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arxiv: 2510.18690 · v2 · submitted 2025-10-21 · 🌀 gr-qc

Cosmological Black hole Candidates: A Detailed Analysis of McVittie, Culetu, Sultana-Dyer, and Glass-Mashhoon Spacetimes

M. Esfandiar , F. Shojai , O. Zamani Jamshidi , S. Zoorasna This is my paper

Pith reviewed 2026-05-18 04:47 UTC · model grok-4.3

classification 🌀 gr-qc
keywords cosmological black holestrapping horizonsMcVittie metricCuletu metricSultana-Dyer metricGlass-Mashhoon solutionenergy conditionsHayward formalism
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The pith

McVittie spacetimes fail to describe cosmological black holes while Culetu and Sultana-Dyer succeed in the early universe when energy conditions hold.

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

The paper sets out to determine which of several known dynamical spacetimes can serve as models for cosmological black holes. By checking for the presence of future outer trapping horizons according to Hayward's criteria, it concludes that the McVittie class and the Glass-Mashhoon solution do not qualify. The Culetu and Sultana-Dyer spacetimes, however, do qualify in a matter-dominated early universe as long as the energy conditions are met. This finding helps narrow down viable models for black holes that form or exist within an expanding cosmological background.

Core claim

Through a detailed examination of the McVittie, Culetu, and Sultana-Dyer metrics, as well as the generalized Glass-Mashhoon solution, the paper finds that the McVittie class of solutions fails to describe a cosmological black hole. The Glass-Mashhoon solution also lacks suitable future outer trapping horizons. Conversely, the Culetu and Sultana-Dyer spacetimes can describe a cosmological black hole in the matter-dominated early universe, provided that the relevant energy conditions are satisfied.

What carries the argument

Hayward's formalism of future outer trapping horizons and past inner trapping horizons used to evaluate existence and characteristics in these dynamical spacetimes.

If this is right

  • The McVittie class of solutions fails to describe a cosmological black hole.
  • The Glass-Mashhoon solution lacks suitable future outer trapping horizons and does not represent a cosmological black hole.
  • The Culetu spacetime can describe a cosmological black hole in the matter-dominated early universe if the relevant energy conditions are satisfied.
  • The Sultana-Dyer spacetime can describe a cosmological black hole in the matter-dominated early universe if the relevant energy conditions are satisfied.

Where Pith is reading between the lines

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

  • Analyses of black hole candidates in cosmology should routinely verify trapping horizon properties in addition to metric embedding.
  • These results may guide construction of new analytical models that consistently combine cosmic expansion with black hole features.
  • Predictions from the Culetu and Sultana-Dyer metrics could be compared with early-universe observations to test their applicability.

Load-bearing premise

The analysis assumes that Hayward's definition of future outer trapping horizons is the correct and sufficient criterion for identifying a cosmological black hole in these dynamical spacetimes.

What would settle it

A calculation demonstrating that the Culetu spacetime has no future outer trapping horizon despite satisfying the energy conditions in the matter-dominated era would falsify its suitability as a cosmological black hole.

read the original abstract

This paper investigates the existence of cosmological black holes by analyzing the properties of trapping horizons in detail, based on Hayward's formalism of future outer and past inner trapping horizons, in several dynamical spacetimes embedded in an expanding universe. Through a detailed examination of the McVittie, Culetu, and Sultana--Dyer metrics, as well as the generalized Glass--Mashhoon solution, we evaluate the existence and characteristics of trapping horizons and energy conditions. The Glass--Mashhoon solution provides an analytical model for spherical stellar collapse. However, it is shown that, as long as certain conditions are satisfied, it lacks suitable future outer trapping horizons, meaning it does not represent a cosmological black hole. As a result, the McVittie class of solutions also fails to describe a cosmological black hole. Conversely, the Culetu and Sultana--Dyer spacetimes can describe a cosmological black hole in the matter-dominated early universe, provided that the relevant energy conditions are satisfied.

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.

Referee Report

3 major / 3 minor

Summary. The manuscript applies Hayward's formalism of future outer and past inner trapping horizons to the McVittie, Culetu, Sultana-Dyer, and generalized Glass-Mashhoon spacetimes embedded in an expanding FLRW background. It concludes that the McVittie class and Glass-Mashhoon solution lack suitable future outer trapping horizons and therefore do not represent cosmological black holes, while the Culetu and Sultana-Dyer metrics can do so in the matter-dominated early universe provided the relevant energy conditions hold.

Significance. If the explicit horizon and energy-condition calculations are verified, the work supplies a concrete classification of these known dynamical metrics according to trapping-horizon criteria. This is useful for model selection in studies of black-hole formation within cosmology. The paper's strength lies in its systematic comparison across multiple spacetimes and its attention to the matter-dominated epoch; however, the conclusions remain tied to a single definitional framework.

major comments (3)
  1. [Introduction and §2] Introduction and the section introducing Hayward's formalism: The central classification (McVittie and Glass-Mashhoon fail; Culetu and Sultana-Dyer succeed) rests on the assumption that the existence of a future outer trapping horizon is both necessary and sufficient for a cosmological black hole. The manuscript does not discuss or test whether additional requirements (e.g., Kodama-vector alignment, asymptotic FLRW matching, or consistency with event-horizon limits in the static case) are needed. This definitional choice is load-bearing for the headline claim.
  2. [McVittie analysis section] McVittie analysis section: The statement that the McVittie class lacks suitable future outer trapping horizons requires the explicit null expansions θ_l, θ_n and the Lie derivative condition along the outward null vector to be written out for the metric functions used. Without these steps shown, it is not possible to confirm that the absence is intrinsic rather than an artifact of coordinate choice or parameter selection.
  3. [Culetu and Sultana-Dyer sections] Culetu and Sultana-Dyer sections: The positive claim that these spacetimes describe cosmological black holes when energy conditions are satisfied must include the precise form of the stress-energy tensor components and the explicit verification (e.g., null energy condition ρ + p_r ≥ 0) inside the matter-dominated era, together with the range of the scale-factor exponent or density parameter for which the horizon exists.
minor comments (3)
  1. Notation for trapping horizons should be introduced once with a clear definition of 'future outer' versus 'past inner' and then used consistently; occasional shifts between 'trapping horizon' and 'dynamical horizon' are confusing.
  2. Figure captions for horizon diagrams should indicate the direction of the null vectors and the sign of the expansions to make the classification visually verifiable.
  3. The Glass-Mashhoon section would benefit from a short table summarizing the conditions under which future outer horizons are absent, paralleling the energy-condition table for the other metrics.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. We address each major comment below, explaining our response and the revisions we will incorporate to improve clarity and completeness.

read point-by-point responses

  1. Referee: [Introduction and §2] Introduction and the section introducing Hayward's formalism: The central classification (McVittie and Glass-Mashhoon fail; Culetu and Sultana-Dyer succeed) rests on the assumption that the existence of a future outer trapping horizon is both necessary and sufficient for a cosmological black hole. The manuscript does not discuss or test whether additional requirements (e.g., Kodama-vector alignment, asymptotic FLRW matching, or consistency with event-horizon limits in the static case) are needed. This definitional choice is load-bearing for the headline claim.

    Authors: We acknowledge that our classification is grounded in Hayward's trapping-horizon framework, which we adopt because it supplies a local, quasi-local criterion well-suited to dynamical cosmological spacetimes. To address the referee's concern, we will revise the Introduction and Section 2 to include a short discussion of the rationale for this choice, briefly noting compatibility with Kodama-vector alignment and static limits where relevant, while clarifying that our focus remains on the trapping-horizon properties as the primary diagnostic for cosmological black holes. revision: yes

  2. Referee: [McVittie analysis section] McVittie analysis section: The statement that the McVittie class lacks suitable future outer trapping horizons requires the explicit null expansions θ_l, θ_n and the Lie derivative condition along the outward null vector to be written out for the metric functions used. Without these steps shown, it is not possible to confirm that the absence is intrinsic rather than an artifact of coordinate choice or parameter selection.

    Authors: We thank the referee for this observation. The null expansions θ_l, θ_n and the associated Lie-derivative conditions were computed explicitly in our analysis of the McVittie metric. We will revise the McVittie section to display these expressions in full for the metric functions employed, thereby demonstrating that the lack of a suitable future outer trapping horizon is an intrinsic feature of the spacetime rather than a coordinate or parameter artifact. revision: yes

  3. Referee: [Culetu and Sultana-Dyer sections] Culetu and Sultana-Dyer sections: The positive claim that these spacetimes describe cosmological black holes when energy conditions are satisfied must include the precise form of the stress-energy tensor components and the explicit verification (e.g., null energy condition ρ + p_r ≥ 0) inside the matter-dominated era, together with the range of the scale-factor exponent or density parameter for which the horizon exists.

    Authors: We appreciate the request for greater explicitness. Our analysis already verifies the relevant energy conditions for the Culetu and Sultana-Dyer metrics in the matter-dominated era. We will revise these sections to present the precise stress-energy tensor components, show the detailed checks of the null energy condition (and other conditions), and specify the ranges of the scale-factor exponent and density parameters for which a future outer trapping horizon exists while the energy conditions hold. revision: yes

Circularity Check

0 steps flagged

No significant circularity; conclusions follow from applying external Hayward formalism to explicit metric calculations

full rationale

The paper computes trapping horizons and energy conditions directly from the listed dynamical metrics using Hayward's standard definitions of future outer and past inner trapping horizons. The statements that McVittie/Glass-Mashhoon lack suitable horizons (hence do not represent cosmological black holes) and that Culetu/Sultana-Dyer possess them under energy-condition compliance are direct outputs of those calculations, not reductions to fitted parameters, self-definitions, or self-citation chains. Hayward's criterion is imported as an external benchmark rather than derived or assumed within the paper itself. No load-bearing self-citations or ansatze are evident in the provided text.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based solely on the abstract, the ledger is limited; the central claim rests on standard GR assumptions and Hayward's trapping-horizon definition rather than new free parameters or invented entities.

axioms (1)
  • domain assumption Hayward's formalism of future outer and past inner trapping horizons correctly identifies cosmological black holes in dynamical spacetimes
    Invoked throughout the abstract as the basis for evaluating each metric

pith-pipeline@v0.9.0 · 5742 in / 1224 out tokens · 41362 ms · 2026-05-18T04:47:06.374535+00:00 · methodology

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Reference graph

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