The DOI matches its Crossref record; 12 of the 12 references it cites were found in a registry or an index.
Choptuik’s Critical Spacetime Exists
Open accessAbstract
About 25 years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, discretely self-similar solution at the threshold of black hole formation. We prove rigorously the existence of a real analytic solution, which we interpret as the solution observed by Choptuik. Using a Fourier-Chebyshev expansion, we find that an important linear differential operator has compact inverse. This is reminiscent of an elliptic operator and explains the real analyticity. Our construction covers an open neighborhood of the past light cone of the singularity. The proof is computer-assisted. Starting from an explicit approximate solution, we show that nearby there is a true solution. We include the complete C source code and a high precision data file with about 80 significant decimal digits (Online Resource), with rigorous error bounds. All computer calculations use integer arithmetic only. We do not study perturbations.
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“However, despite significant efforts, an explicit analytic form of the true critical solution remains unknown, except attempts for analytic approximations [281] and existence proofs [22].”
From Unveiling horizons in quantum critical collapse (2026) · cites this paper as [22]
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Cites
Universality and scaling in gravitational collapse of a massless scalar field
Choptuik, M. W. (1993) · Physical Review Letters · doi:10.1103/physrevlett.70.9 · Cited by 1,439
Local Contractions and a Theorem of Poincare
Sternberg, S. (1957) · American Journal of Mathematics · doi:10.2307/2372437 · Cited by 421
A computer-assisted proof of the Feigenbaum conjectures
Lanford, O. E. (1982) · Bulletin of the American Mathematical Society · doi:10.1090/s0273-0979-1982-15008-x · Cited by 353
The problem of a self-gravitating scalar field
Christodoulou, D. (1986) · Communications in Mathematical Physics · doi:10.1007/bf01205930 · Cited by 321
Examples of Naked Singularity Formation in the Gravitational Collapse of a Scalar Field
Christodoulou, D. (1994) · The Annals of Mathematics · doi:10.2307/2118619 · Cited by 304
Critical Phenomena in Gravitational Collapse
Gundlach, C., & Martín-García, J. M. (2007) · Living Reviews in Relativity · doi:10.12942/lrr-2007-5 · Cited by 297
The formation of black holes and singularities in spherically symmetric gravitational collapse
Christodoulou, D. (1991) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.3160440305 · Cited by 238
Bounded variation solutions of the spherically symmetric einstein‐scalar field equations
Christodoulou, D. (1993) · Communications on Pure and Applied Mathematics · doi:10.1002/cpa.3160460803 · Cited by 182
Understanding critical collapse of a scalar field
Gundlach, C. (1997) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.55.695 · Cited by 147
Critical Behavior in Vacuum Gravitational Collapse in 4 + 1 Dimensions
Bizoń, P., Chmaj, T., & Schmidt, B. G. (2005) · Physical Review Letters · doi:10.1103/physrevlett.95.071102 · Cited by 63
Research works printed but not linked: 5
- Choptuik M.W., Hirschmann E.W., Liebling S.L., Pretorius F.: Critical collapse of the massless scalar field in axisymmetry. Phys. Rev. D. 68(044007), 1–9 (2003)
- Martín-García J.M., Gundlach C.: Global structure of Choptuik’s critical solution in scalar field collapse. Phys. Rev. D 68(024011), 1–25 (2003)
- Martín-García J.M., Gundlach C.: All nonspherical perturbations of the Choptuik spacetime decay. Phys. Rev. D 59(064031), 1–19 (1999)
- Reiterer, M.: Finite codimension stability of some time-periodic hyperbolic equations (via compact resolvents). arXiv:1510.05310 · arXiv
- Reiterer, M., Trubowitz, E.: The graded Lie algebra of general relativity. arXiv:1412.5561 · arXiv
Cited by
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
Metadata
- Authors
- Reiterer, M., & Trubowitz, E.
- Affiliations
- ETH Zurich (CH)
- Corresponding author
- Reiterer, Michael
- ORCID
- Reiterer, Michael
- Venue
- Communications in Mathematical Physics
- Published
- 2019-04-26
- Volume
- 368
- Issue
- 1
- Pages
- 143-186
- ISSN
- 0010-3616 (ISSN-L)
- 1432-0916
- OpenAlex
- W1810141524
- Cited by
- 23
- References
- 12
- Citations by year
- 2026: 5 · 2025: 1 · 2024: 4 · 2023: 2 · 2022: 4 · 2021: 2 · 2020: 3 · 2019: 2
- FWCI
- 1.45
- Type
- journal article
- Language
- en
- Open access
- green
- hdl.handle.net
- Licence
- other-oa
- www.springer.com/tdm (tdm)
- Locations
- Communications in Mathematical Physics (publishedVersion)
- Repository for Publications and Research Data (ETH Zurich) (publishedVersion, other-oa)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- ETH Zürich Research Collection
- Topics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Mathematics › Mathematical Physics › Advanced Mathematical Physics Problems
- Keywords
- gravitational collapse, massless scalar field, spherical symmetry, black hole formation, self-similarity, real analyticity, elliptic operators, computer-assisted proof, rigorous error bounds, integer arithmetic, Einstein field equations
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