The DOI matches its Crossref record; 49 of the 49 references it cites were found in a registry or an index; none of the 8 sources its claims cite is checked for support yet.
Linear waves on constant radius limits of cosmological black hole spacetimes
Open accessAbstract
In this paper we consider the Klein-Gordon equation on spherically symmetric background spacetimes with a constant area radius. The spacetimes under consideration are Nariai and Plebański-Hacyan, and can be considered constant radius limits of Reissner-Nordström-de Sitter spacetimes. We prove boundedness in the case of a non-negative Klein Gordon mass and decay unless the mass is zero. In the latter case we prove decay of solutions that are supported on all harmonic modes with angular momentum l greater or equal to 1. We show that the l=0 modes of solutions to the massless Klein-Gordon equation do not decay. They are subject to conservation laws along degenerate Killing horizons. We apply the estimates in Nariai to give decay of solutions to the massive Klein-Gordon equation on an n-dimensional de Sitter background, using only the vector field method and with no restrictions on the positive Klein-Gordon mass.
Overview
CoChat Summary
TL;DR
The paper studies the Klein-Gordon equation on constant-radius cosmological black-hole limits, Nariai and Plebański-Hacyan, proving boundedness and decay of linear waves, with non-decay for the massless spherically symmetric mode. It also applies the Nariai estimates to obtain decay on \(n\)-dimensional de Sitter for \(\mu=0\).
Key findings
- In the static region of Nariai, non-degenerate energy is bounded for all \(\mu\in\mathbb R\); energy decays faster than any polynomial \(\tau^{-k}\), and single modes \(l\ge 1\) decay exponentially with rate \(c(l)=C l(l+1)\), while \(l=0\) has \(c(0)=C\) (Theorem 4.2).
- Globally in Nariai, energy is bounded; for \(\mu=0\), \(\psi_{l\ge1}\) decays exponentially, while \(\psi_0\) does not decay and tends to a nonzero constant for generic initial data (Theorems 4.3–4.5).
- In Plebański-Hacyan, energy is bounded for all \(\mu\); integrated local energy decay holds with loss of an angular derivative; pointwise decay \(\|\psi\|_{L^\infty(\Sigma_\tau)}\le C\tau^{-1/2}\) holds for \(\psi_{l\ge1}\) when \(\mu=0\), while \(\psi_0\) has non-decay on \(N^+_A,N^+_B\) for generic data (Theorems 4.7–4.10).
- On \(n\)-dimensional de Sitter with \(\mu=0\), exponential pointwise decay is proved using \(s>(n-1)/2\) angular derivatives (Theorem 4.6).
Methods
- Cauchy problem for \((\Box_g-\mu^2)\psi=0\) on fixed Nariai, Plebański-Hacyan, and de Sitter backgrounds; spherical harmonic decomposition reduces the equation to a \(1+1\)-dimensional Klein-Gordon equation with \(m_l^2=r_0^{-2}l(l+1)\).
- Vector field method: stress-energy tensor, energy currents \(J_V\), red-shift vector fields \(N,\bar N\), Morawetz vector field \(V\), global red-shift \(\tilde N\), and boost \(Y\); Stokes’ theorem, integrated local energy decay, pigeonhole/Grönwall arguments, and Sobolev/Poincaré estimates.
- No decay is imposed on the initial data; analysis is carried out in the static region, global coordinates, and a hyperboloidal foliation.
Limitations
- The paper states that Bertotti-Robinson spacetimes are excluded because they are not globally hyperbolic, so the Cauchy problem is not well-posed without additional boundary conditions; rotating Nariai and superradiance are left as open problems.
- The paper notes that the Plebański-Hacyan pointwise decay argument is not very robust because it relies on the boost vector field \(Y\); trapping causes loss of an angular derivative in integrated local energy decay.
- Evident from the text: decay statements in the massless case often require restriction to \(l\ge1\), since \(l=0\) non-decay is a genuine obstruction; also, the de Sitter theorem as stated is for \(\mu=0\), despite the abstract’s wording about the massive case.
Model-written summary. Check it against the paper before relying on it.
Integrity signals
Registry
Cites
Statistics
Verification
Claims the paper makes
0 Valid · 5 Not valid · 10 Not checked · 108 Own results and interpretations
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Hence, each sphere foliating PH is a photon sphere.”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 2 = γ2
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.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Moreover, by d'Alembert's formula there is no pointwise decay of ψ 0 for generic initial data.”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 1.3.4. Decay in n-dimensional de Sitter space.
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.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“By homogeneity of the spacetimes, each null hypersurface in a constant radius spacetime is a degenerate horizon with respect to a suitable Killing vector field.”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 2.2.
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.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“Theorem 3.1.For a fixed µ ∈ R and Ψ ∈ H 2 loc (Σ), Ψ ∈ H 1 loc (Σ), there exists a unique ψ : M → R, with ψ| S ∈ H 2 loc (S), n S ψ| S ∈ H 1 loc (S), for all spacelike submanifolds S ⊂ M with unit future normal n S in M, satisfying”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 3.1. The Cauchy problem.
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.
- Based: passes — The paper gives a reason why it holds.
A result the paper states — it cites no source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“The local method applied to fixed modes ψ = ψ 0 provides in particular a proof, using only vector field methods, of pointwise decay of solutions to the µ = 0 Klein-Gordon equation (1.6) in the static region of dS 2 .”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 1.3.4. Decay in n-dimensional de Sitter space.
plainstates a cause- Asserted: passes — The authors put it forward as true.
- Checkable: passes — It can be checked against evidence.
- Load-bearing: unclear — It is unclear whether the paper’s conclusions depend on it.
- Based: passes — The paper gives a reason why it holds.
83 more claims
Create a free account to see all 123 claims and every source verdict.
- Probably not a substantial claim. 10
- It may state two separate claims as one. 23
- The quote only partly states this claim. 5
Self-check
Percentages, sums and reported differences re-computed from the paper’s own numbers.
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
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 3.2. Spherical harmonic mode decomposition.
“where Y m,l ∈ L 2 (S 2 ) are the spherical harmonics, which form an orthonormal basis under the standard inner product on L 2 (S 2 ).”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 3.2. Spherical harmonic mode decomposition.
“In particular, since Y 0,0 ≡ 1, ψ 0 is a spherically symmetric harmonic mode.”
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
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 4.1.1. Results in Nariai.
“J T [ψ] • n Στ ∼ (∂ t ψ) 2 + (1 -Kx 2 )(∂ x ψ) 2 + | / ∇ψ| 2 + µ 2 ψ 2 ,”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 5.1. Energy boundedness.
“J T [ψ] • n Σ ∼ (∂ t ψ) 2 + (∂ x ψ) 2 + (1 -Kx 2 ) | / ∇ψ| 2 + µ 2 ψ 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
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 2.5.2. Plebański-Hacyan.
“We consider the lightcone C defined by C := {t ≤ |x|}.”
§ LINEAR WAVES ON CONSTANT RADIUS LIMITS OF COSMOLOGICAL BLACK HOLE SPACETIMES > 2.5.2. Plebański-Hacyan.
“C can be foliated by hyperboloids H s = {(t, x) ∈ Q : t 2 -x 2 = s 2 , t ≥ 0}, where s > 0.”
Citations
Cites
The Large Scale Structure of Space-Time
Hawking, S. W., & Ellis, G. F. R. (1973) · doi:10.1017/cbo9780511524646 · Cited by 10,624
Stability of a Schwarzschild Singularity
Regge, T., & Wheeler, J. A. (1957) · Physical Review · doi:10.1103/physrev.108.1063 · Cited by 2,725
Exact Space-Times in Einstein's General Relativity
Griffiths, J. B., & Podolský, J. (2009) · doi:10.1017/cbo9780511635397 · Cited by 872
Stability of the Schwarzschild Metric
Vishveshwara, C. V. (1970) · Physical Review D · doi:10.1103/physrevd.1.2870 · Cited by 583
Friedrich, H. (1986) · Communications in Mathematical Physics · doi:10.1007/bf01205488 · Cited by 475
Uniform Electromagnetic Field in the Theory of General Relativity
Bertotti, B. (1959) · Physical Review · doi:10.1103/physrev.116.1331 · Cited by 411
Semiclassical perdurance of de Sitter space
Ginsparg, P., & Perry, M. J. (1983) · Nuclear Physics B · doi:10.1016/0550-3213(83)90636-3 · Cited by 301
Mode stability of the Kerr black hole
Whiting, B. F. (1989) · Journal of Mathematical Physics · doi:10.1063/1.528308 · Cited by 291
Lectures on black holes and linear wavesPreprint
Dafermos, M., & Rodnianski, I. (2008) · arXiv (Cornell University) · doi:10.48550/arxiv.0811.0354 · Cited by 263
Classification of Near-Horizon Geometries of Extremal Black Holes
Kunduri, H. K., & Lucietti, J. (2013) · Living Reviews in Relativity · doi:10.12942/lrr-2013-8 · Cited by 262
Research works printed but not linked: 8
- M. Dafermos and A. D. Rendall, Strong cosmic censorship for surface-symmetric cosmological spacetimes with collisionless matter, arXiv:0701034.
- M. Dafermos and I. Rodnianski, A Proof of Price's law for the collapse of a selfgravitating scalar field, Invent.Math. 162 (2005), 381-457.
- Quasi-Normal Modes and Exponential Energy Decay for the Kerr-de Sitter Black Hole, Com- munications in Mathematical Physics 306 (2011), 119-163.
- L.P. Eisenhart, Riemannian Geometry, Princeton University Press, Princeton, 1949.
- Decay properties of Klein-Gordon fields on Kerr-AdS spacetimes, Commun.Pure Appl.Math. 66 (2013), 1751-1802.
- H. Nariai, On some static solutions of Einstein's gravitational field equations in a spherically symmetric case, Sci. Rep. Tohoku Univ. Eighth Ser. 34 (1950), 160.
- On a new cosmological solution of Einstein's field equations of gravitation, Sci. Rep. Tohoku Univ. Eighth Ser. 35 (1951), 46.
- I. Robinson, A solution of the Maxwell-Einstein equations, Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys 7 (1959), 351.
Cited by
Explicit formulas and decay rates for the solution of the wave equation in cosmological spacetimes
Natário, J., & Rossetti, F. (2023) · Journal of Mathematical Physics · doi:10.1063/5.0135092 · Cited by 12
Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes
Costa, J. L., Natário, J., & Oliveira, P. (2019) · Annales Henri Poincaré · doi:10.1007/s00023-019-00825-z · Cited by 10
Decay of Solutions to the Klein–Gordon Equation on Some Expanding Cosmological Spacetimes
Natário, J., & Sasane, A. J. (2022) · Annales Henri Poincaré · doi:10.1007/s00023-022-01172-2 · Cited by 7
Metadata
- Authors
- Gajic, D.
- Affiliations
- University of Cambridge (GB)
- Corresponding author
- Gajic, Dejan
- ORCID
- Gajic, Dejan
- Venue
- Advances in Theoretical and Mathematical Physics
- Publisher
- International Press of Boston
- Published
- 2018
- Volume
- 22
- Issue
- 4
- Pages
- 919-1005
- ISSN
- 1095-0753 (ISSN-L)
- 1095-0761
- OpenAlex
- W1593412936
- Cited by
- 3
- References
- 10
- Citations by year
- 2023: 1 · 2022: 1 · 2019: 1
- FWCI
- 0.12
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- Advances in Theoretical and Mathematical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Mathematical Physics Problems
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Klein-Gordon equation, linear waves, Nariai spacetime, Reissner-Nordström-de Sitter spacetime, spherically symmetric spacetimes, decay of solutions, boundedness, conservation laws, vector field method, de Sitter space, angular momentum modes
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-11-12, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-11-12 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
What counts as a claim ›How we measure confidence ›How we check this ›
Are you an author, or do you see a problem with this page? Request a review
How to cite
Cite CoChat for the checks on this page, and the paper for its own findings.
- This page
- CoChat. (2026). Citation check: Linear waves on constant radius limits of cosmological black hole spacetimes. https://app.cochat.ai/papers/10.4310/atmp.2018.v22.n4.a2
- The paper
- Gajic, D. (2018). Linear waves on constant radius limits of cosmological black hole spacetimes. Advances in Theoretical and Mathematical Physics. https://doi.org/10.4310/atmp.2018.v22.n4.a2
Data sources
Bibliographic data: Semantic Scholar, OpenAlex, Crossref.
Contains information from the Semantic Scholar Academic Graph Datasets, made available under the ODC Attribution License (ODC-BY 1.0).
Semantic Scholar’s data is described in: Kinney, R., et al. (2023). The Semantic Scholar Open Data Platform. arXiv:2301.10140.
