Black hole formation by a scalar field

D. E. Afanasev; M. O. Katanaev

arxiv: 2606.10792 · v1 · pith:DZNVK4Q3new · submitted 2026-06-09 · 🌀 gr-qc

Black hole formation by a scalar field

D. E. Afanasev , M. O. Katanaev This is my paper

Pith reviewed 2026-06-27 12:25 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black hole formationscalar fieldLiouville solutiongeneral relativityevent horizongeometric defectmass function

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The pith

If the scalar field potential is exponential and unbounded from below, the Liouville solution describes the formation of a spherically symmetric black hole whose horizon expands at light speed from a point.

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

The paper examines the Liouville solution in general relativity coupled to a scalar field. This solution cannot have its time dependence removed and remains invariant under global Lorentz transformations. When the potential is exponential and unbounded from below, the solution corresponds to black hole formation in which the horizon starts with zero radius at a finite time and expands outward at the speed of light. Comparison with the Schwarzschild metric shows that the enclosed mass grows proportionally to the square of time. A distant observer detects a geometric defect at the moment the horizon appears, analogous to the defect from a monopole or spherical dislocation in spacetime.

Core claim

The Liouville solution describes the formation of a spherically symmetric black hole if the scalar field potential is exponential and unbounded from below. The event horizon appears as a sphere of infinitesimal radius at a finite moment of time and then expands at the speed of light to infinity. A distant observer measures a geometric defect at the appearance point, analogous to that of a monopole or spherical dislocation. The mass function obtained by comparison with the Schwarzschild solution is proportional to the square of time.

What carries the argument

The Liouville solution in general relativity with a scalar field, which is invariant under global Lorentz transformations and has time dependence that cannot be removed by coordinate choice.

If this is right

The event horizon forms at a finite time with initially zero radius.
The horizon expands at the velocity of light.
A geometric defect is measurable by a distant observer at the formation point.
The mass of the resulting black hole is proportional to the square of time.

Where Pith is reading between the lines

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

The construction supplies an explicit analytic example of black hole formation driven by a scalar field.
Unbounded potentials of this type permit horizon formation in finite coordinate time rather than only asymptotically.

Load-bearing premise

The scalar field potential must be chosen to be exponential and unbounded from below.

What would settle it

A direct computation of the metric components showing that the surface identified as the horizon does not expand at the speed of light or that the asymptotic mass does not scale with time squared would falsify the central claim.

Figures

Figures reproduced from arXiv: 2606.10792 by D. E. Afanasev, M. O. Katanaev.

read the original abstract

The Liouville solution in General Relativity with a scalar field is discussed. This solution is invariant with respect to global Lorentz transformations, and dependence on time cannot be removed. If the scalar field potential is exponential and unbounded from below, the Liouville solution describes the formation of spherically symmetric black hole. The event horizon is a sphere, which appears with infinitesimal radius at a finite moment of time and afterwards expands with the velocity of light to infinity. A distant observer can measure the geometric defect at the point where the horizon appears. It is similar to the defect produced by the monopole or spherical dislocation of space-time. Comparison with the Schwarzschild solution yields the mass function which is proportional to the time squared.

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 applies the Liouville solution to scalar-field GR and claims it produces black-hole formation, but the stated global Lorentz invariance clashes with the spherical symmetry and finite-time nucleation at a preferred center.

read the letter

The main takeaway is that this paper takes the known Liouville solution in Einstein-scalar gravity and argues that an exponential potential unbounded from below turns it into a description of spherically symmetric black hole formation. The horizon nucleates with zero radius at finite time, expands at light speed, and the mass extracted by matching to Schwarzschild grows as time squared.

It does something useful by giving an explicit analytic example and working out the mass function from the comparison. That part is straightforward and could save time for anyone hunting for closed-form solutions in this class of models.

The soft spot is the symmetry tension. The abstract states the solution is invariant under global Lorentz transformations yet also spherically symmetric around a center, with the horizon appearing at a definite location and moment. Global Lorentz invariance leaves no room for a preferred origin or absolute time, so the two properties look incompatible on the face of it. The black-hole reading depends on that localized nucleation event, which makes the inconsistency load-bearing. The abstract offers no derivation or reconciliation, so it is not possible to tell whether the full equations actually support both claims at once.

This is for people who work on exact solutions in general relativity with scalar fields. A reader already familiar with Liouville-type metrics might find the mass function worth noting, but the central interpretation needs verification before it can be taken as settled.

I would send it to peer review. The claim is specific enough that referees can check the field equations and see whether the symmetries really coexist or whether the interpretation needs to be adjusted.

Referee Report

1 major / 0 minor

Summary. The manuscript discusses the Liouville solution in general relativity with a scalar field. It claims that for an exponential scalar potential unbounded from below, this solution—which is invariant under global Lorentz transformations with irreducible time dependence—describes the formation of a spherically symmetric black hole. The event horizon appears as a sphere of infinitesimal radius at a finite time and subsequently expands at light speed; comparison with the Schwarzschild metric yields a mass function proportional to the square of time, and distant observers can detect a geometric defect analogous to a monopole.

Significance. If the central claim holds, the work would supply an exact dynamical solution for scalar-driven black-hole formation with explicit Lorentz invariance and a calculable mass function. The geometric defect and light-speed horizon expansion would be distinctive features.

major comments (1)

[Abstract] Abstract: the solution is stated to be 'invariant with respect to global Lorentz transformations' while also being spherically symmetric with an event horizon that 'appears with infinitesimal radius at a finite moment of time' at a specific location before expanding. Global Lorentz invariance precludes any preferred origin or absolute time, yet the described nucleation event and spherical symmetry around a center require precisely such a distinguished point and time. This tension is load-bearing for the black-hole interpretation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting a potential tension in our description of the Liouville solution. We address the comment below and will make revisions to improve clarity.

read point-by-point responses

Referee: [Abstract] Abstract: the solution is stated to be 'invariant with respect to global Lorentz transformations' while also being spherically symmetric with an event horizon that 'appears with infinitesimal radius at a finite moment of time' at a specific location before expanding. Global Lorentz invariance precludes any preferred origin or absolute time, yet the described nucleation event and spherical symmetry around a center require precisely such a distinguished point and time. This tension is load-bearing for the black-hole interpretation.

Authors: The Einstein-scalar equations and the exponential potential are invariant under global Lorentz transformations. The Liouville solution we analyze is a particular solution to these equations that cannot have its time dependence removed by coordinate redefinition and is written in coordinates centered on the nucleation event. Equivalent solutions exist in any boosted frame, each with its own nucleation point; the invariance of the theory therefore allows the family of solutions, while any single member selects a preferred origin. This is analogous to a specific boosted Schwarzschild solution. We agree the abstract wording is imprecise on this distinction and will revise it (and the introduction) to state that the solution is covariant under Lorentz transformations while the chosen realization breaks the symmetry by fixing the center. The black-hole interpretation remains intact in the coordinates used. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation self-contained from field equations

full rationale

The paper presents the Liouville solution (invariant under global Lorentz transformations) with an exponential scalar potential unbounded from below, derives the spherically symmetric black-hole interpretation including the expanding horizon and mass function proportional to time squared via direct comparison to the Schwarzschild metric, and states these as consequences of the field equations. No self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations appear; the central claims rest on the stated potential choice and explicit solution properties without reducing to their own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of the Liouville solution to Einstein's equations with a scalar field and the specific choice of an exponential potential unbounded from below.

axioms (2)

standard math Einstein's field equations coupled to a scalar field
The Liouville solution is stated to be a solution to these equations.
domain assumption The scalar potential is exponential and unbounded from below
This form is required for the solution to describe black hole formation as stated in the abstract.

pith-pipeline@v0.9.1-grok · 5644 in / 1411 out tokens · 37341 ms · 2026-06-27T12:25:48.875463+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

18 extracted references · 2 canonical work pages

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[2]

Christodoulou

D. Christodoulou. The formation of black holes and singu larities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. , 44 (1991) 3, 339-373

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[3]

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

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I. P . C. Heard, D. Wands. Cosmology with positive and nega tive exponential potentials. Class. Quant. Grav., 19 (2002), 5435-5448

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Chakrabarti

S. Chakrabarti. Scalar ﬁeld collapce with an exponentia l potential. Gen. Rel. Grav. , 49:24(2):1–12, 2017. 6

2017
[6]

Mosani, Koushiki, P

K. Mosani, Koushiki, P . S. Joshi, J. V . Trivedi, T. Bhanja. Gravitational collapse of scalar and vector ﬁelds. Phys. Rev. D, 108 (2023) 4, 044049

2023
[7]

Corona, R

D. Corona, R. Giamb´ o. Global Models of Collapsing Scala r Field: Endstate. Symmetry, 16 (2024) 5, 583

2024
[8]

J. L. Lehners. Ekpyrotic and Cyclic Cosmology. Phys. Rept., 465 (2008), 223-263

2008
[9]

Giamb´ o

R. Giamb´ o. Gravitational Collapse of a Spherical Scalar Field arXiv:2307.01796 [gr-qc], 2023

work page arXiv 2023
[10]

D. E. Afanasev, M. O. Katanaev. Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field. Phys. Lett. B, 864 (2025), 139439

2025
[11]

D. E. Afanasev, M. O. Katanaev. Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field. JCAP, 08 (2025), 045. arXiv:2410.03799v2 [gr- qc], 2024

work page arXiv 2025
[12]

E. Ling. The Big Bang is a Coordinate Singularity for k = −1 Inﬂationary FLRW Space- times. F ound. Phys., 50 (2020) 5, 385-428

2020
[13]

Barriola, A

M. Barriola, A. Vilenkin. Gravitational Field of a Glob al Monopole. Phys. Rev. Lett., 63 (1989), 341

1989
[14]

Liouville

L. Liouville. M´ emoire sur l’int´ egration des ´ equations diﬀ´ erentielles du mouvement d’un nombre quelconque de points mat´ eriels.J. Math. Pures Appl., 14:257–299, 1849
[15]

M. O. Katanaev, I. V . V olovich. Theory of defects in solids and three-dimensional gravity. Ann. Phys., 216(1):1–28, 1992

1992
[16]

M. O. Katanaev. Geometric Theory of Defects. Physics – Uspekhi 48 (2005) 675–701

2005
[17]

Peinlev´ e

P . Peinlev´ e. La m´ ecanique classique el la theorie de la relativit´ e.C. R. Acad. Sci. (Paris) , 173:677–680, 1921

1921
[18]

Gullstrand

A. Gullstrand. Allegemeine l¨ osung des statischen eink¨ orper-problems in der Einsteinshen gravitations theorie. Arkiv. Mat. Astron. Fys., 16(8):1–15, 1922. 7

1922

[1] [1]

J. R. Oppenheimer and H. Snyder. On continued gravitatio nsl contraction. Phys. Rev., 56:455–459, 1939

1939

[2] [2]

Christodoulou

D. Christodoulou. The formation of black holes and singu larities in spherically symmetric gravitational collapse. Commun. Pure Appl. Math. , 44 (1991) 3, 339-373

1991

[3] [3]

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

1993

[4] [4]

I. P . C. Heard, D. Wands. Cosmology with positive and nega tive exponential potentials. Class. Quant. Grav., 19 (2002), 5435-5448

2002

[5] [5]

Chakrabarti

S. Chakrabarti. Scalar ﬁeld collapce with an exponentia l potential. Gen. Rel. Grav. , 49:24(2):1–12, 2017. 6

2017

[6] [6]

Mosani, Koushiki, P

K. Mosani, Koushiki, P . S. Joshi, J. V . Trivedi, T. Bhanja. Gravitational collapse of scalar and vector ﬁelds. Phys. Rev. D, 108 (2023) 4, 044049

2023

[7] [7]

Corona, R

D. Corona, R. Giamb´ o. Global Models of Collapsing Scala r Field: Endstate. Symmetry, 16 (2024) 5, 583

2024

[8] [8]

J. L. Lehners. Ekpyrotic and Cyclic Cosmology. Phys. Rept., 465 (2008), 223-263

2008

[9] [9]

Giamb´ o

R. Giamb´ o. Gravitational Collapse of a Spherical Scalar Field arXiv:2307.01796 [gr-qc], 2023

work page arXiv 2023

[10] [10]

D. E. Afanasev, M. O. Katanaev. Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field. Phys. Lett. B, 864 (2025), 139439

2025

[11] [11]

D. E. Afanasev, M. O. Katanaev. Geodesics and Global Properties of the Liouville Solution in General Relativity with a Scalar Field. JCAP, 08 (2025), 045. arXiv:2410.03799v2 [gr- qc], 2024

work page arXiv 2025

[12] [12]

E. Ling. The Big Bang is a Coordinate Singularity for k = −1 Inﬂationary FLRW Space- times. F ound. Phys., 50 (2020) 5, 385-428

2020

[13] [13]

Barriola, A

M. Barriola, A. Vilenkin. Gravitational Field of a Glob al Monopole. Phys. Rev. Lett., 63 (1989), 341

1989

[14] [14]

Liouville

L. Liouville. M´ emoire sur l’int´ egration des ´ equations diﬀ´ erentielles du mouvement d’un nombre quelconque de points mat´ eriels.J. Math. Pures Appl., 14:257–299, 1849

[15] [15]

M. O. Katanaev, I. V . V olovich. Theory of defects in solids and three-dimensional gravity. Ann. Phys., 216(1):1–28, 1992

1992

[16] [16]

M. O. Katanaev. Geometric Theory of Defects. Physics – Uspekhi 48 (2005) 675–701

2005

[17] [17]

Peinlev´ e

P . Peinlev´ e. La m´ ecanique classique el la theorie de la relativit´ e.C. R. Acad. Sci. (Paris) , 173:677–680, 1921

1921

[18] [18]

Gullstrand

A. Gullstrand. Allegemeine l¨ osung des statischen eink¨ orper-problems in der Einsteinshen gravitations theorie. Arkiv. Mat. Astron. Fys., 16(8):1–15, 1922. 7

1922