The self-consistent field method and the macroscopic universe consisting of a fluid and a black holes

Yu. G. Ignat'ev

REVIEW 4 major objections 3 minor 27 references

The self-consistent field method and the macroscopic universe consisting of a fluid and a black holes

T0 review · 4 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Averaged black-hole metric fluctuations generate a vacuum equation of state, so black-hole backreaction can act as a cosmological constant.

desk verdict A physically motivated attempt to derive a cosmological constant from averaged black-hole perturbations, but the central calculation is missing and the few displayed averages contain inconsistent N-scaling and dimensional errors. read the letter →

arxiv 1908.03488 v3 pith:L3XSRK77 submitted 2019-08-09 gr-qc

Yu. G. Ignat'ev This is my paper · ORCID

classification gr-qc MSC 83C0583C5583C5783F05 PACS 04.20.Cv98.80.Cq96.50.S52.27.Ny

keywords macroscopiccosmologyblackholesequationofstateself-consistentfieldmethodmetricfluctuationscosmologicalconstantFriedmannuniversebackreaction

open problems Dark Energy

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 tries to show that a universe filled with a dust fluid and many identical black holes can be described macroscopically by the standard Einstein equations plus a correction term that behaves exactly like a cosmological constant. The argument treats the black holes as localized spherically symmetric metric perturbations, averages their combined effect over positions, and finds that the averaged quadratic fluctuations contribute an effective energy-momentum tensor with pressure equal to minus energy density, $p = -\varepsilon$. If this is right, the backreaction of black holes is a classical mechanism for late-time accelerated expansion, requiring no new physics beyond general relativity. The paper also derives a growth law for the black-hole masses, $m \sim t$, on the non-relativistic stage, meaning the holes can become very massive today.

What carries the argument

The load-bearing mechanism is the self-consistent field method applied to gravity: one particle moves in the average field of all others, and the macroscopic metric is obtained by averaging over the particle-coordinate distribution. The specific technical object is the averaged bilinear combination of metric fluctuations, computed with the averaging rule $\overline{\phi} = \prod_a \frac{1}{V_a}\int d^3 r_a \, \phi(r|x_1,\dots,x_N)$, together with the assumptions that differentiation commutes with averaging and that the averaged Einstein tensor inherits the Friedmann symmetry. From the finite averages $\overline{\xi^2} = \frac{108}{175N}\left(\frac{2\mu_0}{r_0}\right)^2$ and the renormalized derivative averages, the correction to the Einstein tensor is evaluated and converted into the effective stress tensor $\delta T_i^k$. The mass-growth law follows from the separable linearized equations for the local perturbation and the point-mass mass function.

What would settle it

Compute the second-order backreaction of a population of point masses in a gauge-invariant cosmological perturbation scheme and compare the isotropic pressure correction to the energy-density correction: if the averaged correction is not exactly $\delta p = -\delta\varepsilon$, the claim fails. Observationally, if the dark-energy equation of state measured from distance indicators deviates from $w = -1$ in a way that tracks the black-hole abundance, the predicted vacuum term would be excluded.

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Extended reading notes

Core claim

The central claim is that the self-consistent-field averaging of linearized spherically symmetric perturbations produced by $N$ equal point masses in a Friedmann dust background yields an effective correction to the energy-momentum tensor $\delta T_i^k = \frac{9N}{4r_0^2}\left(\frac{2\mu_0}{r_0}\right)^2 \delta_i^k$. Because this correction is proportional to $\delta_i^k$, it has the vacuum equation of state $\varepsilon_g + p_g = 0$; it adds a constant negative pressure to the fluid and acts as a cosmological constant. The paper further asserts that the growing mode of the mass function gives $m(\eta) \sim \eta^2$ in the non-relativistic stage and hence $m \sim t$, so the masses of the black holes grow to very large values at the modern stage. The macroscopic Einstein equations themselves are unchanged; what changes is only the effective equation of state of the fluid.

Load-bearing premise

The main load-bearing premise is that the averaging rule over black-hole positions is a physically meaningful, gauge-invariant operation that commutes with derivatives and preserves the symmetry of the background; if that fails, the claimed vacuum equation of state does not follow.

Editorial extensions

If this is right

The effective stress-energy correction is isotropic and has $\varepsilon_g + p_g = 0$, so it adds a constant negative pressure independent of the fluid density.
This term is mathematically equivalent to a cosmological constant, so it would drive accelerated expansion at late stages without introducing a scalar field.
The mass of a black hole grows as $m \sim t$ in the non-relativistic stage, so black-hole masses can reach very large values today.
The correction does not alter the macroscopic Einstein equations themselves; it only changes the effective equation of state of the fluid.

Reading between the lines

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

If this mechanism is real, the abundance and mass function of black holes could be constrained by the measured value of dark energy, since the correction scales as $N\mu_0^2$.
A natural test is to repeat the averaging in a fully gauge-invariant way; the paper's result would be robust only if the vacuum equation of state survives that check.
The same averaging technique could apply to other localized sources, such as compact objects or topological defects, potentially generating similar effective negative-pressure fluids.
One could look for the predicted mass growth in the mass distribution of supermassive black holes across cosmic time; a growth law $m \sim t$ would be a distinctive signature.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Ignat'ev is after something genuinely worth asking: can the averaged backreaction from many localized mass concentrations in a dust FLRW universe produce a term with p = -ε, and might that plausibly account for late-time acceleration? The setup — self-consistent field method, linearized spherically symmetric perturbations from earlier papers, averaging over uncorrelated source positions — is coherent, and the paper is honest about building on his own previous work. The mass-growth formula (58) tying black-hole mass to cosmological time is also an interesting consequence, though it does not follow from the paper's own equations in the way claimed.

The problem is that the main result is not actually derived. Equation (68), the effective stress tensor, appears after a one-line 'Calculating now...' with no expression for δG^(2) in terms of the averaged bilinears. The preceding averages do not provide enough information: (64) gives ⟨ξ²⟩ with a 1/N, while (68) is proportional to N. For N uncorrelated sources, the mean square of the summed field should scale as N, so (64) and (68) cannot both be right. Equation (67) also has a dimension problem: the right side as printed does not have units of a derivative-squared average. Then the jump to (68) is an unstated step, not a calculation.

There are softer worries too: the averaging rule and the renormalization shift (20)/(63) are not gauge-invariant, and the paper does not address that; Assumption 2 is plausible but strong; and the conclusion that m ~ t in the nonrelativistic stage is inconsistent with m ∝ η² in a dust universe where η ~ t^(1/3). These are not minor quibbles. The p = -ε result is the entire payload, and it is unsupported by anything shown.

I don't see this as salvageable by light editing. It needs a genuine re-derivation of the averaged second-order Einstein tensor, with a clear calculation and consistent N-counting. For a reader interested in backreaction models of dark energy, the paper's premise is worth knowing about, but as a reference it is not usable in its current form.

For peer review: I would send it out once, not desk-reject, because the question is significant and the framework is not a fringe attempt. A competent referee will likely reject on the grounds above, but the exercise is worthwhile. I would not cite it myself until the calculation is fixed.

Referee Report

4 major / 3 minor

Summary. The paper develops a self-consistent-field formalism for a macroscopic FLRW universe filled with dust and a population of point masses interpreted as black holes. Starting from linearized spherically symmetric perturbations, the author derives mass evolution laws for the ultrarelativistic and dust stages, then averages the perturbations using a prescribed coordinate-averaging rule. The central claim is that the averaged second-order corrections to the Einstein tensor produce an effective energy-momentum tensor δT^i_k = (9N/4r0^2)(2μ0/r0)^2 δ^i_k, which has the vacuum equation of state p = -ε and therefore acts like a cosmological constant. The paper also claims that the black-hole mass grows as m ∼ t during the matter-dominated stage.

Significance. If the central calculation were correct, the paper would offer a purely classical backreaction mechanism for cosmic acceleration and an attendant growth law for primordial black holes, with no new physics beyond general relativity. The formal setup, which adapts the self-consistent-field method to gravitational systems, is a reasonable starting point, and the explicit solutions for the mass evolution in Eqs. (54) and (58) are straightforward and clearly presented. However, the paper's main result, Eq. (68), is not derived: it is stated after two unevaluated averages and contains inconsistencies in N-scaling and dimensions in the preceding equations. The strength of the claimed conclusion is therefore not matched by the displayed computation.

major comments (4)

[§2.3, Eq. (68)] The central result δT^i_k = (9N/4r0^2)(2μ0/r0)^2 δ^i_k is asserted without derivation. The text says 'Calculating now the averages from corrections to Einstein tensor' but never displays δG^(2)i_k. A second-order Einstein tensor built from the metric (46)/(61) is a quadratic functional of ξ and its first derivatives, so its average depends on ⟨ξ²⟩, ⟨(∂ηξ)²⟩, ⟨(∇ξ)²⟩ and cross terms; only (64) and (67) are given, and (66) is stated without proof. The jump from those averages to the isotropic form (68) is therefore unsupported.
[§2.2, Eqs. (61) and (64)] The N-scaling in Eq. (64) is inconsistent with the superposition (61). With hik = -a²δik Σ_a ξ_a and statistically independent sources, the renormalized average of (Σ_a ξ_a)² is N times the single-source variance, not 1/N times it. Eq. (64) gives ξ² ∝ 1/N, while Eq. (65) and the final result (68) carry factors of N. Since the claimed effect grows with the number of sources, this is not a harmless normalization convention; it indicates a missing or incorrect combinatorial factor in the averaging step.
[§2.2, Eq. (67)] Eq. (67) has a dimensional inconsistency: the left side ∂αξ∂βξ has dimension L^-2, whereas the right side (6πN/r0^2)(2μ0/r0^2) has dimension N L^-3 (taking ξ dimensionless and μ0/r0 dimensionless). Even allowing for a typo in the factor, the expression is not dimensionally homogeneous. Because this equation is one of the two inputs to the claimed δG^(2), the inconsistency propagates directly into the central result.
[§1.2–§2.2, Eqs. (20) and (63)] The renormalization g(0) → gbar, and especially the constant shift in (63), is assumed to leave the physical macroscopic metric unchanged. If this shift is not merely a gauge or coordinate redefinition, the effective cosmological constant obtained in (68) could be an artifact of the renormalization rather than a genuine backreaction effect. The paper does not establish that the averaging procedure is gauge-invariant or that the shift in (63) is physically equivalent to a scale transformation of the background. This is a load-bearing assumption for the p = -ε conclusion.

minor comments (3)

[General] The manuscript has numerous typographical and grammatical errors that impede reading, such as 'thepory', 'th e', 'av eraging', and the title's 'a black holes'. A careful English-language edit is needed.
[Eq. (53)] The constants in Eq. (53) are not consistently named: the text introduces µ0 and µ1 but then writes m0 = µ1 + µ2, and the subsequent discussion assumes µ1 = 0 while referring to µ0. This should be clarified.
[References [20, 21]] The Isaacson references are cited as Phys. Rev. 66, 1263 (1966) and 1272 (1966); the standard citation is Phys. Rev. 166, 1263 and 1272 (1968). Please verify and correct.

Circularity Check

1 steps flagged · score 6.0 of 10

The central p=−ε result is essentially a restatement of Assumption 2: once the averaged tensor is assumed to inherit the metric's algebraic structure, δT_i^k ∝ δ_i^k and the vacuum equation of state follow by construction rather than by explicit averaging.

self definitional [Section 1.3, Assumption 2; Section 2.3, Eqs. (68) and (70)]
"Assumption 2 The macroscopic average of the Einstein tensor inherits the symmetry properties of the macroscopic metrics: Lξgik=σgik =⇒LξGik=σ1gik ... all symmetrical covariant tensors of the second valency have the same algebraic structure as a metric tensor. Thus it is logical to assume that algebraic structure of macroscopic tensors will also be the same. ... Calculating now the averages from corrections to Einstein tensor, caused by local fluctuations of metrics, let us find the correction ... δT i k = ... = 9N/4r2 0 (2μ0/r0)^2 δi k."

Applying Assumption 2 to the spatially flat FLRW background (37), where g_i^k=δ_i^k, the assertion that the averaged second-order Einstein tensor has 'the same algebraic structure as a metric tensor' directly gives δG^(2)i_k ∝ δ_i^k. Since δT_i^k = −δG^(2)i_k/8π, the displayed result δT_i^k ∝ δ_i^k is already the content of Assumption 2: T_0^0=ε and T_j^i=−pδ_j^i with p+ε=0. Thus the central equation-of-state conclusion (70), ε_g+p_g=0, is not an independent output of the averaging calculation; it is a restatement of the symmetry postulate. No computation of δG^(2) is shown between the averages (64)–(67) and Eq. (68), so the only displayed derivation of the δ_i^k form is the assumption itself.

full rationale

The paper does not fit a parameter to a target dark-energy value; the coefficient in (68) is claimed from the averages (64)–(67). However, the load-bearing physical conclusion ε_g+p_g=0 is not obtained by an explicit calculation of δG^(2). Equation (68) is stated after 'Calculating now the averages', but no second-order Einstein tensor is displayed, and the displayed proportionality to δ_i^k is exactly what Assumption 2 requires: for FLRW, 'same algebraic structure as the metric' means T_i^k ∝ δ_i^k, which is the vacuum equation of state. So the central p=−ε result is circular in the sense that the conclusion is put in through the symmetry postulate rather than derived from the perturbation averages. The mass-growth result (58), by contrast, is an independent solution of (47) and does not reduce to an input. There are also correctness concerns — the averages (64) and (67) are asserted without derivation and appear to have inconsistent N-scaling and dimensions, and the renormalization in §2.2 is assumed to be a harmless constant rescaling — but these are not circularity; they are unverified computational and gauge questions. Since the central new physical claim (cosmological-constant-like backreaction) reduces by construction to Assumption 2, while a secondary result (mass growth) is independent, the circularity score is 6.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The central result rests on the initial mass scale μ0 and the number of sources N, both of which are inputs rather than derived values. The averaging and symmetry assumptions (Assumptions 1 and 2), the localization ansatz, and the metric renormalization are all specific to this paper and are not independently verified.

free parameters (2)

μ0
The amplitude of the growing mode of the black hole mass at the start of the radiation era. The magnitude of the effective vacuum-like term scales as μ0 squared, so the physical size of the claimed effect depends on an unconstrained initial condition.
N
The number of identical black holes inside the conformal volume V = 4πr0^3/3. The effective stress tensor scales with N, but N is not determined by the model.

assumptions (6)

ad hoc to paper Averaging commutes with differentiation and integration (Assumption 1, Eqs. 10-11).
This is a property of the averaging operation that the paper assumes rather than proves, and the physical averaging method is left unspecified.
ad hoc to paper The macroscopic Einstein tensor inherits the symmetries of the macroscopic metric (Assumption 2, Eq. 29).
Used to conclude that the averaged effective stress tensor has FLRW form before computing it; this restricts the possible outcome.
domain assumption Linear perturbation theory with small fluctuations and second-order truncation is sufficient for the macroscopic backreaction (Eqs. 4, 15).
The paper neglects third and higher order terms; the validity of a perturbative backreaction is assumed.
domain assumption Fluctuations are localized inside the sound horizon and vanish with their derivatives at r = rs (Eqs. 49-50).
This localization makes the volume integrals finite and is used to derive the values (63)-(67).
ad hoc to paper The metric renormalization g^(0) to gbar (Eq. 20) removes the linear average and leaves the true macroscopic metric.
The renormalization is called a formal technique; no independent justification is given.
standard math Birkhoff theorem prevents external masses from changing the macroscopic Friedmann energy density.
Used in the conclusion to argue that fluctuations only change the equation of state.

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Cite this review

Pith. "Pith review of The self-consistent field method and the macroscopic universe consisting of a fluid and a black holes." pith.science (2026). https://pith.science/paper/L3XSRK77

@misc{pith2026190803488,
  author       = {Pith},
  title        = {Pith review of: The self-consistent field method and the macroscopic universe consisting of a fluid and a black holes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/L3XSRK77}},
  note         = {Machine review of arXiv:1908.03488}
}

abstract

The article discusses and substantiates a self-consistent approach to the macroscopic description of systems with gravitational interaction. Corrections to the equation of state of the fluid are found based on macroscopic Einstein equations which were obtained by averaging over microscopic spherically symmetric metric fluctuations created by the primary Black Holes in a fluid medium. It is shown that these corrections are effectively equivalent to addition of a fluid to the system with the equation of state $p=-\varepsilon$. In addition, it is shown that, in this case, the mass of Black Holes can grow at a modern stage of evolution to very large values.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

27 extracted references · 27 canonical work pages

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Yu. G. Ignat’ev, Gravitacia i Teoriya Otnositelnosti, Kazan, Kaz an University-Press, Iessue 14, 90 (1978) (in Russian)

work page 1978

[2] [2]

Yu. G. Ignat’ev, Gravitacia i Teoriya Otnositelnosti, Kazan, Kaz an University-Press, Iessue 20, 50 (1983) (in Russian)

work page 1983

[3] [3]

Yu. G. Ignat’ev, Grav. and Cosmol., 13 59 (2007)

work page 2007

[4] [4]

Ignatyev (Ignat’ev)

Yurii G. Ignatyev (Ignat’ev). Relativistic Kinetic Theory of None quilibrium Processes in Gravitational Fields. Kazan, Foliant-Press (2010) (in Russian)

work page 2010

[5] [5]

Yu. G. Ignat’ev and A. A. Popov, Sov. Phys. J., 32 391 (1989)

work page 1989

[6] [6]

Yu. G. Ignat’ev and A. A. Popov, Actrophys. and Space Sci. 163 153 (1990)

work page 1990

[7] [7]

Yu. G. Ignat’ev and A. A. Popov, Phys. Lett. A. 220 1996

work page 1996

[8] [8]

Yu. G. Ignat’ev and N. Elmakhi, Russ. Phys. J. 51 74 (2008)

work page 2008