Peeling, Smoothness, and Polyhomogeneous Infinity

Peeling is a hierarchy of Weyl-curvature falloffs along outgoing null directions. In its classical form it follows from a sufficiently differentiable conformal completion and regular rescaled Weyl tensor; it does not follow from the bare statement that the metric approaches Minkowski space. Polyhomogeneous completions replace pure powers of inverse radius by powers multiplied by logarithms. They can preserve the leading radiative behavior while lowering differentiability and modifying subleading charges. The hypotheses, tetrad choice, and order of every logarithm therefore belong in any peeling claim.

Required background. The Riemann Curvature Tensor supplies the Weyl tensor, conformal curvature, and algebraic types used in the hierarchy.

Helpful background. Pseudo-Riemannian Metrics and Signature fixes null-tetrad normalization; Affine and Variational Geodesics supplies the outgoing affine parameter; Causality and Global Structure provides conformal infinity; Formulations of General Relativity supplies characteristic data and Bianchi evolution; and Exact Solutions of Einstein’s Equations provides flat and Schwarzschild checks.

The classical Weyl peeling hierarchy

Work near a smooth component of future null infinity and choose an outgoing Newman–Penrose tetrad $(\widehat\ell^a,\widehat n^a,\widehat m^a,\widehat{\bar m}{}^a)$ for the physical metric, normalized by

$$\widehat g_{ab}\widehat\ell^a\widehat n^b=-1, \qquad \widehat g_{ab}\widehat m^a\widehat{\bar m}{}^b=1.$$

Use the site’s Weyl-scalar definitions, in particular $\Psi_2=-C(\widehat\ell,\widehat m,\widehat{\bar m},\widehat n)$; the convention card collects the complete five-scalar package.

Take $r$ to be an affine or asymptotically affine parameter increasing along the outgoing congruence, and parallel-propagate the tetrad to the required order. With the convention in which $\Psi_4$ carries outgoing radiation at $\mathscr I^+$, classical peeling is

Weyl scalar	Leading order	Leading interpretation
$\Psi_4$	$O(r^{-1})$	Transverse outgoing radiation; algebraic type N term
$\Psi_3$	$O(r^{-2})$	Next, type III term
$\Psi_2$	$O(r^{-3})$	Coulombic mass and type II/D content
$\Psi_1$	$O(r^{-4})$	Subleading longitudinal content
$\Psi_0$	$O(r^{-5})$	Fastest-decaying incoming-oriented component

Authors who interchange the two real null legs interchange the scalar labels, so the tetrad orientation must accompany the table. The invariant content is an expansion whose successive coefficients become algebraically more special toward leading order, not five coordinate components considered in isolation.

Schwarzschild is an important nonradiative check. In an adapted principal tetrad,

$$\Psi_2=+\frac{GM}{r^3}, \qquad \Psi_0=\Psi_1=\Psi_3=\Psi_4=0.$$

It satisfies the hierarchy although its Bondi news vanishes. Peeling is therefore a statement about organized curvature decay, not a statement that radiation is present. Minkowski spacetime has all five scalars zero and satisfies every order trivially.

From conformal regularity to peeling

Let $g_{ab}=\Omega^2\widehat g_{ab}$ with $\Omega\sim r^{-1}$ at $\mathscr I^+$. The mixed-index Weyl tensor is conformally invariant:

$$C^a{}_{bcd}[g]=C^a{}_{bcd}[\widehat g].$$

For a smooth vacuum conformal completion, the unphysical Weyl tensor vanishes at $\mathscr I^+$, and the rescaled field

$$K^a{}_{bcd}:=\Omega^{-1}C^a{}_{bcd}[g]$$

extends regularly to the boundary. The vacuum Bianchi identity becomes a regular propagation equation for $K^a{}_{bcd}$. This geometric route and its hypotheses are explained in Frauendiener 2004, §2.3, pp. 17–18.

The different powers can be seen directly. Choose a regular unphysical tetrad $(\ell,n,m,\bar m)$ with $\ell$ transverse to $\mathscr I^+$ and $n$ tangent to its generators. An adapted physical tetrad has leading scaling

$$\widehat\ell^a=\Omega^2\ell^a, \qquad \widehat n^a=n^a, \qquad \widehat m^a=\Omega m^a.$$

These scalings preserve the physical normalization because $\widehat g=\Omega^{-2}g$. Lowering the first Weyl index gives $\widehat C_{abcd}=\Omega^{-2}C_{abcd}[g]=\Omega^{-1}K_{abcd}$, with the last $K$ lowered using $g$. The contraction defining $\Psi_0$ contains two $\widehat\ell$ and two $\widehat m$ legs, hence scales as $\Omega^{-1}\Omega^4\Omega^2=\Omega^5$. The contraction defining $\Psi_4$ contains two $\widehat n$ and two $\widehat{\bar m}$ legs, hence scales as $\Omega$. Counting the legs of the other contractions gives

$$\Psi_j=O(\Omega^{5-j})=O(r^{j-5}), \qquad j=0,1,2,3,4.$$

This derivation shows exactly where bounded rescaled curvature and a controlled tetrad enter. An arbitrarily growing boost of the tetrad would change the component powers without changing the curvature tensor.

Differentiability determines how much of that reasoning is legal. In one commonly used definition, a $C^4$ unphysical metric plus the vacuum equations makes $K^a{}_{bcd}$ at least $C^1$ at $\mathscr I^+$. Under weaker differentiability, only the leading falloffs associated with $\Psi_4$, $\Psi_3$, and $\Psi_2$ may survive; the classical estimates for $\Psi_1$ and $\Psi_0$ need not. The precise qualification appears in Ashtekar and Khera 2024, §2.1, pp. 5–6.

The implication is chiefly one-way:

$$\text{sufficient conformal differentiability and field equations} \quad\Longrightarrow\quad \text{peeling to the corresponding order}.$$

Verifying several leading powers along one congruence does not construct a conformal manifold, prove angular regularity, control a corner at $i^0$, or establish completeness of $\mathscr I$. Even all five scalar orders require control of tetrad rotations and uniformity in retarded time and angle before they can stand in for a geometric theorem.

Polyhomogeneous null infinity

A polyhomogeneous field admits an ordered asymptotic expansion of the form

$$F(u,r,x^A) \sim \sum_{p=p_0}^{\infty} \sum_{q=0}^{N_p} F_{pq}(u,x^A)\,r^{-p}\bigl[\log(r/r_0)\bigr]^q,$$

with smooth coefficient fields in the remaining variables, a fixed reference length $r_0\gt 0$, and a specified remainder class. The logarithm has a dimensionless argument. Using the dimensionless defining function $\Omega=r_0/r$ turns a logarithmic term into

$$r^{-p}\bigl[\log(r/r_0)\bigr]^q =r_0^{-p}\Omega^p(-\log\Omega)^q.$$

For $q\gt 0$ this is not smooth at $\Omega=0$. For a positive integer $p$, $\Omega^p\log\Omega$ extends as a $C^{p-1}$ function when set to zero at the boundary, but its $p$th derivative diverges logarithmically. Changing $r_0$ mixes coefficients at the same inverse-radius order; it does not remove the highest nonzero logarithmic power. A logarithm first appearing at high $p$ may therefore leave several leading curvature orders intact while obstructing a higher-order Taylor expansion.

Polyhomogeneity is more structured than an uncontrolled error term. The powers and maximum log degrees must form a prescribed index set, the expansion must be stable under the differentiations used in the equations, and the remainder must be uniform in $(u,x^A)$ in the stated region. It is also tied to an admissible Bondi or conformal gauge. A singular coordinate transformation can manufacture logs, whereas logs that survive all allowed smooth conformal and coordinate changes represent a genuine regularity obstruction.

Chruściel, MacCallum, and Singleton showed formal consistency of a Bondi–Sachs characteristic hierarchy with such expansions. In their class, the leading $r^{-1}$ and $r^{-2}$ Riemann coefficients retain the type N and type III behavior of smooth peeling, while logarithms can enter at $r^{-3}$; peeling agrees with the smooth case through $O(r^{-2-\epsilon})$ for $0\leq\epsilon\lt 1$ Chruściel, MacCallum, and Singleton 1995, §3, pp. 12–15. This is a result for their stated Bondi gauge, regularity, topology, and formal characteristic expansions—not a theorem that arbitrary logarithmic metrics have controlled radiation.

What logarithms preserve and what they change

Under the hypotheses of that polyhomogeneous Bondi–Sachs analysis, the usual mass-loss equation persists because the potentially contributing leading logarithmic coefficients are constant in retarded time. Yet the authors explicitly caution that if the leading part of the Bondi metric function itself contains a logarithm, interpreting the formal coefficient as a finite physical mass requires additional work Chruściel, MacCallum, and Singleton 1995, §3, pp. 13–14. Likewise, standard Newman–Penrose constants need not remain constant in a generic polyhomogeneous expansion.

Logs should not be equated with incoming radiation. Smooth examples can contain both incoming and outgoing radiation, while polyhomogeneous examples can have a well-defined outgoing null boundary. The radiation question is answered by news, flux, and data on $\mathscr I^-$; the differentiability question is answered by the conformal expansion. The two interact through the field equations but are not synonyms.

There is a particularly delicate source of logarithms at the corner where spatial and null infinity meet. Friedrich’s representation blows up the point $i^0$ to a cylinder; its critical two-spheres are where that cylinder meets $\mathscr I^-$ and $\mathscr I^+$. Transport equations can generate a logarithm at a new order even when every lower order is smooth. Paetz proves, in a formal conformal-field-equation setting, that radiation vanishing to every order at a critical set is sufficient for absence of these logs. With the additional assumptions of constant ADM mass aspect and vanishing dual ADM mass aspect, the same condition is necessary for a smooth critical-set representation Paetz 2018, Theorem 1.1 and Remark 1.2, pp. 5–6.

The qualification matters: the converse is not a general no-log theorem for arbitrary data, and the cited remark leaves the corresponding nonlinear characteristic existence problem open. Stationarity near spatial infinity is a useful sufficient mechanism in several constructions, not an established necessary condition in all asymptotic classes.

Common pitfalls

Peeling is tetrad-sensitive but not arbitrary. A badly boosted tetrad changes component powers. State an adapted normalization and allowed asymptotic tetrad transformations before comparing scalars.

A logarithm is not automatically a curvature divergence. A term such as $\Omega^5\log\Omega$ has several continuous derivatives and may affect only high-order data. Determine its order before diagnosing the leading radiation field.

No-log is not no-radiation. Smooth null infinity permits nonzero news. Only special corner theorems, under additional mass and regularity hypotheses, relate infinite-order radiation behavior to smoothness at the critical sets.

Exercises

Suppose an unphysical metric contains a nonzero term $s_{ab}(u,x^A)\Omega^3\log\Omega$, with $s_{ab}$ smooth, and a calculation in an adapted tetrad gives $\Psi_4=O(r^{-1})$, $\Psi_3=O(r^{-2})$, and $\Psi_2=O(r^{-3})$. Determine the differentiability contributed by the logarithmic factor at $\Omega=0$. May one invoke the $C^4$ conformal argument to conclude the classical orders for $\Psi_1$ and $\Psi_0$?

Solution

Let $f(\Omega)=\Omega^3\log\Omega$ for $\Omega\gt 0$ and set $f(0)=0$. Direct differentiation gives

$$f'=3\Omega^2\log\Omega+\Omega^2, \qquad f''=6\Omega\log\Omega+5\Omega, \qquad f'''=6\log\Omega+11.$$

The first two derivatives extend continuously to zero, but the third diverges. The term is $C^2$ and not $C^3$ unless its coefficient vanishes or cancels against another contribution. Therefore the full unphysical metric is not $C^4$ on the information given, and the $C^4$ theorem cannot be invoked. The three calculated leading orders are compatible with partial peeling, but they do not determine $\Psi_1=O(r^{-4})$ or $\Psi_0=O(r^{-5})$; those must be established separately in the weaker regularity class.

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.
• Chruściel, Piotr T., Malcolm A. H. MacCallum, and David B. Singleton. “Gravitational Waves in General Relativity XIV: Bondi Expansions and the ‘Polyhomogeneity’ of Scri.” Philosophical Transactions of the Royal Society of London A 350, no. 1692 (1995): 113–141. DOI.
• Frauendiener, Jörg. “Conformal Infinity.” Living Reviews in Relativity 7 (2004): 1. 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.