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Kirchhoff integral approach to the calculation of Green’s functions beyond the normal neighborhood
Open accessAbstract
We propose a new method for investigating the global properties of the retarded Green's function ${G}_{R}({x}^{\ensuremath{'}},x)$ for fields propagating on an arbitrary globally hyperbolic spacetime. Our method combines the Hadamard form for ${G}_{R}$ (this form is only valid within a normal neighborhood of $x$) together with Kirchhoff's integral representation for the field in order to calculate ${G}_{R}$ outside the maximal normal neighborhood of $x$. As an example, we apply this method to the case of a scalar field on a black hole toy-model spacetime, the Pleba\ifmmode \acute{n}\else \'{n}\fi{}ski-Hacyan spacetime, ${\mathbb{M}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{S}}^{2}$. The method allows us to determine in an exact manner that the singularity structure of the ``direct'' term in the Hadamard form for ${G}_{R}({x}^{\ensuremath{'}},x)$ changes from a form $\ensuremath{\delta}(\ensuremath{\sigma})$ to ``$1/\ensuremath{\pi}\ensuremath{\sigma}$'' after the null geodesic joining $x$ and ${x}^{\ensuremath{'}}$ has crossed a caustic point, where $\ensuremath{\sigma}$ is the world function. Furthermore, there is a change of form from a Heaviside step function $H(\ensuremath{-}\ensuremath{\sigma})$ to a ``$\ensuremath{-}\mathrm{ln}|\ensuremath{\sigma}|/\ensuremath{\pi}$'' in the ``tail'' term, which has not been explicitly noted before in the literature. We complement the results from the Kirchhoff integral method with an analysis for large-$\ensuremath{\ell}$ of the Green's function modes. This analysis allows us to determine the singularity structure after null geodesics have crossed an arbitrary number of caustics, although it raises a causality issue which the Kirchhoff integral method resolves. Because of the similarity in the caustic structure of the spacetimes, we expect our main results for wave propagation to be valid also on Schwarzschild spacetime.
Overview
CoChat Summary
TL;DR
The paper introduces a Kirchhoff integral method to compute the retarded Green’s function beyond the normal neighborhood by iterating the Hadamard form. Applied to a scalar field on Plebański-Hacyan spacetime \(M_2 \times S^2\), it gives exact changes in the singularity structure after caustic crossing and is complemented by a large-\(\ell\) mode analysis.
Key findings
- For \(\pi < \eta < 2\pi\), after one caustic crossing, the “direct” term changes from \(\delta(\sigma)\) to \(P.V.(1/\pi\sigma)\), and the “tail” term acquires a singular term \(-\ln|\sigma|/\pi\): \(G_R = U(2\pi-\gamma)P.V.(1/\pi\sigma) - \nu_0(2\pi-\gamma)\ln|\sigma|/\pi + O(1)\).
- The large-\(\ell\) analysis reproduces the four-fold singularity structure for arbitrary caustic crossings: \(\delta(\sigma) \to P.V.(1/\pi\sigma) \to -\delta(\sigma) \to -P.V.(1/\pi\sigma) \to \delta(\sigma)\cdots\), and for the tail term \(\theta(-\sigma) \to -\ln|\sigma| \to -\theta(-\sigma) \to \ln|\sigma| \to \theta(-\sigma)\cdots\).
- At angular separation \(\gamma=\pi\), the four-fold structure degenerates to a two-fold structure: \(\delta \to -\delta \to \delta \cdots\).
- The Kirchhoff method resolves the causality issue raised by the naive \(P.V.(1/\pi\sigma)\) term; the large-\(\ell\) method does not resolve it. The authors expect the main wave-propagation results to also hold on Schwarzschild spacetime because of similar caustic structure.
Methods
- The Kirchhoff integral representation is applied to the retarded Green’s function itself, using the Hadamard form for both propagator and data, and iterated to reach points outside the maximal normal neighborhood.
- The method is applied to a scalar field on the black-hole toy-model Plebański-Hacyan spacetime \(M_2 \times S^2\), with detailed evaluation of the \(U\times U\), \(U\times V\), and \(V\times V\) contributions.
- A complementary large-\(\ell\) asymptotic analysis of the Green’s function modes is performed, using multipole decomposition and distributional identities; this yields global singularity structure but requires restricting to a particular value of \(m^2+\xi R\).
Limitations
- Stated: the Kirchhoff method is in principle valid for any globally hyperbolic spacetime, but in practice requires knowledge of the Hadamard form; applying it beyond the second caustic becomes considerably more involved.
- Stated: the large-\(\ell\) asymptotic analysis must restrict to a particular value of \(m^2+\xi R\), and it raises a causality issue that the Kirchhoff method resolves.
- Stated: the result \(G_R = U(2\pi-\gamma)P.V.(1/\pi\sigma) - \nu_0(2\pi-\gamma)\ln|\sigma|/\pi + O(1)\) diverges at \(\gamma=\pi\) and does not apply on a neighborhood of \(\gamma=\pi\); the large-\(\ell\) result applies for \(\gamma\) bounded away from \(0\) and \(\pi\). The authors also note they could not go further to find \(\nu_2\) from the recurrence relations.
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0 Valid · 0 Not valid · 11 Not checked · 90 Own results and interpretations
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In the paper
“This analysis allows us to determine the singularity structure after null geodesics have crossed an arbitrary number of caustics, although it raises a causality issue which the Kirchhoff integral method resolves.”
- Also stated in: § I. INTRODUCTION · plain
“the large-ℓ asymptotic analysis uncovers the singularity structure for an arbitrary number of caustic crossings”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > Abstract
plainstates a cause- Asserted: passes — The authors put it forward as true.
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- Also stated in: § I. INTRODUCTION · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The method allows us to determine in an exact manner that the singularity structure of the 'direct' term in the Hadamard form for GR(x , x) changes from a form δ(σ) to '1/πσ' after the null geodesic joining x and x has crossed a caustic point, where σ is the world function.”
- Also stated in: § I. INTRODUCTION · boosted
“That is, we show that, after the wavefront has crossed one caustic, the 'direct' term in the Green's function changes its singularity character from δ(σ) to P.V. (1/πσ), as expected”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > Abstract
boostedstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § I. INTRODUCTION · boosted
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Furthermore, there is a change of form from a θ(-σ) to a '-ln |σ|/π' in the 'tail' term, which has not been explicitly noted before in the literature.”
- Also stated in: § I. INTRODUCTION · plain
“and that the 'tail' term gives rise to a singular term ln |σ| in the same region.”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > Abstract
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
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- Also stated in: § I. INTRODUCTION · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“G R (x, x ) = U (2π -γ)P V 1 πσ 1 (x, x ) -ν 0 (2π -γ) ln |σ 1 (x, x )| π + O(1),(2)”
- Also stated in: § I. INTRODUCTION · plain
“σ 1 = - 1 2 η 2 + 1 2 (2π -γ) 2 ,(3)”
- Also stated in: § I. INTRODUCTION · plain
“and the O(1) term corresponds to a function that remains finite throughout the region π < η < 2π.”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > I. INTRODUCTION
plain- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: passes — The paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
- Also stated in: § I. INTRODUCTION · plain
A result the paper reports from its own data — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The large-ℓ analysis that we carry out here yields not only the fourfold structure for the 'direct' term (±δ(σ) → ∓P.V. (1/πσ)) but also that for the 'tail' term (±θ(-σ) → ∓ ln |σ| /π), in both cases for any number of caustic crossings.”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > I. INTRODUCTION
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Each contradiction between two passages listed here was confirmed by an independent check.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > 5. Summary: the U × V contribution.
“From (89), with γ = π (giving β = α -π and H(α -β) ≡ 1), we have”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > 5. Summary: the U × V contribution.
“γ 1 = α, γ 2 = β = π -α,”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > V. SINGULARITY STRUCTURE VIA MODE SUM APPROXIMATION
“D(s) = -2 ln |s| -2 ∞ k=0 ln 1 - s 2 4k 2 π 2 ,(148)”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > Additional information
“sin x = x ∞ k=1 1 - x 2 k 2 π 2 .”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > 5. Summary: the U × V contribution.
“1 -m 2 -4ξR + 1 (2π -θ) 2 - cot(2π -θ) 2π -θ ln |τ -2π + θ|”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > B. PH: M2 × S 2
“ν 0 (γ) = 1 8 U (γ) 1 -4m 2 -4ξR + 1 γ 2 - cot γ γ . (23”
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > 5. Summary: the U × V contribution.
“G R (x, x ) = U (2π -γ)P V 1 πσ(x, x ) -ν 0 (2π -γ) ln |σ(x, x )| π + O(1).(126)”
§ A Kirchhoff integral approach to the calculation of Green's functions beyond the normal neighbourhood > I. INTRODUCTION
“σ 1 = - 1 2 η 2 + 1 2 (2π -γ) 2 ,(3)”
Citations
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Cited by
Self-force and Green function in Schwarzschild spacetime via quasinormal modes and branch cut
Casals, M., Dolan, S. R., Ottewill, A. C., & Wardell, B. (2013) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.88.044022 · Cited by 67
Self-citationSelf-force via Green functions and worldline integration
Wardell, B., Galley, C. R., Zenginoğlu, A., Casals, M., Dolan, S. R., & Ottewill, A. C. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.89.084021 · Cited by 49
Self-citationCaustic echoes from a Schwarzschild black hole
Zenginoğlu, A., & Galley, C. R. (2012) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.86.064030 · Cited by 45
Communication through quantum fields near a black hole
Jonsson, R. H., Aruquipa, D. Q., Casals, M., Kempf, A., & Martín-Martínez, E. (2020) · Physical review. D/Physical review. D. · doi:10.1103/physrevd.101.125005 · Cited by 36
Self-citationScalar Green function of the Kerr spacetime
Yang, H., Zhang, F., Zimmerman, A., & Chen, Y. (2014) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.89.064014 · Cited by 29
Metadata
- Authors
- Casals, M., & Nolan, B. C.
- Affiliations
- University College Dublin (IE)
- Dublin City University (IE)
- Venue
- Physical Review D
- Publisher
- American Physical Society (APS)
- Published
- 2012-07-25
- Volume
- 86
- Issue
- 2
- ISSN
- 0556-2821 (ISSN-L)
- 1550-7998
- 1550-2368
- OpenAlex
- W2087330173
- Semantic Scholar
- b08284842df52bc4e6e3d5bb003be6736ce990e3
- Cited by
- 19
- References
- 22
- Citations by year
- 2026: 1 · 2025: 1 · 2024: 1 · 2023: 3 · 2020: 1 · 2019: 1 · 2018: 1 · 2017: 1 · 2016: 1 · 2015: 1 · 2014: 4 · 2013: 1 · 2012: 2
- FWCI
- 0.95
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- Irish Research Council for Science, Engineering and Technology
- Institut Périmètre de physique théorique
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum and Classical Electrodynamics
- Keywords
- retarded Green's function, globally hyperbolic spacetimes, caustics, null geodesics, world function, black hole spacetimes, scalar field, causality, wave propagation
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