Conformal Null Infinity

Null infinity is a geometric boundary only when the physical spacetime admits a sufficiently regular conformal completion. The definition packages falloff, topology, causal character, generator completeness, and conformal gauge. A Penrose diagram or a formal change of radial coordinate does not by itself prove that the rescaled metric extends smoothly or that every escaping null geodesic reaches the boundary.

Required background. Analysis for Mathematical Relativity supplies extension regularity, and exterior algebra and differential forms supplies boundary orientations and flux forms.

Helpful background. The Einstein equation supplies conformal field equations; Causality and Global Structure supplies causal boundaries; PDEs and Green Functions for Gravitation supplies asymptotic evolution; and Gravitational Radiation and Waveforms supplies the observable interpretation.

Conformal completion

Let $(\widetilde M,\widetilde g)$ denote the physical spacetime. A conformal completion consists of a manifold-with-boundary $M$, a metric $g$ on $M$, a nonnegative defining function $\Omega$, and an embedding of $\widetilde M$ onto the interior such that

$$g_{\mu\nu}=\Omega^2\widetilde g_{\mu\nu}, \qquad \Omega=0, \qquad d\Omega\ne0$$

on the relevant boundary component. The differentiability of $g$ and $\Omega$ is part of the definition. Rescaling by a smooth positive function,

$$\Omega' = \omega\Omega, \qquad g'=\omega^2g,$$

leaves the physical metric unchanged. Quantities at infinity must therefore be invariant or transform under a declared conformal gauge.

Let $n_\mu=\nabla_\mu\Omega$. The inverse physical metric is $\widetilde g^{\mu\nu}=\Omega^2g^{\mu\nu}$, while the causal character of the unphysical boundary is determined by $g^{\mu\nu}n_\mu n_\nu$ at $\Omega=0$. For vacuum Einstein equations with $\Lambda=0$, the conformal transformation of the Ricci tensor and regularity imply

$$g^{\mu\nu}n_\mu n_\nu=0 \quad\text{on }\mathscr I.$$

Thus $\mathscr I$ is null and $n^\mu$ is tangent as well as normal to it. Its integral curves are the null generators. With $\Lambda>0$ the analogous boundary is spacelike; with $\Lambda<0$ it is timelike. The null conclusion is not purely kinematic—it uses the field equation and asymptotic class.

Future and past components

Future null infinity $\mathscr I^+$ is the component reached by future-directed outgoing null geodesics; $\mathscr I^-$ is reached by past-directed incoming ones. A cut is a spacelike two-sphere intersecting each generator once in the standard asymptotically flat topology. Choosing a retarded time $u$ labels cuts but is not canonical: conformal rescalings and BMS transformations alter the representative.

A common asymptotic-flatness definition asks for a neighborhood of $\mathscr I$ whose completion is sufficiently differentiable, has topology $\mathbb R\times S^2$, and obeys vacuum equations or controlled matter decay near the boundary. Asymptotic simplicity is stronger: every inextendible physical null geodesic has two endpoints on $\mathscr I$. Exact Minkowski spacetime is asymptotically simple; a black-hole spacetime is not, because null geodesics entering the singular interior do not reach $\mathscr I^+$. One instead uses weak asymptotic simplicity or a regular exterior neighborhood. Frauendiener sets out these distinctions and conformal gauges in Frauendiener 2004, §§2.1–2.4.

The Minkowski check

In outgoing coordinates $u=t-r$,

$$d\widetilde s^2=-du^2-2\,du\,dr+r^2d\omega^2.$$

Set $x=1/r$ and $\Omega=x$. Since $dr=-x^{-2}dx$,

$$ds^2 =\Omega^2d\widetilde s^2 =-x^2du^2+2\,du\,dx+d\omega^2.$$

This metric is smooth and nondegenerate at $x=0$. There, $n=dx$ is null because $g^{xx}=x^2\to0$, while $d\Omega=dx\ne0$. The boundary is generated by varying $u$ at fixed angle. This calculation checks the conformal powers and signs; it does not prove a completion for a general radiating metric.

Regularity is a dynamical condition

Einstein’s equation written directly for $g$ and $\Omega$ contains terms that appear singular as $\Omega\to0$. Friedrich’s conformal field equations promote rescaled curvature variables and rewrite the system regularly under suitable gauge and data compatibility. Hyperboloidal small-data theorems then evolve data whose slice meets $\mathscr I^+$ at a finite boundary and prove a smooth future completion Friedrich 1986, hyperboloidal stability statement, pp. 587–590.

Generic asymptotically Euclidean Cauchy data need not yield this smoothness at spatial infinity. Compatibility conditions and logarithmic terms can propagate to null infinity. The next page therefore distinguishes full peeling from polyhomogeneous completions; smoothness cannot be inserted as an innocent coordinate assumption.

Matter requires its own conformal weights. Maxwell fields in four dimensions are conformally well behaved, while massive fields generally are not. A stress tensor that decays too slowly can change the boundary equations or prevent the vacuum neighborhood used to define Bondi quantities.

Completeness versus regularity

A smooth null boundary, the physical affine length of an escaping ray, and the future extent of $\mathscr I^+$ are three distinct issues. For the same null geodesic in $g=\Omega^2\widetilde g$, affine parameters satisfy

$$\frac{d\widetilde s}{ds}=C\Omega^{-2},\qquad C\gt0.$$

If the ray meets a regular boundary transversely at $s=s_*$, then $\Omega\sim c(s_*-s)$ with $c\ne0$, and the physical affine length diverges as $(s_*-s)^{-1}$. In the Minkowski example a ray at fixed $u$ therefore takes infinite physical affine length to reach $x=0$.

By contrast, future completeness of null infinity asks whether the family of cuts continues indefinitely in an appropriately normalized future parameter. The boundary generators vary $u$ at $x=0$; they are not the outgoing physical rays, which have fixed $u$. Truncating Minkowski’s boundary at $u=u_*$ does not shorten the affine length of any outgoing ray with $u\lt u_*$. It does remove future cuts. A conformal-gauge condition or a physical limiting definition is needed to make generator completeness invariant, as explained in the weak-censorship formulation.

This distinction is essential in weak cosmic censorship, which commonly asks for completeness of a constructed $\mathscr I^+$, and in scattering, which asks whether radiation data on that boundary capture all asymptotic energy.

Check the defining function

For Minkowski space in $(u,r)$ coordinates, repeat the compactification with $\Omega'=a(u,\vartheta) /r$, where $a>0$ is smooth. Does the boundary remain null, and which boundary data change?

Solution

$\Omega'=a\Omega$ and $g'=a^2g$, so this is an allowed conformal-gauge change. On $\mathscr I^+$, $d\Omega'=a\,d\Omega$ because the term $\Omega\,da$ vanishes. Hence $g'^{-1}(d\Omega',d\Omega')=g^{-1}(d\Omega,d\Omega)=0$ there, and the boundary remains null. Its intrinsic conformal two-metric, generator normalization, retarded-time representative, and expansion change with $a$. A physical flux or charge must combine fields with the proper conformal weight so that its meaning does not depend on this representative.

References

• Frauendiener, Jörg. “Conformal Infinity.” Living Reviews in Relativity 7 (2004): 1. DOI.
• Friedrich, Helmut. “On the Existence of $n$-Geodesically Complete or Future Complete Solutions of Einstein’s Field Equations with Smooth Asymptotic Structure.” Communications in Mathematical Physics 107 (1986): 587–609. DOI.