=-3/a$ in the site’s convention, and meets a spacelike cut of null infinity after compactification.
4. Hyperboloidal Initial Data and Constraints states a selected CMC conformal construction on a compact manifold with boundary. Elliptic solvability, shear-free matching, boundary gauge, and higher compatibility conditions decide whether the rescaled Weyl field is smooth or logarithms appear.
5. Null-Infinity Fixing and Gauge Choices coordinates lapse and shift so $t(\Omega)=0$, then separately chooses a preferred conformal gauge. It explains why scri fixing is not a physical radiation boundary condition.
6. Hyperboloidal Layers and Cauchy Matching joins an unchanged interior Cauchy chart to an exterior hyperboloidal compactification. An explicit map checks monotonicity, metric derivatives, finite characteristic speeds, and continuous outgoing phase.

The smooth conformal vacuum equations and their hyperboloidal initial-value interpretation originate in Friedrich 1983, §§ 2–4, pp. 449–463. The direct rescaled-Einstein approach, preferred conformal gauge, and fixed scri construction are developed in Zenginoğlu 2008, §§ 2–4, pp. 3–8.

One chain of regularity conditions

Each stage solves a different problem. Conformal rescaling makes infinity finite, but curvature transformations expose negative powers of $\Omega$. The regular conformal field equations trade those divisions for additional constrained variables. Hyperboloidal slices reach the desired boundary, but their physical constraints must be solved with compatible asymptotics. Scri fixing holds the boundary at a coordinate location, while a preferred conformal gauge sets its expansion to zero. A layer preserves a useful interior chart, but its interface must be a smooth coordinate transition for the chosen evolved variables.

None of the adjacent statements is an equivalence. A finite radial coordinate does not imply a smooth conformal metric. A smooth $g$ does not automatically make $d=\Omega^{-1}C[g]$ smooth. A hyperboloidal slice is not necessarily a global Cauchy surface for the whole spacetime. A fixed null boundary does not invite freely prescribed physical data. A smooth test-field layer does not establish regular nonlinear Einstein evolution.

For the proof-critical data problem, Andersson’s CMC construction makes the distinction precise. Smooth free data on a compact manifold can produce a degenerate elliptic solution with an $x^3\ln x$ term. Shear freedom cancels the leading null-shear obstruction, while further boundary equations control smoothness of the rescaled curvature; see Andersson 2002, §§ 4.2–4.3, pp. 188–190.

Consistency checks

Power check. With $\widetilde g=\Omega^{-2}g$, a physical unit normal is $\widetilde n^\mu=\Omega n^\mu$, not $\Omega^{-1}n^\mu$. Every curvature and hypersurface formula should agree with that direction of rescaling.

Minkowski check. Outgoing Minkowski coordinates with $\Omega=1/r$ give $ds^2=-\Omega^2du^2+2\,du\,d\Omega+d\omega_2^2$. The boundary metric is nondegenerate and $(\nabla\Omega)^2=\Omega^2$.

Shear check. For future CMC data with inward boundary normal, the regular unphysical tracefree curvature obeys $B^{\mathrm{TF}}_{AB}=-\lambda^{\mathrm{TF}}_{AB}$ at the cut. The initial-data page normalizes $\Sigma=-1$, so its free tensor has $A|_S=B|_S$. Restoring another curvature scale requires restoring the free-data powers as well. Changing normal orientation changes the geometric shear relation’s sign.

Characteristic check. In a fixed-scri gauge, one radial null direction is tangent to $\Omega=0$ and the other exits the domain. If a physical characteristic enters from beyond scri, the lapse-shift orientation or the claimed boundary has been misidentified.

Layer check. The spatial Jacobian must stay positive, metric and first derivatives must agree at the interface, and the outgoing phase must not acquire a kink or infinite compression.

Review the chapter

In Minkowski spacetime take

$$t=\tau+\sqrt{r^2+a^2}, \qquad \Omega=\frac{a}{\sqrt{r^2+a^2}}.$$

1. Explain why the $\tau$ slices are physically spacelike yet reach future null infinity.
2. State the physical/unphysical metric relation and the boundary conditions on $\Omega$.
3. Give the site-sign mean curvature and the two radial characteristic speeds at fixed scri.
4. State which extra conditions are needed before this model motivates a smooth vacuum Einstein evolution for general data.
5. Explain why replacing the whole construction by spatial compactification alone fails for outgoing oscillatory waves.

Solution

The slope is $dt/dr=r/\sqrt{r^2+a^2}\lt 1$, so the induced physical metric is positive definite, while $t-r\to\tau$ makes the end approach constant retarded time. The metrics obey $g=\Omega^2\widetilde g$; the physical region has $\Omega\gt 0$, and regular scri has $\Omega=0$, $d\Omega\ne0$, smooth nondegenerate $g$, and $(\nabla\Omega)^2=0$.

With the future normal and site convention, $\widetilde K_{ij}=-\widetilde h_{ij}/a$ and $\widetilde K=-3/a$. In the compact coordinate used here, the radial speeds at $\Omega=0$ are $0$ and $-2/a$: one is tangent, and the other exits the domain $\Omega\ge0$.

For general vacuum data one must also solve the physical constraints through a conformal elliptic construction, satisfy shear-free and higher boundary compatibility conditions, choose a regular conformal evolution system and gauges, and assume or prove the needed smooth asymptotic structure. Spatial compactification alone maps infinitely many oscillations into a finite interval; the hyperboloidal time transformation keeps outgoing retarded phase finite and prevents that blueshift.

References

• Andersson, Lars. “Construction of Hyperboloidal Initial Data.” In The Conformal Structure of Spacetimes: Geometry, Analysis, Numerics, edited by Jörg Frauendiener and Helmut Friedrich, Lecture Notes in Physics 604, 183–194. Springer, 2002. arXiv:gr-qc/0205083.
• Friedrich, Helmut. “Cauchy Problems for the Conformal Vacuum Field Equations in General Relativity.” Communications in Mathematical Physics 91 (1983): 445–472. DOI.
• Zenginoğlu, Anıl. “Hyperboloidal Evolution with the Einstein Equations.” Classical and Quantum Gravity 25 (2008): 195025. DOI. Open PDF.
• Zenginoğlu, Anıl. “Hyperboloidal Layers for Hyperbolic Equations on Unbounded Domains.” Journal of Computational Physics 230 (2011): 2286–2302. DOI. Open preprint.