From spatial to null infinity: Connecting initial data to peeling

Berend Schneider; Neev Khera

REVIEW 3 major objections 4 minor 53 references

From spatial to null infinity: Connecting initial data to peeling

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Gravitational wave fall-off at infinity is governed by a parity-time symmetry of the initial data.

desk verdict A genuinely new PT-symmetry route to Ψ2 and Ψ1 peeling in full GR, but the particular-solution estimates that carry the central claim are asserted rather than proved. read the letter →

arxiv 2507.07977 v1 pith:WTHHCQYK submitted 2025-07-10 gr-qc hep-thmath-phmath.MP

Berend Schneider , Neev Khera This is my paper

classification gr-qchep-thmath-phmath.MP MSC 83C3083C0583C40 PACS 04.20.-q04.20.Ha04.30.-w

keywords peelingWeylscalarsnullinfinityspatialPTsymmetryFriedrichcoordinatesasymptoticflatnessgravitationalwaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper establishes a direct link between the symmetry of initial data near spatial infinity and the peeling of gravitational radiation at null infinity. Peeling is the property that the curvature components—the Weyl scalars $\Psi_k$—fall off at definite rates toward infinity; it underpins gravitational-wave extraction and the very definitions of mass, angular momentum, and radiation. Within a broad class of asymptotically regular spacetimes, the authors show that $\Psi_4$ and $\Psi_3$ always peel, while $\Psi_2$ peels at the standard $1/r^3$ rate exactly when the leading-order initial data is symmetric under parity combined with time reversal (PT), and $\Psi_1$ peels at $1/r^4$ when in addition the subleading data is antisymmetric under PT; $\Psi_0$ does not fully peel, falling as $r^{-5}\log r$. The result matters because it converts an abstract condition at infinite distance into a concrete, checkable property of the initial data used in numerical and post-Newtonian studies of compact binaries.

What carries the argument

Three pieces of machinery carry the argument. The first is the Friedrich coordinate system $(\lambda,R)$, which places spatial infinity at $R\to\infty$ and null infinity at $\lambda = \pm 1$, so a single $1/R$ expansion covers the initial-data region, future null infinity, and past null infinity at once. The second is the PT transformation $(\lambda,R,\theta,\phi)\mapsto(-\lambda,R,\pi-\theta,\pi+\phi)$, combining time reversal with the antipodal map on the sphere; the paper shows that PT symmetry or antisymmetry of the data at a given order in $1/R$ is exactly what removes the logarithmically growing pieces of the solution. The third is the mode decomposition of the massless spin-$s$ equations in terms of Jacobi polynomials and Jacobi functions of the second kind, together with the regularity condition $g_{ab}-\eta_{ab}=O(1-\lambda^2)$ that guarantees the source integrals converge and that $\Psi_4$ has a well-defined radiation field. The coefficients of the Jacobi functions of the second kind serve as the bookkeeping device: their vanishing is equivalent to the required PT parity, and the particular-solution analysis shows that only finitely many orders in $1/R$ can ever violate peeling.

What would settle it

Compute the Weyl scalar $\Psi_2$ at future null infinity for an explicit spacetime that satisfies the expansion (4.1) and the regularity condition (4.4) but whose leading-order initial data is PT-antisymmetric: if $\Psi_2$ still decays as $r^{-3}$, the characterization is wrong. Conversely, an initial data set with PT-symmetric leading term whose $\Psi_2$ contains a $\log r/r^3$ piece would also refute the claim. A more local check is the integral $P_{2,1\ell m}(1)$ of equation (4.13a): the paper's parity argument requires it to vanish for PT-symmetric sources, so an explicit PT-symmetric $h_{ab}$ where it does not vanish would break the peeling claim for $\Psi_1$.

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Extended reading notes

Core claim

The central discovery is a characterization of the peeling theorem in terms of a discrete symmetry of initial data. For any vacuum spacetime that is asymptotically regular—admitting an expansion $g_{ab} = \eta_{ab} + h^{(1)}_{ab}/R + \cdots$ in Friedrich coordinates and obeying $g_{ab} - \eta_{ab} = O(1-\lambda^2)$—the Weyl scalars at future null infinity satisfy: $\Psi_4 = \Psi_4^\circ r^{-1} + o(r^{-1})$ and $\Psi_3 = \Psi_3^\circ r^{-2} + o(r^{-2})$ unconditionally; $\Psi_2 = \Psi_2^\circ r^{-3} + o(r^{-3})$ if the leading $1/R$ part of the initial data is PT-symmetric; $\Psi_1 = \Psi_1^\circ r^{-4} + o(r^{-4})$ if, in addition, the subleading $1/R^2$ part is PT-antisymmetric; and $\Psi_0 = O(r^{-5}\log r)$. The proof solves the Bianchi identities order by order in $1/R$: each order becomes an inhomogeneous spin-2 wave equation, with homogeneous solutions built from Jacobi functions and particular solutions given by integrals over lower-order sources. The decisive step is that the coefficients of the Jacobi functions of the second kind—the only place where logarithms near null infinity can arise—are forced to vanish precisely by the PT parity conditions on the data.

Load-bearing premise

The chain of results rests on the regularity assumption $g_{ab}-\eta_{ab}=O(1-\lambda^2)$ (equation (4.4)); if it fails—for instance, for initial data containing irreducible logarithmic terms—the stated peeling rates for $\Psi_2$ and $\Psi_1$ are not established.

Editorial extensions

If this is right

Peeling of $\Psi_2$ can be guaranteed from the initial slice alone: data that is PT-symmetric at order $1/R$ produces the standard $1/r^3$ fall-off at future null infinity.
When the subleading data is PT-antisymmetric, $\Psi_1$ peels and the angular momentum aspect at future null infinity is well defined, removing a known obstruction to defining angular momentum.
Leading-order PT symmetry yields an antipodal matching of the Bondi mass aspect between past and future null infinity at spatial infinity, and, when the three-momentum vanishes, a matching of angular momentum aspects.
Only finitely many terms in the $1/R$ expansion can violate peeling, so a finite-order truncation of the metric expansion is sufficient to make statements about the full asymptotic series.
Asymptotically regular spacetimes with irreducible logarithmic initial data are excluded from the analysis, which the paper identifies as a limitation for scattering-type spacetimes.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the conjecture that (4.4) follows from Weyl regularity is correct, then the PT criteria would extend to every spacetime with a regular Weyl curvature, making the peeling characterization nearly universal for isolated systems.
The same PT bookkeeping should apply to other asymptotic observables, such as the news function and the shear, so a similar 'PT parity of initial data' diagnostic could predict log anomalies in waveform extraction from simulations.
A direct numerical test is available: take a binary black hole initial dataset, decompose its leading $1/R$ pieces under PT, and compare the predicted $\Psi_2$ and $\Psi_1$ fall-offs with Cauchy-characteristic extraction of the evolved spacetime; a mismatch would pinpoint where the assumptions of this paper break.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should read this one. The paper connects initial data symmetries at spatial infinity to peeling of the Weyl scalars at null infinity, and the full-GR part is genuinely new. The scalar case was in Fuentealba–Henneaux, and the linear spin-s generalization is natural, but translating those ideas into the nonlinear Bianchi identities, with a particular solution that has to be controlled, goes beyond anything in the cited literature. The PT-symmetry criterion for Ψ2 and Ψ1 is elegant and likely to be useful for numerical relativity and for thinking about angular momentum at null infinity. The antipodal mapping of charge aspects is a nice payoff. The paper is also honest: it flags the exclusion of logarithmic initial data, the non-perturbative caveat, and the fact that the key regularity condition (4.4) is conjectured rather than derived.

The soft spot is exactly where you expect it. In Section IV.C the bounds (4.14) on the particular-solution integrals are the load-bearing part of the proof, but the paper only verifies them explicitly for S_{2,1}, where the source is quadratic in h_ab and the O((1−λ)^2) estimate goes through. The sentence that says (4.4) “also guarantees” (4.14a,b,d,e) is a leap. For general k,n the source contains connection terms Γ∂Ψ and ΓΓ; the derivatives of h_ab do not have to vanish at λ=1, so a product like h·∂Ψ is only O(1−λ), not O((1−λ)^2). For Ψ1 at n=1 that is exactly the borderline where an O(1−λ) source would make the integral (4.13a) diverge logarithmically, producing a log r/r^4 term and breaking the claimed peeling. The stress-test note is right: this is a gap, not a proof. I do not see a workaround in the text, and no induction argument is supplied for higher n.

That said, the paper is worth a serious referee. The gap is serious but probably fillable, and the framework—Friedrich coordinates, finitely many violating orders, PT as a selection rule—is useful even if the final theorem needs conditions. I would cite it once the estimates are nailed down, and I would bring it to a reading group now to argue about exactly this Section IV.C. Send it out.

For your records: the central claim is conditional, not wrong. The authors are not sloppy; they know where the weak point is. The referee report should ask for a complete proof or a clear inductive argument for (4.14).

Referee Report

3 major / 4 minor

Summary. The paper develops a unified asymptotic expansion of the metric in Friedrich coordinates (λ, R) in a neighborhood of spatial infinity and uses it to evolve initial data toward null infinity in a 1/R expansion. After treating the massless scalar and free spin-s fields, the authors consider the full Einstein equations at first orders and split the solutions into homogeneous spin-2 solutions and particular solutions built from Green's functions. Their main result, Eq. (4.25), states that for 'asymptotically regular' spacetimes (metric expansions (4.1)–(4.3) plus regularity (4.4)), Ψ4 and Ψ3 always peel, Ψ2 peels if the leading-order initial data is PT-symmetric, Ψ1 additionally peels if the subleading data is PT-antisymmetric, and Ψ0 = O(r^{-5} log r). The paper also derives an antipodal mapping of mass and angular momentum aspects between I+ and I- under these symmetries. The central novelty is the claim of a direct, coordinate-based connection between PT symmetries of initial data near spatial infinity and peeling at null infinity.

Significance. The scalar and spin-s sections are a clean, self-contained derivation of PT-parity peeling criteria, with explicit Jacobi-function solutions and asymptotics; these parts are likely to be useful independently of the gravity application. The gravity section addresses an important question—whether peeling is an acceptable assumption for astrophysical spacetimes—and gives a concrete, falsifiable class of initial-data symmetries that would produce standard fall-offs. The treatment is honest about its restrictions: logarithmic initial data are excluded, and the key regularity condition (4.4) is conjectured rather than derived. However, because the proof that the particular solution satisfies the required fall-offs is only asserted, the paper currently establishes a plausible framework rather than a complete theorem. If the missing estimates are supplied, the result would be a substantial contribution to the spatial/null-infinity matching literature.

major comments (3)

[§IV.C, Eqs. (4.13)–(4.14)] The claim that the regularity assumption (4.4) guarantees the bounds (4.14a)–(4.14e) is not demonstrated. The explicit argument after Eq. (4.21) treats only the source S_{2,1}, which is quadratic in h_ab, and uses the factor (1−λ)^2 from (4.4). For general k and n the source S_{k,n} also contains terms of the form Γ∂Ψ and ΓΓ, i.e., products of h_ab (only O(1−λ) by (4.4)) with derivatives of lower-order Weyl scalars or of h_ab; these derivatives are not controlled at λ=1. If such a term enters P_{k,n}, the integral (4.13a) can diverge logarithmically and the corresponding Weyl scalar acquires a log r factor at the order that fixes its peeling rate. Since the abstract's Ψ1 statement and Eq. (4.25d) depend directly on this estimate, the central claim is conditional on an unverified bound.
[§IV.D, Eq. (4.25e)] The unconditional statement Ψ0 = O(r^{-5} log r) is not derived. The text preceding (4.14) states that the bounds (4.14) would guarantee peeling for Ψ1, and category 1 explicitly excludes k=0 ('all peel for k≠0'). The homogeneous part indeed generically has a log r/r^5 term, but no argument is given that the particular solution cannot produce a term with a larger power of log r at the r^{-5} order. Since (4.25e) is part of the paper's summary, the authors should either prove the bound or state it as an assumption or expectation.
[§IV.C, Eq. (4.17)] The parity computation P_{2,1}(1)=0 relies on a definite PT transformation law for S_{2,1}, but the paper never defines the PT action on the source S_{k,n}. Equation (4.20) states only that a gauge exists in which S_{k,1}(h_ab) = PT[S_{k,1}(h_ab)], with the gauge transformation referenced to [33]; no explicit check of the sign in (4.17) is given. This makes the single explicit verification of the peeling bounds incomplete and should be repaired regardless of the more general estimate.

minor comments (4)

[§IV.C, Eq. (4.15)] The notation O_1(1−λ) is not defined, and the phrase 'instead of (4.14b)' appears to be a typo for (4.14c), since O(1−λ) is stronger than O(1/log(1−λ)).
[§V, first and third paragraphs] The statement that (4.4) is satisfied for any isolated astrophysical spacetime is stronger than anything proven in the paper; the subsequent discussion of PN/PM logarithmic terms shows that the condition is not automatic and depends on an unproved coordinate transformation.
[§IV.A, definition of O∞] The O∞ definition is given for scalar functions and then applied to tensor components in a fixed Cartesian frame; since the authors later use PT and gauge transformations, they should state whether O∞ bounds are invariant under the allowed gauge changes.
[General] A short table collecting the PT parity of the initial-data coefficients E_{nℓm}, O_{nℓm} at orders n=0,1,2 and the corresponding peeling statements would improve readability; currently the connection is implicit in Eqs. (2.7)–(2.9), (3.22)–(3.23), and (4.8).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PT-symmetry peeling criteria are derived from explicit solutions of the linearized equations and Green's function estimates, not from a fitted or renamed version of the target fall-offs.

full rationale

The advertised result (Eqs. (4.25c)-(4.25d)) is a genuine derivation rather than a disguised input. In the homogeneous spin-2 sector, the solution (3.12) has coefficients (2.7)-(2.9) fixed by the initial data (3.10), and the peeling conditions (4.8) are read off from the explicit Jacobi asymptotics; the equivalence of B_{k,n}=0 to PT parity is established in Sec. III.B. The particular-solution sector uses the Green's function representation (4.11)-(4.13); Eq. (4.17) shows P_{2,1}(1)=0 under the PT symmetry of S_{k,1}, and Eq. (4.21) with (4.4) gives the S_{2,1}=O((1-lambda)^2) estimate. No parameter is fitted and later called a prediction, and no uniqueness theorem is imported from the authors' prior work. The statement 'This regularity assumption also guarantees that Eqs. (4.14a), (4.14b), (4.14d) and (4.14e) are satisfied' (Sec. IV.C, after Eq. (4.21)) is asserted rather than proved for all sources; however, this is a rigor gap or omitted estimate, not a circular step, because the regularity condition (4.4) is a metric falloff assumption independent of the target Psi2/Psi1 peeling rates. The Psi4 and Psi0 peeling statements are built into the definition of asymptotic regularity via (4.4), but the paper presents them as consequences of that definition, not as the derived central claim. Self-citations [35,36] are contextual references for the antipodal charge matching and are not load-bearing. Hence the derivation chain is self-contained with respect to the paper's main claim despite the unproven bound.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claim relies on the asymptotic regularity assumptions (4.1) and (4.4), which restrict the class of spacetimes. Standard mathematical tools (Jacobi functions, GHP formalism) are used. No new physical entities are introduced.

assumptions (6)

domain assumption The spacetime metric admits an asymptotic expansion in powers of R^{-1} around a Minkowski background (Eq 4.1).
Restricts to data with analytic 1/R expansion, excluding logarithmic terms. Introduced in Section IV.A.
domain assumption The metric satisfies gab - eta_ab = O(1 - lambda^2) (Eq 4.4).
Ensures a well-defined radiation field at null infinity and is used to prove the particular solution peels. Conjectured in the paper to follow from Weyl regularity, but not proven.
domain assumption The initial data on the t=0 slice can be expanded as in Eq (2.2) or (3.10) with analytic coefficients.
The authors note this can be relaxed, but it is assumed for the analysis.
standard math Standard properties of Jacobi polynomials and functions (orthogonality, asymptotics, raising and lowering operators) from Appendix A.
Used to solve the wave equations and compute asymptotics.
standard math The GHP formalism and the massless free-field equations describe the Weyl scalars (Eq 3.3 to 3.7).
Background formalism for linearized fields and Bianchi identities.
domain assumption The Bianchi identities can be expanded around flat space so that the Weyl scalars satisfy flat wave equations with sources (Eq 4.5 to 4.7).
Relies on the asymptotic expansion of the metric and curvature.

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Pith. "Pith review of From spatial to null infinity: Connecting initial data to peeling." pith.science (2026). https://pith.science/paper/WTHHCQYK

@misc{pith2026250707977,
  author       = {Pith},
  title        = {Pith review of: From spatial to null infinity: Connecting initial data to peeling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WTHHCQYK}},
  note         = {Machine review of arXiv:2507.07977}
}

abstract

The asymptotic structure of space-time is studied by imposing conditions on the asymptotics of the metric. These conditions are weak enough to include large classes of physically relevant isolated space-times, but have a rich enough structure to be able to define important physically meaningful quantities like mass, angular momentum, and gravitational waves. By using a unified expansion of the metric in a neighborhood of spatial infinity that includes a piece of null infinity, we connect the asymptotic expansions of solutions to Einstein's equations in the different asymptotic regimes. Within the class of space-times under consideration, we find a connection between the peeling properties of the Weyl scalars and symmetries of initial data near spatial infinity. In particular, we show that for initial data that to leading order is symmetric under parity + time reversal, $\Psi_2$ has the usual $1/r^3$ fall-off rate at null infinity. If, in addition, the subleading part of the data is antisymmetric under parity + time reversal, then $\Psi_1$ has the usual $1/r^4$ fall-off rate at future null infinity.

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Works this paper leans on

53 extracted references · 29 canonical work pages

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M. Henneaux and C. Troessaert, The asymptotic structure of gravity at spatial infinity in four spacetime dimensions, Proceedings of the Steklov Institute of Mathematics 309, 127 (2020)

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Thus, they only contribute to 1/r2 and higher

We only have to consider the leading order in the 1 /R expansion because when the order in 1 /R increases, the solutions (2.6a) also increase the powers of 1 − λ2. Thus, they only contribute to 1/r2 and higher. We will later see similar results for the linear spin-s fields, and for gravity, where only a finite number of terms in the expansion will contribute

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This is not a feature of the standard wave equation but rather a consequence of the initial data (2.2)

Notice that the solution (2.6) either peels or violates peeling at both I + and I − simultaneously. This is not a feature of the standard wave equation but rather a consequence of the initial data (2.2). This can be seen by considering spherically symmetric solutions to the standard wave equation (2.3), which are given by ϕ = r−1f (u) + r−1g(v) in double ...

work page
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Similar to the scalar field case, there exist solutions that peel at either I + or I −. For example, the solutions (see Appendix B for a derivation) ϕk,ℓm = r−1−s−ℓ(r2∂v)ℓ+s−kVℓm(v) + r−1−s−ℓ(r2∂u)ℓ−s+kUℓm(u) , (3.26) where Vℓm(v) and Uℓm(u) are arbitrary, may fail to peel. Here r = (v − u)/2. In the simple case of k = 0, ℓ = s, Vℓm = 0, and Uℓm = |u|nδℓ,...

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In this case not all ϕk peel, but the situation is improved compared to (3.25a)

Instead of imposing the PT -parity condition at all orders n ≤ s, we can impose it for n ≤ w < sfor some w. In this case not all ϕk peel, but the situation is improved compared to (3.25a). In particular, we get that along I − we have ϕk = O 1 rk+1 for k ≤ s + w , (3.29a) ϕk = O log r rk+1 for k > s+ w . (3.29b)

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From this we can compute, for example, PT [Ψ1](λ, R, θ, ϕ) = PT [Cabcdlamblcnd](λ, R, θ, ϕ) = Cabcd(−na)(− ¯mb)(−nc)(−ld)(−λ, R, π− θ, π+ ϕ) = Ψ3(−λ, R, π− θ, π+ ϕ)

Under a PT transformation the tetrad vectors transform as PT ∗[la] = PT ∗[∂a v ] = −∂a u = −na , (3.30a) PT ∗[ma] = PT ∗ h 1√ 2r ∂a θ + i sin θ ∂a ϕ i = 1√ 2r −∂a θ + i sin(π − θ) ∂a ϕ = − ¯ma , (3.30b) and since PT = PT −1, PT ∗[na] = −la and PT ∗[ ¯ma] = −ma. From this we can compute, for example, PT [Ψ1](λ, R, θ, ϕ) = PT [Cabcdlamblcnd](λ, R, θ, ϕ) = C...

work page
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The P (α,β) ν≥0 , Q(α,β) ν≥2 , (1 − λ)−αP (−α,β) ν+α<α and (1 + λ)−βP (α,−β) ν+β<β solutions to the Jacobi equation all peel for k ̸= 0. The conditions (4.14a), (4.14b), and (4.14d) state that if their coefficients are finite, then the corresponding components of the particular solution have the same (peeling) fall-off rate

work page
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The solution (1 + λ)−βP (α,−β) ν+β<β corresponds to a Weyl scalar having the fall-off rate (1−λ)n+3, so that the integral Jk,nℓm (4.13d) must blow up no faster than (1−λ)2−k−n

work page

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The conditions in category 1 and 2 are conditions on every coefficient in the 1 /R expansion

The solution Q(α,β) ν=1 corresponds to a Weyl scalar Ψ1 that falls off as (1 − λ)4 log(1 − λ), so that the corresponding particular solution only peels if its coefficient Pk,1ℓm (4.13a) decays at least as fast as 1 / log(1 − λ). The conditions in category 1 and 2 are condition...
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The Jacobi equation then has the following 2 linearly independent solutions: P (α,−β) β+ν (x) (1 + x)β , (A17) and P (−α,β) α+ν (x) (1 − x)α

Case 2: negative integer ν In this case ν <0 is a negative integer and ν >−α − β − 1, so ν cannot be transformed to a positive number using the symmetries of the Jacobi equation. The Jacobi equation then has the following 2 linearly independent solutions: P (α,−β) β+ν (x) (1 +...
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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

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Henneaux and C

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Thus, they only contribute to 1/r2 and higher

We only have to consider the leading order in the 1 /R expansion because when the order in 1 /R increases, the solutions (2.6a) also increase the powers of 1 − λ2. Thus, they only contribute to 1/r2 and higher. We will later see similar results for the linear spin-s fields, and for gravity, where only a finite number of terms in the expansion will contribute

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[3] [2]

This is not a feature of the standard wave equation but rather a consequence of the initial data (2.2)

Notice that the solution (2.6) either peels or violates peeling at both I + and I − simultaneously. This is not a feature of the standard wave equation but rather a consequence of the initial data (2.2). This can be seen by considering spherically symmetric solutions to the standard wave equation (2.3), which are given by ϕ = r−1f (u) + r−1g(v) in double ...

work page

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Similar to the scalar field case, there exist solutions that peel at either I + or I −. For example, the solutions (see Appendix B for a derivation) ϕk,ℓm = r−1−s−ℓ(r2∂v)ℓ+s−kVℓm(v) + r−1−s−ℓ(r2∂u)ℓ−s+kUℓm(u) , (3.26) where Vℓm(v) and Uℓm(u) are arbitrary, may fail to peel. Here r = (v − u)/2. In the simple case of k = 0, ℓ = s, Vℓm = 0, and Uℓm = |u|nδℓ,...

work page

[5] [4]

In this case not all ϕk peel, but the situation is improved compared to (3.25a)

Instead of imposing the PT -parity condition at all orders n ≤ s, we can impose it for n ≤ w < sfor some w. In this case not all ϕk peel, but the situation is improved compared to (3.25a). In particular, we get that along I − we have ϕk = O 1 rk+1 for k ≤ s + w , (3.29a) ϕk = O log r rk+1 for k > s+ w . (3.29b)

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[6] [5]

From this we can compute, for example, PT [Ψ1](λ, R, θ, ϕ) = PT [Cabcdlamblcnd](λ, R, θ, ϕ) = Cabcd(−na)(− ¯mb)(−nc)(−ld)(−λ, R, π− θ, π+ ϕ) = Ψ3(−λ, R, π− θ, π+ ϕ)

Under a PT transformation the tetrad vectors transform as PT ∗[la] = PT ∗[∂a v ] = −∂a u = −na , (3.30a) PT ∗[ma] = PT ∗ h 1√ 2r ∂a θ + i sin θ ∂a ϕ i = 1√ 2r −∂a θ + i sin(π − θ) ∂a ϕ = − ¯ma , (3.30b) and since PT = PT −1, PT ∗[na] = −la and PT ∗[ ¯ma] = −ma. From this we can compute, for example, PT [Ψ1](λ, R, θ, ϕ) = PT [Cabcdlamblcnd](λ, R, θ, ϕ) = C...

work page

[7] [6]

The P (α,β) ν≥0 , Q(α,β) ν≥2 , (1 − λ)−αP (−α,β) ν+α<α and (1 + λ)−βP (α,−β) ν+β<β solutions to the Jacobi equation all peel for k ̸= 0. The conditions (4.14a), (4.14b), and (4.14d) state that if their coefficients are finite, then the corresponding components of the particular solution have the same (peeling) fall-off rate

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[8] [7]

The solution (1 + λ)−βP (α,−β) ν+β<β corresponds to a Weyl scalar having the fall-off rate (1−λ)n+3, so that the integral Jk,nℓm (4.13d) must blow up no faster than (1−λ)2−k−n

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