Analytic example of critical behaviour in scalar field collapse
Open accessAbstract
An exact one parameter ( ) family of solutions representing scalar field collapse is shown to exhibit a type of critical behaviour which has been discussed by Choptuik. The three possible evolutions are outlined. For supercritical evolution (when black holes form) it is shown that a quantity related to the mass of the black hole exhibits a power law dependence on , for near critical evolution . The solution supports the conjecture that black hole formation initially occurs at infinitesimal mass.
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In the supercritical regime α > 0 of the Roberts solution, the black hole mass grows without bound as v → ∞, and the entire spacetime eventually gets trapped, failing to be asymptotically flat.
“In the supercritical regime α > 0, the black hole mass grows without bound as v → ∞, and the entire spacetime eventually gets trapped, failing to be asymptotically flat [126,184].”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [126]
The paper’s discussion of supercritical evolution (p. 1259) says the black hole has infinite mass and the entire spacetime ultimately becomes trapped. But its summary assigns black-hole formation to α < 0; for α > 0, the scalar field disperses and leaves flat space. The claim reverses the parameter sign. (LLM, conf=0.88)
Experimental — accuracy not yet measuredA class of CSS solutions for the Einstein-scalar system was first introduced by Roberts in the context of cosmic censorship, and later rediscovered by Brady and by Oshiro, Nakamura, and Tomimatsu in the study of critical collapse.
“A class of such CSS solutions for the Einstein-scalar system (2.2) was first introduced by Roberts in the context of cosmic censorship [122], and later rediscovered by Brady and by Oshiro, Nakamura, and Tomimatsu in the study of critical collapse [126,282].”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [126]
One cannot simply read off the power-law scaling from the total mass with marginally non-zero α (which would suggest δ = 0.5).
“As a result, one cannot simply read off the power-law scaling from the total mass with marginally non-zero α (which would suggest δ = 0.5) [126,282].”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [126]
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References 17 · Cited by 107
Research works printed but not linked: 3
- Choptuik M W 1989 Frontiers in Numerical Relativity ed C R Evans. L S Finn and D W Hobill (Cambridge: Goldwinh A and Piran T 1987 Phys. Rev. D 36 3575
- Penmse R 1979 An Einsrein Centenary Survey ed S W Hawking and W Israel (Cambridge: Cambridge 171
- Gundlach C. Price R and Pullin 1 1994 Phyr, Rev. D 49 890 Math. Phys. 109 591,613 Cambridge University Press) (Cambridge: Cambridge University Press) University Press)
- Authors
- Brady, P. R.
- Affiliations
- Newcastle University (GB)
- Corresponding author
- Brady, Patrick R
- ORCID
- Brady, Patrick R
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1994-05-01
- Volume
- 11
- Issue
- 5
- Pages
- 1255-1260
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2034934686
- Cited by
- 107
- References
- 17
- Citations by year
- 2026: 4 · 2025: 1 · 2024: 1 · 2023: 2 · 2022: 1 · 2021: 6 · 2020: 1 · 2019: 7 · 2018: 3 · 2017: 3 · 2016: 2 · 2015: 2 · 2013: 1 · 2012: 2
- FWCI
- 8.17
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- Funders
- University of Alberta
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Cosmology and Gravitation Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- scalar field collapse, critical behavior, exact solutions, black hole formation, Choptuik scaling, power-law dependence, black hole mass, gravitational collapse
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