Matching Spatial and Null Infinity

Past and future null infinity carry a priori independent BMS groups and independent charges. Relating incoming data on $\mathscr I^-$ to outgoing data on $\mathscr I^+$ requires a controlled passage through spatial infinity $i^0$, where a nonzero ADM mass prevents an ordinary smooth conformal point. The successful constructions resolve directions at $i^0$, impose continuity and falloff conditions strong enough to compare null and spacelike limits, and select an antipodal identification. Charge matching at the corner then yields flux balance on finite cuts; a global scattering conservation law needs further control at timelike infinity and, when present, at horizons.

Required background. The Riemann Curvature Tensor supplies the electric and magnetic Weyl data used in the matching conditions.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes null orientations; Affine and Variational Geodesics supplies generator completeness; Causality and Global Structure provides conformal boundaries; Formulations of General Relativity supplies characteristic and Cauchy evolution; and Exact Solutions of Einstein’s Equations provides Minkowski and Kerr checks.

The corner problem at spatial infinity

On $\mathscr I^-$, a BMS supertranslation is specified by a smooth function on the sphere of incoming generators. On $\mathscr I^+$, another smooth function labels outgoing generators. The field equations intrinsic to either null boundary do not identify those two spheres. In Minkowski spacetime the smooth conformal geometry at $i^0$ makes the identification look automatic, but that picture is exceptional: for nonzero ADM mass the unphysical metric is not $C^1$ at the conformal point in the ordinary sense.

The Ashtekar–Hansen construction retains a continuous metric with regular direction-dependent derivatives. Spacelike directions approaching $i^0$ form a three-dimensional unit hyperboloid $\mathcal H$ in $T_{i^0}M$. It is the hyperboloid of unit spacelike vectors and has Lorentzian induced signature. Fields such as suitably rescaled electric and magnetic Weyl tensors have direction-dependent limits that become smooth fields on $\mathcal H$.

For null matching, one conformally completes this hyperboloid to a cylinder $\mathcal C$. Its two boundary spheres $\mathcal N^-$ and $\mathcal N^+$ represent null directions approaching $i^0$ along $\mathscr I^-$ and $\mathscr I^+$. A reflection isometry of $\mathcal H$ extends conformally to $\mathcal C$ and maps $\mathcal N^-$ to $\mathcal N^+$; on the angular sphere it is the antipodal map

$$A:(\theta,\phi)\longmapsto(\pi-\theta,\phi+\pi).$$

This direction-space construction, including the two possible orders of taking null and spacelike limits, is developed in Prabhu 2019, §§2.1 and 4, pp. 12–16 and 23–31.

The cylinder $\mathcal C$ should not be silently identified with Friedrich’s conformal-field-equation cylinder. Both blow up spatial infinity, but their ambient structures and purposes differ: the former compactifies the space of Ashtekar–Hansen directions for charge matching, while the latter turns the spatial-infinity point into a total-characteristic cylinder for evolution and transport equations. At its critical spheres logarithms may obstruct smooth extension to null infinity Paetz 2018, §§3–5, pp. 27–60.

Conformal frames must also be matched

The usual Bondi conformal frame makes the null generator divergence-free on $\mathscr I$. That frame is ill adapted to viewing $\mathscr I^+$ as a null cone with vertex $i^0$, because generators of a cone have nonzero expansion. A gluing construction therefore uses a conformal factor whose null normal has controlled positive divergence near $i^0$, compares fields there, and only then translates back to a divergence-free Bondi frame. The relation sends the Bondi affine parameter to $-\infty$ at $i^0$.

This is not bookkeeping. A tensor component that has a finite direction-dependent limit in the expanding frame can acquire a definite inverse-power falloff in Bondi retarded time. Treating the two conformal frames as identical can assign the wrong weight to news or curvature. The incompatibility and explicit frame conversion are given in Ashtekar and Khera 2024, §3.1, pp. 16–18.

Even a spacetime that is asymptotically flat separately at null and spatial infinity need not satisfy the required continuity between them. One must specify a common completion or a gluing rule, the topology on directions, the rescaling weights of fields, and the order in which limits are taken.

Supertranslations and supermomentum

Prabhu’s nonlinear supermomentum result begins with an Ashtekar–Hansen spacetime and imposes null regularity at $i^0$. Let $\rho:=\Sigma^{-1}$ be a defining function for the boundary spheres of $\mathcal C$, so $\rho\to0$ at $\mathcal N^\pm$, and let $n^a$ generate $\mathscr I$ while $l^a$ is the auxiliary null normal to its cuts. The assumptions include regular direction-dependent limits of

$$\Sigma^{-3}\Omega^{-1}C_{abcd}l^a n^b l^c n^d$$

from both null and spacelike directions. They also require, for some $\epsilon\gt 0$, the Bondi news $N_{ab}$ and the projected radiative Weyl field $R_{ab}$ to obey

$$N_{ab}=O\!\left(\rho^{1+\epsilon}\right), \qquad R_{ab}=O\!\left(\rho^{1+\epsilon}\right)$$

as $\rho\to0$ along each generator of $\mathscr I$; equivalently, these are $O(\Sigma^{-(1+\epsilon)})$ falloffs. Here $R_{ab}$ is the two-cut projection of $\Omega^{-1}C_{abcd}$ with two factors of $n^a$. These hypotheses make the supermomentum limits and the flux of all BMS supermomenta finite; the weaker conditions needed only for four-momentum do not suffice. The exact null-regularity definition and its role are stated in Prabhu 2019, Definition 4.1 and Remark 4.1, pp. 23–27.

A BMS parameter $f^\pm$ is rescaled to a boundary value $F^\pm=\lim_{i^0}\Sigma f^\pm$ on $\mathcal N^\pm$. Not every interpolation through $\mathcal C$ has zero Spi-supermomentum flux. Selecting the totally fluxless supertranslations imposes, in Prabhu’s orientation and general rescaling,

$$\Upsilon\circ F^-=-\varsigma F^+,$$

where $\Upsilon$ is the reflection and $\varsigma$ is its conformal factor. This is a choice of the diagonal symmetry subalgebra, not another restriction on the spacetime. If both null generators are instead oriented toward the future and the boundary spheres use the same unit-sphere representative, the conventional statement is simply

$$f^+(x^A)=f^-(A x^A).$$

The asymptotic Einstein equations then antipodally match the relevant electric-Weyl data, and the supermomentum charges on limiting cuts at $i^0$ are equal. The construction and its orientation-dependent sign are explicit in Prabhu 2019, §4.1, pp. 27–31.

Lorentz charges require stronger assumptions

Supertranslations form a normal subgroup of BMS, so a “pure” Lorentz generator is not canonical until supertranslation freedom has been fixed. The Lorentz matching construction chooses an Ashtekar–Hansen conformal gauge, removes residual translations, and imposes continuity of a rescaled trace-free Schouten component. The resulting Lorentz vector fields are even under reflection on $\mathcal H$ and hence antipodally matched on $\mathcal N^\pm$.

For the charges, Prabhu and Shehzad use the following sufficient conditions:

1. enough null-infinity regularity for the required rescaled Weyl components, stated there through peeling;
2. vanishing leading magnetic Weyl field $B_{ab}$ on $\mathcal H$;
3. reflection-odd parity of the subleading magnetic field $\beta_{ab}$;
4. the null-regularity conditions used for supermomentum;
5. continuity at $\mathcal N^\pm$ of the relevant trace-free rescaled Schouten projection; and
6. continuity at $\mathcal N^\pm$ of the particular rescaled Weyl component entering the Lorentz charge.

Under these hypotheses the Lorentz charge computed from $\mathscr I^-$, from $\mathcal H$, and from $\mathscr I^+$ has the same limiting value. Combined with supermomentum matching, this identifies all BMS charges for this class. The assumptions are summarized—and explicitly described as sufficient, possibly not necessary—in Prabhu and Shehzad 2022, §6, pp. 34–35.

The scope is narrower than the slogan “BMS charges always match.” The condition $B_{ab}=0$ holds for stationary or axisymmetric asymptotically flat spacetimes, and the Kerr–Newman family satisfies the full listed regularity class. Validity for a large generic class of nonlinear radiative solutions remains open. In polyhomogeneous settings, the curvature component needed for the standard Lorentz charge may fail to exist, so a modified charge and a new proof would be required.

From corner equality to flux conservation

Let $S^-$ and $S^+$ be finite cuts of past and future null infinity, and let $f^-$ and $f^+$ be matched parameters. With consistent orientations, charge balance has the schematic form

$$Q^+[f^+;S^+]-Q^-[f^-;S^-] =\mathcal F^+[f^+;i^0\!\to S^+] +\mathcal F^-[f^-;S^-\!\to i^0].$$

Equality of the charges at the corner is what removes an extra spatial-infinity term. It does not say charges on arbitrary finite cuts are equal; radiation between the cuts changes them. Prabhu gives the exact finite-cut supermomentum balance in Prabhu 2019, §4.1, pp. 30–31, and the corresponding all-BMS statement appears in Prabhu and Shehzad 2022, §5, pp. 32–34.

To turn this into “total incoming flux equals total outgoing flux,” one must send the other cuts to $i^-$ and $i^+$. That step needs falloff strong enough that endpoint charges vanish or have controlled limits. Massive bodies reach timelike infinity rather than null infinity. Collapse may produce a black hole, in which case horizon charges and fluxes can enter the complete conservation law. These are additional scattering assumptions, not consequences of matching at $i^0$.

An alternative gluing framework can select a single preferred Poincaré group common to null and spatial infinity under weaker continuity conditions, while deliberately not proving conservation of every supermomentum. Ashtekar and Khera emphasize this separation in Ashtekar and Khera 2024, §§3–5, pp. 16–30. “A common Poincaré subgroup” and “a diagonal full BMS group” are different conclusions.

Common pitfalls

Antipodal matching is not equal-angle matching. An incoming ray labeled by $x^A$ continues through the origin to the outgoing direction $A x^A$. Orientation conventions may add a sign to the symmetry parameter.

Separate regularity is not corner continuity. Smooth fields on each open component of $\mathscr I$ and direction-dependent fields on $\mathcal H$ can still approach incompatible limits at $\mathcal N^\pm$.

The global conservation law has more boundaries than $i^0$. Timelike infinity, massive matter, and event horizons must be included or excluded by explicit hypotheses before total incoming and outgoing fluxes are equated.

Exercises

Use the future-directed convention $f^+(x^A)=f^-(A x^A)$ and expand

$$f^-(x^A)=\sum_{\ell,m}a^-_{\ell m}Y_{\ell m}(x^A).$$

Find $a^+_{\ell m}$. In particular, match $f^-=Y_{00}+2Y_{10}$. Does this matching imply equality of the corresponding charges on arbitrary finite cuts?

Solution

Spherical harmonics obey $Y_{\ell m}(A x^A)=(-1)^\ell Y_{\ell m}(x^A)$. Therefore

$$a^+_{\ell m}=(-1)^\ell a^-_{\ell m},$$

and the example becomes $f^+=Y_{00}-2Y_{10}$. The even monopole is unchanged, while the dipole reverses under the antipodal relabeling. If one uses the convention in which the generator on $\mathscr I^-$ is past-directed, the extra orientation sign must be included as well. Matching gives equality of the limiting corner charges at $i^0$ under the stated regularity conditions. At finite cuts, their difference equals the intervening flux; it vanishes only when that matched flux does.

References

• Ashtekar, Abhay, and Neev Khera. “Unified Treatment of Null and Spatial Infinity III: Asymptotically Minkowski Space-times.” Journal of High Energy Physics 2024, no. 2 (2024): 210. DOI.
• Paetz, Tim-Torben. “On the Smoothness of the Critical Sets of the Cylinder at Spatial Infinity in Vacuum Spacetimes.” Journal of Mathematical Physics 59, no. 10 (2018): 102501. DOI.
• Prabhu, Kartik, and Ibrahim Shehzad. “Conservation of Asymptotic Charges from Past to Future Null Infinity: Lorentz Charges in General Relativity.” Journal of High Energy Physics 2022, no. 8 (2022): 29. DOI.
• Prabhu, Kartik. “Conservation of Asymptotic Charges from Past to Future Null Infinity: Supermomentum in General Relativity.” Journal of High Energy Physics 2019, no. 3 (2019): 148. DOI.