Radiative Limits at Future Null Infinity

Radiation from an isolated system does not approach a fixed spatial point: it travels outward along null directions while its amplitude falls as $1/r$. Future null infinity, denoted $\mathscr I^+$, packages the limit $r\to\infty$ with retarded time and angle held fixed into a finite geometric boundary. A radiative limit is meaningful when a conformally rescaled metric extends regularly enough to that boundary and the rescaled radiation fields have finite limits there. This page uses conformal completion to define those limits; rigorous peeling theorems and detector-specific waveform reconstruction remain separate subjects.

Helpful background. Einstein Equations supplies the vacuum field equation used at the boundary; Waves, Green Functions, and Geometric Optics supplies outgoing characteristics and retarded time; Causality and Global Structure develops conformal boundaries as spacetime geometry; Gravitational Radiation and Waveforms supplies the radiative observables used here; and Formulations of General Relativity distinguishes characteristic data, constraints, and gauge.

The conformal boundary for outgoing radiation

Let $(\mathcal M,g_{ab})$ be the physical asymptotically flat spacetime. A conformal completion consists of a manifold with boundary $\widehat{\mathcal M}$, a function $\Omega$, and an unphysical metric

$$\widehat g_{ab}=\Omega^2 g_{ab}$$

such that the physical region has $\Omega>0$, while

$$\Omega=0, \qquad \widehat\nabla_a\Omega\ne0 \quad\text{on }\mathscr I^+.$$

For an isolated four-dimensional system, the relevant component has topology $\mathbb R\times S^2$: each generator represents one asymptotic null direction, and a parameter along the generator plays the role of retarded time. In vacuum near the boundary, the conformally transformed Einstein equation implies

$$\left.\widehat g^{ab}\widehat\nabla_a\Omega\widehat\nabla_b\Omega\right|_{\mathscr I^+}=0.$$

Thus $\mathscr I^+$ is itself a null hypersurface. The nonvanishing gradient is essential: without it, $\Omega=0$ need not define a regular boundary at all. The construction and its Bondi-frame specialization are given in Mädler and Winicour 2018, §4, pp. 17–20, PDF.

The completion is not unique. Replacing $\Omega$ by $\Omega'=\omega\Omega$ with a smooth positive $\omega$ changes the unphysical representative but not the physical metric. A preferred Bondi conformal frame can be chosen so that the generators of $\mathscr I^+$ are affinely parametrized, expansion-free, and shear-free, and its cross-sections carry the unit-sphere metric $q_{AB}$. Those conditions fix useful structure without pretending that a choice of retarded-time origin or angular frame is physical.

Minkowski space as the normalization check

The flat-space completion makes every boundary condition concrete. In outgoing coordinates $u=t-r$,

$$ds^2=-du^2-2\,du\,dr+r^2q_{AB}dx^A dx^B.$$

Set $\Omega=1/r$, so $dr=-d\Omega/\Omega^2$. The rescaled metric is

$$d\widehat s^2 =-\Omega^2du^2+2\,du\,d\Omega+q_{AB}dx^A dx^B.$$

It is regular at $\Omega=0$, and $d\Omega$ plainly remains nonzero. In the $(u,\Omega)$ block, the inverse metric has $\widehat g^{\Omega\Omega}=\Omega^2$; hence the normal becomes null precisely at the boundary. The vector $\widehat g^{ab}\nabla_b\Omega$ reduces there to the tangent $\partial_u$, showing directly the special null-hypersurface fact that the normal direction is also tangent. This check also fixes the sign of the $du\,d\Omega$ term for the inherited $(-,+,+,+)$ signature.

Finite data at the boundary

In a Bondi frame, write the angular metric as

$$g_{AB}=r^2q_{AB}+rC_{AB}+O(1), \qquad q^{AB}C_{AB}=0.$$

The physical perturbation $C_{AB}/r$ vanishes pointwise as $r\to\infty$, but the coefficient $C_{AB}(u,x^A)$ survives on $\mathscr I^+$. Choose a complex unit-sphere dyad with $m^A\bar m_A=1$ and define the rescaled complex strain

$$H(u,x^A) =C_{AB}\bar m^A\bar m^B =\lim_{r\to\infty}r\bigl(h_+-ih_\times\bigr).$$

The news tensor and news scalar in this volume are

$$N_{AB}=\partial_u C_{AB}, \qquad \mathcal N=N_{AB}\bar m^A\bar m^B=\partial_u H.$$

These are limits along outgoing null hypersurfaces, not limits on a spacelike slice. Holding $t$ fixed while sending $r$ to infinity drives $u=t-r$ to the infinite past and asks a different question. Mädler and Winicour instead define $N^{\rm MW}_{AB}=\tfrac12\partial_u C_{AB}$; therefore every equation imported from that review requires the explicit translation $N^{\rm MW}_{AB}=N_{AB}/2$.

The finite coefficient alone is not yet an observable independent of frame. The shear $C_{AB}$ changes inhomogeneously under a supertranslation, whereas its retarded-time derivative has a geometrically defined conformal limit. A waveform comparison must consequently specify a Bondi frame even though the presence of news is not removable by an allowed asymptotic coordinate choice.

Regularity and its limits

A smooth conformal metric supports a Taylor expansion in $\Omega$ and makes the leading Bondi fields ordinary tensors on $\mathscr I^+$. Weaker asymptotics may contain logarithms, producing polyhomogeneous expansions and weakened peeling. The existence of finite $C_{AB}$ and $N_{AB}$ should therefore not be advertised as a proof that every Weyl component obeys the strongest classical peeling hierarchy. Mädler and Winicour explicitly separate the formal $1/r$ expansion from more general asymptotics Mädler and Winicour 2018, §3, pp. 10–16, PDF.

Nor does conformal completion turn finite-radius numerical data directly into Bondi data. Extraction requires a controlled areal radius, tetrad, retarded time, conformal frame, and limiting procedure. Those reconstruction choices lie beyond this page; here $\mathscr I^+$ supplies the geometric target against which they must be checked.

Exercise

Show that the statement “$\mathscr I^+$ is null” is unchanged by the conformal replacement $\Omega'=\omega\Omega$, $\widehat g'_{ab}=\omega^2\widehat g_{ab}$, where $\omega$ is smooth and positive at the boundary.

Solution

On $\mathscr I^+$, the term proportional to $\Omega\nabla_a\omega$ vanishes, so $\nabla_a\Omega'=\omega\nabla_a\Omega$. The inverse metric transforms as $\widehat g'^{ab}=\omega^{-2}\widehat g^{ab}$. Consequently $\widehat g'^{ab}\nabla_a\Omega'\nabla_b\Omega'=\widehat g^{ab}\nabla_a\Omega\nabla_b\Omega=0$ at the boundary. Positivity of $\omega$ also preserves $d\Omega\ne0$, so both regularity and nullness survive the change of conformal representative.

References

• Thomas Mädler and Jeffrey Winicour, “Bondi–Sachs Formalism,” Scholarpedia 11 (2016) 33528, updated arXiv version (2018), arXiv:1609.01731v3, Open PDF.