Critical Behavior of Black Hole Formation in a Scalar Wave Collapse
Open accessAbstract
We present a self-similar model of spherically symmetric collapse of a massless scalar field with a parameter p. The black hole formation is explicitly shown to occur only in the strong-field implosin of p > 1. The field evolution in the critical limit p →1 is compared with numerical results found by Choptuik.
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A 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 [282]
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 [282]
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Self-similar scalar field collapse: Naked singularities and critical behavior
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Global structure of Choptuik’s critical solution in scalar field collapse
Martín-García, J. M., & Gundlach, C. (2003) · Physical Review D · doi:10.1103/physrevd.68.024011 · Cited by 46 · see citing works
Collapse of a scalar field in 2 + 1 gravity
Hirschmann, E. W., Wang, A., & Wu, Y. (2004) · Classical and Quantum Gravity · doi:10.1088/0264-9381/21/7/006 · Cited by 46 · see citing works
Frolov, A. V. (1997) · Physical Review D · doi:10.1103/physrevd.56.6433 · Cited by 37 · see citing works
Critical collapse in (2+1)-dimensional AdS spacetime: quasi-CSS solutions and linear perturbations
Clément, G., & Fabbri, A. (2002) · Nuclear Physics B · doi:10.1016/s0550-3213(02)00183-9 · Cited by 30 · see citing works
Critical phenomena of collapsing massless scalar wave packets
Wang, A., & de Oliveira, H. P. (1997) · Physical Review D · doi:10.1103/physrevd.56.753 · Cited by 28 · see citing works
Critical collapse beyond spherical symmetry: General perturbations of the Roberts solution
Frolov, A. V. (1999) · Physical Review D · doi:10.1103/physrevd.59.104011 · Cited by 26 · see citing works
Continuous self-similarity breaking in critical collapse
Frolov, A. V. (2000) · Physical Review D · doi:10.1103/physrevd.61.084006 · Cited by 26 · see citing works
Analytical treatment of critical collapse in (2+1)-dimensional AdS spacetime: a toy model
Clément, G., & Fabbri, A. (2001) · Classical and Quantum Gravity · doi:10.1088/0264-9381/18/17/318 · Cited by 19 · see citing works
Predictability and semiclassical approximation at the onset of black hole formation
Bose, S., Parker, L., & Peleg, Y. (1996) · Physical Review D · doi:10.1103/physrevd.54.7490 · Cited by 17 · see citing works
Critical scalar field collapse in <b> AdS <sub>3</sub> </b> : an analytical approach
Baier, R., Stricker, S. A., & Taanila, O. (2014) · Classical and Quantum Gravity · doi:10.1088/0264-9381/31/2/025007 · Cited by 16 · see citing works
An extreme critical spacetime: echoing and black-hole perturbations
Hayward, S. A. (2000) · Classical and Quantum Gravity · doi:10.1088/0264-9381/17/19/304 · Cited by 14 · see citing works
Kinematics of discretely self-similar spherically symmetric spacetimes
Gundlach, C., & Martín-García, J. M. (2003) · Physical Review D · doi:10.1103/physrevd.68.064019 · Cited by 8 · see citing works
Comment on `An extreme critical spacetime: echoing and black-hole perturbations'
Clément, G., & Hayward, S. A. (2001) · Classical and Quantum Gravity · doi:10.1088/0264-9381/18/21/401 · Cited by 8 · see citing works
Unveiling horizons in quantum critical collapse
Tomašević, M., & Wu, C.-H. (2026) · Journal of High Energy Physics · doi:10.1007/jhep04(2026)151 · Cited by 5 · see citing works
- Authors
- Oshiro, Y., Nakamura, K., & Tomimatsu, A.
- Affiliations
- Nagoya University (JP)
- ORCID
- Nakamura, K.
- Venue
- Progress of Theoretical Physics
- Publisher
- Oxford University Press (OUP)
- Published
- 1994-06-01
- Volume
- 91
- Issue
- 6
- Pages
- 1265-1270
- ISSN
- 0033-068X (ISSN-L)
- 1347-4081
- OpenAlex
- W2028663900
- Cited by
- 82
- References
- 0
- Citations by year
- 2026: 3 · 2025: 1 · 2022: 1 · 2021: 3 · 2020: 1 · 2019: 1 · 2016: 1 · 2015: 2 · 2013: 2 · 2012: 1
- FWCI
- 6.74
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Progress of Theoretical Physics (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- CERN Document Server (European Organization for Nuclear Research) (submittedVersion)
- arXiv (Cornell University)
- 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 › Pulsars and Gravitational Waves Research
- Keywords
- black hole formation, massless scalar field, self-similar solutions, spherical collapse, critical phenomena, implosion, strong-field limit, critical exponents
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