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Dynamical axisymmetric compact objects in General Relativity

T0 review · 3 major / 3 minor · reviewed 2026-08-03 · deepseek-v4-flash

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Pith's one-line read This paper constructs the first exact axisymmetric, non-vacuum, asymptotically-FLRW black/white hole solution of the Einstein–scalar system, and locates its dynamical horizons with the mean curvature vector.

desk verdict New exact axisymmetric dynamical scalar-FLRW black hole solution, worth refereeing; the extended Fonarev proof has a fixable factor typo, and the horizon numerics need more detail. read the letter →

arxiv 2512.19542 v3 pith:WXQATX2H submitted 2025-12-22 gr-qc hep-th

Jibril Ben Achour , Adolfo Cisterna , Mokhtar Hassaine This is my paper · ORCID
classification gr-qchep-th
keywords primordialblackholesexactsolutionsofEinsteinequationsEinstein-scalarsystemaxisymmetricspacetimesdynamicalhorizonsmeancurvaturevectorKodamaZipoy-Voorheesmetric
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper sets out to establish the first exact, non-vacuum, axisymmetric solution of the Einstein–scalar system that is asymptotically FLRW and contains a dynamical black (or white) hole. To achieve it, the authors extend the Fonarev solution-generating map so it can act on static axisymmetric seeds obtained through the Buchdahl transformation, producing a time-dependent conformal factor and a shifted scalar profile that solve the field equations with a Liouville potential. Their main example is a dynamical Zipoy–Voorhees geometry, and they show that its trapping and anti-trapping horizons are located by the zero-norm surfaces of the mean curvature vector, the natural generalization of the Kodama vector beyond spherical symmetry. A sympathetic reader would care because exact analytic benchmarks like this are rare and are needed to test numerical collapse simulations, model primordial black holes, and develop tools for non-spherical dynamical horizons.

What carries the argument

The load-bearing object is the extended Fonarev theorem, which combines two transformations: the Buchdahl map turns a static axisymmetric vacuum seed into an Einstein-scalar seed by raising the ignorable-coordinate metric component to powers and taking the scalar proportional to its logarithm; the Fonarev step then multiplies the whole metric by e^{2μ(a)}, with μ(a)=ξ2 ln(Ca+B), and shifts the scalar by (ξ1/κ) μ(a), so the pair solves the field equations with a Liouville potential V=V0 e^{ξ3 φ} subject to the constraints (3.26). The second tool is the mean curvature vector, built from the traces of the extrinsic curvatures of the two normals to a closed 2-surface; its norm is proportional to

What would settle it

Substitute the metric (4.4)–(4.6) and the scalar (4.5) directly into the Einstein and scalar field equations with the Liouville potential, verifying every component symbolically under the constraints (3.26); any non-vanishing component away from the known singularities would falsify the solution. Separately, an independent numerical computation of θ+θ− from the null expansions (4.22)–(4.23) should reproduce the reported S-shaped horizon curves and the critical times (t1, r1) and (t2, r2).

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Extended reading notes

Core claim

The central claim is that the time-dependent Zipoy–Voorhees metric (4.4)–(4.6) with scalar (4.5) is an exact solution of the Einstein–scalar system with a Liouville potential under the constraints (3.26), and that it is the first exact non-vacuum asymptotically FLRW axisymmetric black/white hole solution of that system. The solution reduces to FLRW at large r, has a time-like singularity at r=2M (ring-like for δ>1) and a space-like singularity at t=0, and its apparent horizons are given by the zero-locus of the mean-curvature-vector norm (4.20), equivalently θ+θ−=0. In the contracting branch (C<0, t<0, ξ2>0) the geometry contains a future trapping/black-hole horizon; in the expanding branch

Load-bearing premise

The whole construction depends on the proof of the extended Fonarev theorem being exactly right: the step that fixes the scalar profile in terms of the metric deformation must be a genuine solution of the field equations, because if the constraints (3.26) are not satisfied by an independent substitution, the metric (4.4)–(4.6) does not solve Einstein's equations and the black-hole interpretation fails.

Editorial extensions

If this is right

  • The map generates an entire family of asymptotically FLRW axisymmetric scalar-sourced spacetimes: any static axisymmetric vacuum seed satisfying the Buchdahl conditions gives a dynamical solution, with the Zipoy–Voorhees geometry as one representative.
  • For the explicit solution, horizon trajectories are given analytically by 2ξ2/t = ±R(r,θ), and the distinction between black-hole and cosmological horizons is fixed by the sign of ξ2 together with the Lie derivative of the null expansions.
  • The black-hole branch lives in a contracting FLRW background; the time-reversed branch is a white hole in an expanding background, so the solution provides an exact analogue of black-hole formation in a contracting universe.
  • The mean curvature vector provides a foliation-independent diagnostic for (anti-)trapped regions that extends the Kodama construction and applies to non-spherical dynamical geometries generally.
  • The metric is an analytic benchmark that can be used to test numerical collapse codes and to study dynamical-horizon thermodynamics and semi-classical evaporation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the theorem is correct, the same two-step construction should work for other static axisymmetric vacuum seeds in the Weyl family, offering a way to build a catalog of dynamical compact objects with different multipole structures; this is not demonstrated in the paper.
  • The MCV zero-norm criterion suggests a practical numerical recipe — compute the norm from the two-surface normals and track its zero-set — which may be more robust than conventional foliation-by-foliation apparent-horizon searches in non-spherical collapse.
  • An independent symbolic substitution of (4.4)–(4.6) and (4.5) into the field equations would settle the construction, since the proof's integration step fixes the scalar profile through a proportionality condition that the paper does not exhibit explicitly.
  • A natural testable extension is to use the MCV's zero-locus as the basis for a quasi-local compaction function and to compute primordial-black-hole formation thresholds beyond spherical symmetry; the paper announces this as a companion project.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 3 minor

Summary. The paper develops a solution-generating technique for the self-interacting Einstein-scalar system, combining Buchdahl-type transformations with a generalized Fonarev map to produce non-stationary, axisymmetric solutions. The central result is the extended Fonarev theorem of §3.2.1, whose proof is presented in §3.2.2. The method is then applied to the static Zipoy–Voorhees seed, yielding the time-dependent metric (4.4)–(4.6) with the scalar (4.5), claimed to be asymptotically FLRW and to describe a dynamical axisymmetric black or white hole. The paper also reviews Anco's mean-curvature vector as a generalization of the Kodama vector and uses its zero-locus to identify trapping and anti-trapping horizons. The novelty rests on the exactness of the new solution and on the reliability of the horizon classification.

Significance. If the construction is valid, this is a useful and nontrivial step: it provides an explicit, non-spherically-symmetric, time-dependent solution of the Einstein-scalar system motivated by primordial black hole physics, and it demonstrates a practical method for locating dynamical horizons beyond spherical symmetry. Strengths of the manuscript include the fully explicit metric and scalar field, parameter constraints derived from the field equations rather than fitted, and the fact that the spherical δ=1 limit and the massless HMN limit reduce to previously known solutions. Credit is also explicitly given to Anco for the MCV construction. However, the significance is contingent on repairing the proof of the generating theorem and on documenting the numerical horizon classification; as it stands, the central exactness claim is not fully established.

major comments (3)
  1. [§3.2.2, Eqs. (3.45b)–(3.48)] The proof of the extended Fonarev theorem contains a factor-ξ0 error. For the Buchdahl seed φ=ξ0 ln(g^aa), one has ∂_kφ=(ξ0/β)(g^aa)^{-β}∂_k(g^aa)^β, so the constant P defined by ∂_kφ=P(g^aa)^{-β}∂_k(g^aa)^β equals ξ0/β. Equation (3.45b) therefore reduces to ˙μ=κξ1P ˙Ψ=κ(ξ0ξ1/β)˙Ψ, not κξ0ξ1P ˙Ψ. With the stated ξ1=β/ξ0, the correct conclusion is ˙μ=κ˙Ψ and Ψ=μ/κ. The printed condition ξ0ξ1P=1 is inconsistent with the given P: for the Buchdahl profile it gives ξ0ξ1P=ξ0, not 1. This step is load-bearing because the new solution (4.4)–(4.6) is generated through this theorem. The theorem may be salvageable by removing the superfluous ξ0 or redefining P, but as written the proof does not establish the exactness of the solution. Please correct the proof or provide an independent direct substitution of (4.4)–(4.6).
  2. [§4.2.2, Eq. (4.36) and following paragraph] The stated condition R<0 for past anti-trapping horizons is inconsistent with Eq. (4.36). For the C<0 branch with t<0 and ξ2<0, Eq. (4.36) reads 2ξ2/t=+R. Since the left-hand side is positive, one must have R>0, not R<0. The text's statement that 'such a horizon is defined only for R(r_h,θ_h)<0' corresponds instead to the θ+=0 condition with ξ2<0. This sign error affects the white-hole branch analysis and should be corrected.
  3. [§4.2.3, Fig. 1 and Eq. (4.42)] The identification of black-hole versus cosmological horizon segments relies on the numerical sign of L_{l_-}θ_+ evaluated along the horizon-locus curves, but the numerical procedure is not described. No algorithm, grid parameters, precision, or representative values of the critical points (t_1,r_1) and (t_2,r_2) are provided, and the figure alone is not sufficient to reproduce the computation. Since the physical interpretation — black hole vs contracting cosmological horizon, and horizon production/annihilation — depends on these numerical signs, please provide reproducible numerical data or analytic criteria for the sign of (4.42).
minor comments (3)
  1. [Eq. (3.52)] The expression for ξ1 appears to be misprinted: β/ξ0 = sqrt(2κβ^2/(1-β^2)), not sqrt(2κβ/(1-β^2)).
  2. [Throughout] The acronym for the Husain–Martinez–Nuñez solution is written both as HNM and HMN; please use one consistently.
  3. [§4.2, Eqs. (4.4)–(4.5)] The Buchdahl transform is applied with a=t, where the seed component is ¯g_tt=-f^δ. Raising a negative metric component to a real power β is branch-dependent, and the scalar profile φ=ξ0 ln(¯g_tt) formally involves the logarithm of a negative quantity. The paper silently writes f^{δβ} and δξ0 ln f. This is a standard convention in FJNW-type solutions, but it should be stated explicitly, e.g., by working with |¯g_aa|, to make the theorem mathematically precise for non-integer β.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity; the central derivation is self-contained, though the proof of the extended Fonarev theorem contains a coefficient inconsistency that is a correctness issue rather than a circularity.

full rationale

I found no circular step of the kinds defined in the instructions. The extended Fonarev theorem is proved in-paper from Buchdahl's classical theorem [113] and an explicit ansatz; the parameter constraints (3.26) are derived from the reduced Einstein equations, not fitted to a target prediction. The new metric (4.4)-(4.6) is then obtained by applying that theorem, and the MCV horizon analysis is explicitly credited to Anco [96] and checked against the null-expansion product, so it is not a renamed or self-imported result. The self-citations that appear, e.g. [114] for applying Buchdahl's method to axisymmetric seeds, are motivational rather than load-bearing: the needed fact is re-derived in §3.2.2 using [113]. There is, however, a notable algebraic flaw in the proof that should be flagged even though it is not circular. In §3.2.2 the paper states: '∂_kφ = P(¯g_aa)^{-β}∂_k(¯g_aa)^β ... equation (3.45b) transforms into ˙µ = κξ_0ξ_1P ˙Ψ, of which the solution, using ξ_0ξ_1P = 1, is Ψ(a) = µ(a)/κ.' For the Buchdahl profile φ = ξ_0 ln(¯g_aa), one has ∂_kφ = (ξ_0/β)(¯g_aa)^{-β}∂_k(¯g_aa)^β, so P = ξ_0/β. Combined with ξ_1 = β/ξ_0 from (3.26a), this gives ξ_0ξ_1P = ξ_0, not 1. The printed integration condition is therefore not satisfied by the seed used in the theorem. This is a mathematical inconsistency in the supporting derivation, not a reduction of the claimed output to an input, a fitted parameter disguised as a prediction, or a load-bearing self-citation. Because the central existence claim rests on this theorem, the inconsistency is a serious correctness risk that would need direct substitution or a corrected P/ξ_0 normalization, but it does not by itself make the paper circular.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The paper introduces no new fields or entities; it combines two known transformations. The only 'fit' parameters are the labels (β,δ,M,C,B) of the solution family, none fitted to data. The main burden is the theorem's algebra, which contains a likely typo at the proportionality step.

free parameters (5)
  • β (scalar hair parameter) = free parameter, |β|≤1; β=±√3/2 for V0=0
    Labels the scalar charge/hair of the Buchdahl transform; it is a family label, not fitted to data, but the theorem constrains ξ1, ξ2, ξ3, V0 in terms of it.
  • δ (Zipoy-Voorhees deformation) = free parameter, δ>0, δ≠1 for non-Schwarzschild
    Quadrupole deformation of the static seed; an integration/family constant, not fitted to data.
  • M (mass parameter) = m=δM
    ADM mass label of the ZV seed; constant of the family.
  • C = free real constant in a(t)=(Ct)^ξ2
    Sets the cosmological expansion rate; free parameter, not fitted.
  • B = set B=0 WLOG
    Shifts the big-bang singularity; a gauge/sector choice.
assumptions (6)
  • domain assumption Buchdahl theorem: from a static/axisymmetric vacuum seed with an ignorable coordinate a and no off-diagonal components, (g_aa)^β dxa^2 + (g_aa)^{1-β} h_ij dx^i dx^j plus φ=ξ0 ln(g_aa) solves the massless Einstein-scalar equations.
    Used in §3.1.1 and §3.2.2 as the seed for the extended Fonarev construction; cited to [113,114].
  • domain assumption Fonarev's theorem for spherically symmetric seeds: conformal factor e^{2μ} with μ=ξ2 ln(Ct+B) plus scalar shift maps static Einstein-scalar solutions to solutions with Liouville potential under constraints (3.17-3.19).
    Reviewed in §3.1.2; basis for the extended version; cited to [43].
  • domain assumption The a-coordinate is ignorable: ∂_a g_{μν}=0 and g_{ia}=0.
    Restricts the seed class (eq. 3.22/3.28); holds for ZV with a=t.
  • domain assumption The mean curvature vector of Anco [96] coincides with the Kodama vector in spherical symmetry and its norm's vanishing locates (anti-)trapping horizons.
    Used in §2.2; the paper proves the spherical limit (2.32-2.35) and cites [7,84,85] for the general claim.
  • standard math Quasi-local trapping-horizon theory: θ±=0 for null normals orthogonal to a fixed 2-surface defines apparent horizons.
    Standard framework (Hayward, Ashtekar-Krishnan); used in §2.2.2 and §4.2.2.
  • domain assumption The seed ZV geometry's singularity structure (ring/string) does not prevent the conformal extension from being a valid solution in r>2M.
    The solution is defined on r>2M, outside the seed singularity; relies on the known global structure of ZV [127].

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Cite this review

Pith. "Pith review of Dynamical axisymmetric compact objects in General Relativity." pith.science (2026). https://pith.science/paper/WXQATX2H

@misc{pith2026251219542,
  author       = {Pith},
  title        = {Pith review of: Dynamical axisymmetric compact objects in General Relativity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WXQATX2H}},
  note         = {Machine review of arXiv:2512.19542}
}
read the original abstract

The search for exact solutions describing asymptotically FLRW compact objects in General Relativity remains a challenging problem. Progress has largely been limited to the spherically symmetric case, with notable exceptions such as the Kerr--de Sitter and Thakurta solutions. In this work, we present two new results that advance the description of axisymmetric compact objects embedded in a cosmological background. First, we introduce a new solution-generating technique that allows for the construction of nonstationary, axisymmetric solutions of the self-interacting Einstein-scalar system. Using this method, we obtain the first exact solution that can describe a dynamical axisymmetric compact object in a FLRW cosmology. We then outline how a detailed analysis of its properties, particularly dynamical trapping (or anti-trapping) horizons, can be carried out. For this purpose, we employ the mean curvature vector (MCV), which provides a natural extension of the Kodama vector beyond spherical symmetry. The norm of the MCV defines a foliation-independent, though embedding-dependent, quantity that can be used to identify trapped, anti-trapped, and untrapped regions, and to characterise the causal structure of the geometry without relying on specific symmetry assumptions. The embedding dependence must be treated carefully, as it determines the extent to which the analysis can be performed analytically while minimising the use of numerical methods. Overall, the solution-generating approach and the associated analysis tools offer a framework to further investigate dynamical axisymmetric compact objects, including black holes in cosmological settings and scenarios involving dynamical scalar accretion.

Figures

Figures reproduced from arXiv: 2512.19542 by the authors.

Figure 1
Figure 1. S−curve for the ZV-HMN solution with δ = 2. There, in the first panel, we consider different values of θ, which effectively parametrize the degree of axisymmetry of our solution. The resulting curve is the standard so-called S-curve that arises in this class of spacetimes [128] and was first obtained in the HMN solution [42]. Details of a representative curve are shown in the second panel of [PITH_FULL_IMAGE:figure… view at source ↗

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