arxiv: 2509.03584 · v2 · submitted 2025-09-03 · 🌀 gr-qc · astro-ph.CO· hep-th

Unveiling horizons in quantum critical collapse

Marija Toma\v{s}evi\'c , Chih-Hung Wu This is my paper

Pith reviewed 2026-05-18 18:59 UTC · model grok-4.3

classification 🌀 gr-qc astro-ph.COhep-th

keywords critical gravitational collapsequantum correctionsweak cosmic censorshipsemiclassical gravitymass gapType I and Type II collapseBoulware statenaked singularity

0 comments p. Extension

The pith

Quantum corrections in critical collapse generate a finite mass gap that turns potential naked singularities into black holes.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper investigates what happens when quantum effects from collapsing matter are included in the most delicate regime of gravitational collapse, where classical solutions can produce naked singularities. Using an anomaly-based one-loop calculation on exactly solvable critical solutions, the authors solve the semiclassical Einstein equations in both 2+1 and 3+1 dimensions. Regularity picks out a specific quantum state whose backreaction grows and creates a mass gap. This gap converts the classical Type II behavior into a quantum Type I outcome in which black holes form above a minimum mass, thereby upholding weak cosmic censorship. A reader would care because the result offers a concrete way quantum mechanics can resolve the censorship issue dynamically before full quantum gravity is required.

Core claim

In Einstein gravity minimally coupled to a free massless scalar field, the one-loop semiclassical equations are solved analytically for critical solutions using the anomaly method. Regularity conditions select a Boulware-like quantum state that encodes vacuum polarization from the collapsing matter. Horizon-tracing analysis that includes both classical and quantum modes shows that the quantum corrections produce a growing mode, resulting in a finite mass gap. This establishes a phase transition from classical Type II to quantum-modified Type I critical behavior and supplies a quantum-level enforcement of the weak cosmic censorship conjecture.

What carries the argument

Horizon-tracing analysis that tracks both classical and quantum modes to extract the finite mass gap induced by semiclassical backreaction.

If this is right

Classical Type II critical solutions that permit naked singularities are replaced by quantum Type I behavior with a nonzero minimum black-hole mass.
Vacuum polarization encoded in the selected quantum state supplies the backreaction that enforces weak cosmic censorship.
The same mass-gap mechanism appears in both the 2+1- and 3+1-dimensional critical solutions examined.
Regularity at the horizon uniquely determines the Boulware-like state for these time-dependent geometries.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

If the mass gap survives in more realistic matter models, similar quantum enforcement of censorship may operate inside evaporating black holes.
Numerical evolution of the semiclassical equations with the identified growing mode could be used to measure the size of the gap as a function of the critical parameter.
The approach might be extended to non-minimally coupled or massive fields to test whether the Type I transition remains generic.

Load-bearing premise

The one-loop semiclassical approximation using the anomaly-based method remains valid and sufficient for non-conformal matter fields in explicitly time-dependent critical spacetimes near arbitrarily high curvatures.

What would settle it

An explicit computation of the semiclassical stress-energy tensor along a critical solution that yields no growing quantum mode or no finite mass gap would falsify the claimed phase transition.

read the original abstract

Critical gravitational collapse offers a unique window into regimes of arbitrarily high curvature, culminating in a naked singularity arising from smooth initial data -- thus providing a dynamical counterexample to weak cosmic censorship. Near the critical regime, quantum effects from the collapsing matter are expected to intervene before full quantum gravity resolves the singularity. Despite its fundamental significance, a self-consistent treatment has so far remained elusive. In this work, we perform a one-loop semiclassical analysis using the robust anomaly-based method in the canonical setup of Einstein gravity minimally coupled to a free, massless scalar field. Focusing on explicitly solvable critical solutions in both 2+1 and 3+1 dimensions, we analytically solve the semiclassical Einstein equations and provide definitive answers to several long-standing questions. We find that regularity uniquely selects a Boulware-like quantum state, encoding genuine vacuum polarization effects from the collapsing matter. Remarkably, the resulting quantum corrections manifest as a growing mode. Horizon-tracing analyses, incorporating both classical and quantum modes, reveal the emergence of a finite mass gap, signaling a phase transition from classical Type II to quantum-modified Type I behavior, thereby providing a quantum enforcement of the weak cosmic censorship. The most nontrivial aspect of our analysis involves dealing with non-conformal matter fields in explicitly time-dependent critical spacetimes. Along the way, we uncover intriguing and previously underexplored features of quantum field theory in curved spacetime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes one-loop backreaction on solvable critical collapse solutions and finds a quantum mass gap that flips Type II to Type I behavior, but the result rests on the anomaly method holding for non-conformal fields at high curvature.

read the letter

The main thing to know is that this work takes known critical solutions in 2+1 and 3+1 dimensions, adds semiclassical corrections from the trace anomaly for a massless scalar, selects a Boulware-like state for regularity, and reports a growing mode that produces a finite mass gap and a shift to Type I scaling, which they interpret as quantum enforcement of weak cosmic censorship.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a one-loop semiclassical analysis of critical gravitational collapse using the anomaly-based method for Einstein gravity minimally coupled to a free massless scalar field. Focusing on explicitly solvable critical solutions in 2+1 and 3+1 dimensions, the authors analytically solve the semiclassical Einstein equations. They claim that regularity selects a Boulware-like quantum state, producing a growing mode whose inclusion in horizon-tracing analyses yields a finite mass gap. This signals a phase transition from classical Type II to quantum-modified Type I behavior, interpreted as providing a quantum enforcement of the weak cosmic censorship conjecture. The work emphasizes handling non-conformal fields in time-dependent critical spacetimes.

Significance. If the results hold, the paper would offer an analytical window into how quantum vacuum polarization can intervene in critical collapse to prevent naked singularities, addressing a long-standing question at the interface of general relativity and quantum field theory in curved spacetime. The analytical solvability for critical solutions and the focus on regularity-selected states are notable strengths that could stimulate further work on semiclassical backreaction in dynamical high-curvature regimes.

major comments (2)

[Abstract and semiclassical equations section] Abstract and the section deriving the semiclassical equations: the assertion that the anomaly-based method yields definitive results for non-conformal (minimal, ξ=0) coupling in explicitly time-dependent critical spacetimes is load-bearing for the mass-gap claim. The full renormalized <Tμν> contains additional state-dependent and curvature-squared contributions whose regularization is nontrivial in self-similar or near-critical backgrounds; without explicit derivation or bounds showing these terms remain subdominant before curvatures become Planckian, the growing mode and resulting finite mass gap cannot be considered robust.
[Horizon-tracing analyses] Horizon-tracing analyses section: the reported emergence of a finite mass gap and the Type II to quantum Type I transition rests on the one-loop approximation remaining valid arbitrarily close to the would-be singularity. The manuscript must supply either an error estimate or a demonstration that higher-order or non-perturbative corrections do not erase the mass gap in the regime where classical curvature diverges; absent this, the conclusion of quantum enforcement of weak cosmic censorship is not yet supported.

minor comments (2)

[Quantum state selection] Clarify the precise boundary conditions or mode expansions used to define the Boulware-like state selected by regularity; this would aid reproducibility of the state choice.
[Abstract] The abstract mentions 'definitive answers to several long-standing questions'—a brief enumerated list of those questions and where they are resolved would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the robustness of our semiclassical treatment for non-conformal fields and the validity of the one-loop approximation. We address each major comment below and indicate the revisions we will incorporate.

read point-by-point responses

Referee: [Abstract and semiclassical equations section] Abstract and the section deriving the semiclassical equations: the assertion that the anomaly-based method yields definitive results for non-conformal (minimal, ξ=0) coupling in explicitly time-dependent critical spacetimes is load-bearing for the mass-gap claim. The full renormalized <Tμν> contains additional state-dependent and curvature-squared contributions whose regularization is nontrivial in self-similar or near-critical backgrounds; without explicit derivation or bounds showing these terms remain subdominant before curvatures become Planckian, the growing mode and resulting finite mass gap cannot be considered robust.

Authors: We agree that additional justification is required to establish the subdominance of state-dependent and curvature-squared terms for minimal coupling in time-dependent critical backgrounds. In the revised version, we will expand the semiclassical equations section with an explicit discussion of the regularization procedure for the full renormalized stress-energy tensor. Using the specific self-similar form of the critical solutions, we derive bounds showing that the extra contributions are suppressed relative to the anomaly terms in the regularity-selected Boulware-like state, remaining subdominant until curvatures approach the Planck scale. This strengthens the support for the growing mode and finite mass gap. revision: yes
Referee: [Horizon-tracing analyses] Horizon-tracing analyses section: the reported emergence of a finite mass gap and the Type II to quantum Type I transition rests on the one-loop approximation remaining valid arbitrarily close to the would-be singularity. The manuscript must supply either an error estimate or a demonstration that higher-order or non-perturbative corrections do not erase the mass gap in the regime where classical curvature diverges; absent this, the conclusion of quantum enforcement of weak cosmic censorship is not yet supported.

Authors: We acknowledge the need for a clearer statement on the regime of validity of the one-loop approximation. In the revision, we will add a dedicated paragraph in the horizon-tracing analyses section providing an order-of-magnitude error estimate: higher-order corrections are parametrically suppressed by powers of the Planck length over the local curvature radius and only become comparable when curvatures reach Planckian values, at which point the semiclassical framework itself ceases to apply. A complete non-perturbative demonstration lies outside the semiclassical approach and would require a full quantum gravity treatment, which is beyond the scope of this work. We will explicitly qualify our conclusions as holding within the one-loop semiclassical regime. revision: partial

Circularity Check

0 steps flagged

Analytical solution of semiclassical equations yields mass gap without reduction to inputs

full rationale

The paper derives the finite mass gap and Type II to quantum Type I transition by analytically solving the semiclassical Einstein equations with the anomaly-based stress tensor for a minimally coupled scalar. Regularity at the origin selects the Boulware-like state, which sources a growing mode; horizon-tracing then produces the mass gap as an output of the dynamics. This chain depends on solving the differential system with stated boundary conditions rather than any fitted parameter, self-referential definition, or load-bearing self-citation. The classical critical solutions serve as external input, and no uniqueness theorem or ansatz is smuggled in via prior work by the same authors. The derivation is therefore self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The analysis rests on the semiclassical Einstein equations with anomaly method for non-conformal fields and the assumption that regularity selects a unique quantum state; no free parameters or new entities are explicitly introduced in the abstract.

axioms (2)

domain assumption One-loop semiclassical approximation with anomaly method captures quantum effects before full quantum gravity resolves the singularity.
Invoked to perform the analysis in the near-critical regime.
domain assumption Regularity conditions uniquely determine the Boulware-like quantum state for the collapsing matter.
Used to fix the quantum state and derive corrections.

pith-pipeline@v0.9.0 · 5779 in / 1472 out tokens · 51965 ms · 2026-05-18T18:59:00.496203+00:00 · methodology

discussion (0)

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IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

one-loop semiclassical analysis using the robust anomaly-based method... ⟨Taa⟩=ℏ/24π(R−6(∇ϕ)²+6□ϕ)
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

finite mass gap, signaling a phase transition from classical Type II to quantum-modified Type I behavior

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Forward citations

Cited by 2 Pith papers

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Reference graph

Works this paper leans on

300 extracted references · 300 canonical work pages · cited by 1 Pith paper · 175 internal anchors

[1]

Penrose,Gravitational collapse: The role of general relativity,Riv

R. Penrose,Gravitational collapse: The role of general relativity,Riv. Nuovo Cim.1(1969) 252

work page 1969
[2]

S. W. Hawking,Nature of space and time,hep-th/9409195

work page internal anchor Pith review Pith/arXiv arXiv
[3]

K. S. Virbhadra, D. Narasimha and S. M. Chitre,Role of the scalar field in gravitational lensing,Astron. Astrophys.337(1998) 1 [astro-ph/9801174]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[4]

K. S. Virbhadra and G. F. R. Ellis,Gravitational lensing by naked singularities,Phys. Rev. D65(2002) 103004

work page 2002
[5]

K. S. Virbhadra and C. R. Keeton,Time delay and magnification centroid due to gravitational lensing by black holes and naked singularities,Phys. Rev. D77(2008) 124014 [0710.2333]

work page internal anchor Pith review Pith/arXiv arXiv 2008
[6]

K. S. Virbhadra,Distortions of images of Schwarzschild lensing,Phys. Rev. D106(2022) 064038 [2204.01879]

work page arXiv 2022
[7]

R. Giamb` o,Gravitational collapse of a spherical scalar field,New Frontiers in Gravitational Collapse and Spacetime Singularities, Springer Nature Singapore582(2024) 141–173 [2307.01796]

work page arXiv 2024
[8]

S. L. Liebling and C. Palenzuela,Dynamical boson stars,Living Rev. Rel.26(2023) 1 [1202.5809]

work page internal anchor Pith review Pith/arXiv arXiv 2023
[9]

Christodoulou,The Problem of a Self-gravitating Scalar Field,Commun

D. Christodoulou,The Problem of a Self-gravitating Scalar Field,Commun. Math. Phys. 105(1986) 337

work page 1986
[10]

Christodoulou,The formation of black holes and singularities in spherically symmetric gravitational collapse,Commun

D. Christodoulou,The formation of black holes and singularities in spherically symmetric gravitational collapse,Commun. Pure Appl. Math.44(1991) 339

work page 1991
[11]

Christodoulou,Bounded variation solutions of the spherically symmetric einstein-scalar field equations,Commun

D. Christodoulou,Bounded variation solutions of the spherically symmetric einstein-scalar field equations,Commun. Pure Appl. Math.46(1993) 1131

work page 1993
[12]

Christodoulou,Examples of naked singularity formation in the gravitational collapse of a scalar field,Annals Math.140(1994) 607

D. Christodoulou,Examples of naked singularity formation in the gravitational collapse of a scalar field,Annals Math.140(1994) 607

work page 1994
[13]

Christodoulou,On the global initial value problem and the issue of singularities, Classical and Quantum Gravity16(1999) A23

D. Christodoulou,On the global initial value problem and the issue of singularities, Classical and Quantum Gravity16(1999) A23. – 113 –

work page 1999
[14]

The Formation of Black Holes in General Relativity

D. Christodoulou,The Formation of Black Holes in General Relativity, in12th Marcel Grossmann Meeting on General Relativity, pp. 24–34, 5, 2008,0805.3880, DOI

work page internal anchor Pith review Pith/arXiv arXiv 2008
[15]

Black hole formation from a complete regular past

M. Dafermos,Black hole formation from a complete regular past,Commun. Math. Phys. 289(2009) 579 [gr-qc/0310040]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[16]

M. W. Choptuik,Universality and scaling in gravitational collapse of a massless scalar field,Phys. Rev. Lett.70(1993) 9

work page 1993
[17]

The Choptuik spacetime as an eigenvalue problem

C. Gundlach,The Choptuik space-time as an eigenvalue problem,Phys. Rev. Lett.75 (1995) 3214 [gr-qc/9507054]

work page internal anchor Pith review Pith/arXiv arXiv 1995
[18]

Understanding critical collapse of a scalar field

C. Gundlach,Understanding critical collapse of a scalar field,Phys. Rev. D55(1997) 695 [gr-qc/9604019]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[19]

Kinematics of discretely self-similar spherically symmetric spacetimes

C. Gundlach and J. M. Mart ´ ın-Garc ´ ıa,Kinematics of discretely self-similar spherically symmetric spacetimes,Physical Review D68(2003) [gr-qc/0306001]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[20]

J. M. Martin-Garcia and C. Gundlach,Global structure of Choptuik’s critical solution in scalar field collapse,Phys. Rev. D68(2003) 024011 [gr-qc/0304070]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[21]

A. V. Frolov and U.-L. Pen,The Naked singularity in the global structure of critical collapse space-times,Phys. Rev. D68(2003) 124024 [gr-qc/0307081]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[22]

Reiterer and E

M. Reiterer and E. Trubowitz,Choptuik’s critical spacetime exists,Commun. Math. Phys. 368(2019) 143 [1203.3766]

work page arXiv 2019
[23]

Critical phenomena in gravitational collapse (Physics Reports)

C. Gundlach,Critical phenomena in gravitational collapse,Phys. Rept.376(2003) 339 [gr-qc/0210101]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[24]

C. R. Evans and J. S. Coleman,Observation of critical phenomena and self-similarity in the gravitational collapse of radiation fluid,Phys. Rev. Lett.72(1994) 1782 [gr-qc/9402041]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[25]

Non-Universality of Critical Behaviour in Spherically Symmetric Gravitational Collapse

D. Maison,Nonuniversality of critical behavior in spherically symmetric gravitational collapse,Phys. Lett. B366(1996) 82 [gr-qc/9504008]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[26]

Charge scaling and universality in critical collapse

C. Gundlach and J. M. Martin-Garcia,Charge scaling and universality in critical collapse, Phys. Rev. D54(1996) 7353 [gr-qc/9606072]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[27]

Fine-Structure of Choptuik's Mass-Scaling Relation

S. Hod and T. Piran,Fine structure of Choptuik’s mass scaling relation,Phys. Rev. D55 (1997) 440 [gr-qc/9606087]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[28]

J. M. Martin-Garcia and C. Gundlach,All nonspherical perturbations of the Choptuik space-time decay,Phys. Rev. D59(1999) 064031 [gr-qc/9809059]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[29]

Bartnik and J

R. Bartnik and J. McKinnon,Particlelike solutions of the einstein-yang-mills equations, Phys. Rev. Lett.61(1988) 141

work page 1988
[30]

Bizon,Colored black holes,Phys

P. Bizon,Colored black holes,Phys. Rev. Lett.64(1990) 2844

work page 1990
[31]

Seidel and W.-M

E. Seidel and W.-M. Suen,Oscillating soliton stars,Phys. Rev. Lett.66(1991) 1659

work page 1991
[32]

R. S. Hamade, J. H. Horne and J. M. Stewart,Continuous selfsimilarity and S duality, Class. Quant. Grav.13(1996) 2241 [gr-qc/9511024]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[33]

D. M. Eardley, E. W. Hirschmann and J. H. Horne,S duality at the black hole threshold in gravitational collapse,Phys. Rev. D52(1995) R5397 [gr-qc/9505041]

work page internal anchor Pith review Pith/arXiv arXiv 1995
[34]

E. W. Hirschmann and D. M. Eardley,Criticality and bifurcation in the gravitational collapse of a selfcoupled scalar field,Phys. Rev. D56(1997) 4696 [gr-qc/9511052]. – 114 –

work page internal anchor Pith review Pith/arXiv arXiv 1997
[35]

T. Hara, T. Koike and S. Adachi,Renormalization group and critical behavior in gravitational collapse,gr-qc/9607010

work page internal anchor Pith review Pith/arXiv arXiv
[36]

Critical Behaviour and Universality in Gravitational Collapse of a Charged Scalar Field

S. Hod and T. Piran,Critical behavior and universality in gravitational collapse of a charged scalar field,Phys. Rev. D55(1997) 3485 [gr-qc/9606093]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[37]

M. H. P. M. van Putten,Approximate black holes for numerical relativity,Phys. Rev. D54 (1996) R5931 [gr-qc/9607074]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[38]

S. L. Liebling and M. W. Choptuik,Black hole criticality in the Brans-Dicke model,Phys. Rev. Lett.77(1996) 1424 [gr-qc/9606057]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[39]

Echoing and scaling in Einstein-Yang-Mills critical collapse

C. Gundlach,Echoing and scaling in Einstein Yang-Mills critical collapse,Phys. Rev. D55 (1997) 6002 [gr-qc/9610069]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[40]

M. W. Choptuik, T. Chmaj and P. Bizon,Critical behavior in gravitational collapse of a Yang-Mills field,Phys. Rev. Lett.77(1996) 424 [gr-qc/9603051]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[41]

M. W. Choptuik, E. W. Hirschmann and S. L. Liebling,Instability of an ’approximate black hole’,Phys. Rev. D55(1997) 6014 [gr-qc/9701011]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[42]

P. R. Brady, C. M. Chambers and S. M. C. V. Goncalves,Phases of massive scalar field collapse,Phys. Rev. D56(1997) R6057 [gr-qc/9709014]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[43]

G. Rein, A. D. Rendall and J. Schaeffer,Critical collapse of collisionless matter: A Numerical investigation,Phys. Rev. D58(1998) 044007 [gr-qc/9804040]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[44]

D. W. Neilsen and M. W. Choptuik,Critical phenomena in perfect fluids,Class. Quant. Grav.17(2000) 761 [gr-qc/9812053]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[45]

S. L. Liebling,Multiply unstable black hole critical solutions,Phys. Rev. D58(1998) 084015 [gr-qc/9805043]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[46]

Critical Collapse of Skyrmions

P. Bizon and T. Chmaj,Critical collapse of Skyrmions,Phys. Rev. D58(1998) 041501 [gr-qc/9801012]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[47]

On Equivalence of Critical Collapse of Non-Abelian Fields

P. Bizon, T. Chmaj and Z. Tabor,On equivalence of critical collapse of nonAbelian fields, Phys. Rev. D59(1999) 104003 [gr-qc/9901039]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[48]

S. L. Liebling,Critical phenomena inside global monopoles,Phys. Rev. D60(1999) 061502 [gr-qc/9904077]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[49]

M. W. Choptuik, E. W. Hirschmann and R. L. Marsa,New critical behavior in Einstein-Yang-Mills collapse,Phys. Rev. D60(1999) 124011 [gr-qc/9903081]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[50]

Choptuik scaling in six dimensions

D. Garfinkle, C. Cutler and G. C. Duncan,Choptuik scaling in six-dimensions,Phys. Rev. D60(1999) 104007 [gr-qc/9908044]

work page internal anchor Pith review Pith/arXiv arXiv 1999
[51]

Scalar field collapse in three-dimensional AdS spacetime

V. Husain and M. Olivier,Scalar field collapse in three-dimensional AdS space-time,Class. Quant. Grav.18(2001) L1 [gr-qc/0008060]

work page internal anchor Pith review Pith/arXiv arXiv 2001
[52]

Gravitational collapse in 2+1 dimensional AdS spacetime

F. Pretorius and M. W. Choptuik,Gravitational collapse in (2+1)-dimensional AdS space-time,Phys. Rev. D62(2000) 124012 [gr-qc/0007008]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[53]

S. H. Hawley and M. W. Choptuik,Boson stars driven to the brink of black hole formation, Phys. Rev. D62(2000) 104024 [gr-qc/0007039]

work page internal anchor Pith review Pith/arXiv arXiv 2000
[54]

Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry

I. Olabarrieta and M. W. Choptuik,Critical phenomena at the threshold of black hole formation for collisionless matter in spherical symmetry,Phys. Rev. D65(2002) 024007 [gr-qc/0107076]. – 115 –

work page internal anchor Pith review Pith/arXiv arXiv 2002
[55]

J. M. Martin-Garcia and C. Gundlach,Selfsimilar spherically symmetric solutions of the massless Einstein-Vlasov system,Phys. Rev. D65(2002) 084026 [gr-qc/0112009]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[56]

Spherically symmetric scalar field collapse in any dimension

M. Birukou, V. Husain, G. Kunstatter, E. Vaz and M. Olivier,Scalar field collapse in any dimension,Phys. Rev. D65(2002) 104036 [gr-qc/0201026]

work page internal anchor Pith review Pith/arXiv arXiv 2002
[57]

R. S. Millward and E. W. Hirschmann,Critical behavior of gravitating sphalerons,Phys. Rev. D68(2003) 024017 [gr-qc/0212015]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[58]

Critical collapse of a massive vector field

D. Garfinkle, R. B. Mann and C. Vuille,Critical collapse of a massive vector field,Phys. Rev. D68(2003) 064015 [gr-qc/0305014]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[59]

J. F. Ventrella and M. W. Choptuik,Critical phenomena in the Einstein massless Dirac system,Phys. Rev. D68(2003) 044020 [gr-qc/0304007]

work page internal anchor Pith review Pith/arXiv arXiv 2003
[60]

Formation and decay of Einstein-Yang-Mills black holes

O. Rinne,Formation and decay of Einstein-Yang-Mills black holes,Phys. Rev. D90(2014) 124084 [1409.6173]

work page internal anchor Pith review Pith/arXiv arXiv 2014
[61]

J. V. Rocha and M. Tomaˇ sevi´ c,Self-similarity in Einstein-Maxwell-dilaton theories and critical collapse,Phys. Rev. D98(2018) 104063 [1810.04907]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[62]

Universality and Scaling at the Onset of Quantum Black Hole Formation

A. Strominger and L. Thorlacius,Universality and scaling at the onset of quantum black hole formation,Phys. Rev. Lett.72(1994) 1584 [hep-th/9312017]

work page internal anchor Pith review Pith/arXiv arXiv 1994
[63]

Phase Transition in Spherically Symmetric Gravitational Collapse of a Massless Scalar Field

Y. Kiem,Phase transition in spherically symmetric gravitational collapse of a massless scalar field,hep-th/9407100

work page internal anchor Pith review Pith/arXiv arXiv
[64]

J. G. Zhou, H. J. W. Mueller-Kirsten and M.-Z. Yang,New look at the critical behavior near the threshold of black hole formation in the Russo-Susskind-Thorlacius model,Phys. Rev. D51(1995) R314

work page 1995
[65]

Choptuik Scaling and Quantum Effects in 2D Dilaton Gravity

Y. Peleg, S. Bose and L. Parker,Choptuik scaling and quantum effects in 2-d dilaton gravity,Phys. Rev. D55(1997) 4525 [gr-qc/9608040]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[66]

S. Bose, L. Parker and Y. Peleg,Predictability and semiclassical approximation at the onset of black hole formation,Phys. Rev. D54(1996) 7490 [hep-th/9606152]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[67]

Spherical Collapse of a Mass-Less Scalar Field With Semi-Classical Corrections

S. Ayal and T. Piran,Spherical collapse of a massless scalar field with semiclassical corrections,Phys. Rev. D56(1997) 4768 [gr-qc/9704027]

work page internal anchor Pith review Pith/arXiv arXiv 1997
[68]

Chiba and M

T. Chiba and M. Siino,Disappearance of black hole criticality in semiclassical general relativity,Modern Physics Letters A12(1997) 709

work page 1997
[69]

P. R. Brady and A. C. Ottewill,Quantum corrections to critical phenomena in gravitational collapse,Phys. Rev. D58(1998) 024006 [gr-qc/9804058]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[70]

Berczi, P

B. Berczi, P. M. Saffin and S.-Y. Zhou,Gravitational collapse with quantum fields,Phys. Rev. D104(2021) L041703 [2010.10142]

work page arXiv 2021
[71]

J. N. Guenther, C. Hoelbling and L. Varnhorst,Semiclassical gravitational collapse of a radially symmetric massless scalar quantum field,Phys. Rev. D105(2022) 105010 [2010.13215]

work page arXiv 2022
[72]

Berczi, P

B. Berczi, P. M. Saffin and S.-Y. Zhou,Gravitational collapse of quantum fields and Choptuik scaling,JHEP02(2022) 183 [2111.11400]

work page arXiv 2022
[73]

Hoelbling, J

C. Hoelbling, J. N. Guenther and L. Varnhorst,Real time dynamics of a semiclassical gravitational collapse of a scalar quantum field,PoSLA TTICE2021(2022) 156 [2111.15562]. – 116 –

work page arXiv 2022
[74]

Varnhorst, C

L. Varnhorst, C. Hoelbling and J. N. Gunether,Real time evolution of scalar fields in semiclassical gravity,PoSLA TTICE2022(2023) 391

work page 2023
[75]

Moitra,Self-similar gravitational dynamics, singularities and criticality in 2D,JHEP06 (2023) 194 [2211.01394]

U. Moitra,Self-similar gravitational dynamics, singularities and criticality in 2D,JHEP06 (2023) 194 [2211.01394]

work page arXiv 2023
[76]

Critical behaviour in quantum gravitational collapse

V. Husain,Critical behaviour in quantum gravitational collapse,0808.0949

work page internal anchor Pith review Pith/arXiv arXiv
[77]

Dynamical Singularity Resolution in Spherically Symmetric Black Hole Formation

J. Ziprick and G. Kunstatter,Dynamical Singularity Resolution in Spherically Symmetric Black Hole Formation,Phys. Rev. D80(2009) 024032 [0902.3224]

work page internal anchor Pith review Pith/arXiv arXiv 2009
[78]

Benitez, R

F. Benitez, R. Gambini, L. Lehner, S. Liebling and J. Pullin,Critical collapse of a scalar field in semiclassical loop quantum gravity,Phys. Rev. Lett.124(2020) 071301 [2002.04044]

work page arXiv 2020
[79]

Ben ´ ıtez, R

F. Ben ´ ıtez, R. Gambini, S. L. Liebling and J. Pullin,Criticality in the collapse of spherically symmetric massless scalar fields in semiclassical loop quantum gravity,Phys. Rev. D104(2021) 024008 [2106.00674]

work page arXiv 2021
[80]

S. M. Christensen and S. A. Fulling,Trace anomalies and the hawking effect,Phys. Rev. D 15(1977) 2088

work page 1977

Showing first 80 references.