An extreme critical spacetime: echoing and black-hole perturbations
Open accessAbstract
A homothetic, static, spherically symmetric solution to the massless Einstein-Klein-Gordon equations is described. There is a curvature singularity which is central, null, bifurcate, massless and marginally trapped. The spacetime is therefore extreme in the sense of lying at the threshold between black holes and naked singularities, just avoiding both. A linear perturbation analysis reveals two types of dominant mode. One breaks the continuous self-similarity by periodic terms reminiscent of discrete self-similarity, with an echoing period within a few per cent of the value observed numerically in near-critical gravitational collapse. The other dominant mode explicitly produces a black hole, white hole, eternally naked singularity or regular dispersal, the latter indicating that the background is critical. The black hole is not static but has constant area, the corresponding mass being linear in the perturbation amplitudes, explicitly determining a unit critical exponent. It is argued that a central null singularity may be a feature of critical gravitational collapse.
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The spectrum of ω_A modes forms two lines in the complex plane with a fixed real part, with dominant ω_A = 1 ± √3 i, implying an echoing period Δ = 2π/√3 ≈ 3.62, much closer to Choptuik's numerically observed value Δ ≈ 3.44.
“The spectrum of ω A modes form two lines in the complex plane that has a fixed real part, with the dominant ω A = 1 ± √ 3i, implying an echoing period ∆ = 2π/ √ 3 ≈ 3.62, which is much closer to Choptuik's numerically observed value ∆ ≈ 3.44.”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [260]
Section 4 (eqs. 31–32) gives the dominant mode as k = 1 ± √3i and its echoing period as 2π/√3 ≈ 3.62; it also says this is within about 5% of Choptuik’s 3.44. The figure 2 caption says the ω_A modes form two lines in the complex plane of k, but identifies ω_A as zero—not 1 ± √3i—for that dominant mode. Page numbers are not available in the supplied copy. (LLM, conf=0.88)
Experimental — accuracy not yet measuredThe ω_A modes break CSS through periodic oscillations, reminiscent of DSS behavior.
“These ω A modes break CSS through periodic oscillations, reminiscent of DSS behavior.”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [260]
Spherically symmetric linear perturbations reveal two families of dominant modes, labeled ω_A and ω_B.
“Spherically symmetric linear perturbations were studied in [260], revealing two families of dominant modes, labeled ω A and ω B .”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [260]
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References 14 · Cited by 14
Research works printed but not linked: 1
- Penrose R 1977 Space-time singularities Proc. 1st Marcel Grossmann Meeting on General Relativity ed R Ruffini (Amsterdam: North-Holland)
- Authors
- Hayward, S. A.
- Affiliations
- Pennsylvania State University (US)
- Corresponding author
- Hayward, Sean A
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 2000-09-19
- Volume
- 17
- Issue
- 19
- Pages
- 4021-4030
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2950968458
- Cited by
- 14
- References
- 14
- Citations by year
- 2026: 1 · 2021: 2 · 2015: 1 · 2012: 1
- FWCI
- 1.34
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- National Science Foundation (9800973)
- 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 › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Keywords
- black hole perturbations, Einstein-Klein-Gordon equations, null singularity, marginally trapped surfaces, naked singularities, white holes, discrete self-similarity, critical exponents, spherically symmetric spacetimes
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