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REVIEW 3 major objections 5 minor 16 references

Static horizons in cosmology

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read The paper constructs black hole and white hole metrics with static event horizons embedded in expanding cosmologies, contradicting earlier no-go theorems.

desk verdict Clean exact solutions and a genuinely different ansatz, but the headline claim of static event horizons in cosmology is not established—these are apparent horizons with tuned late-time behavior. read the letter →

arxiv 2504.20701 v3 pith:DFZBBQWB submitted 2025-04-29 gr-qc astro-ph.COhep-phhep-th

Ida M. Rasulian , Amjad Ashoorioon This is my paper · ORCID
classification gr-qcastro-ph.COhep-phhep-th MSC 83C5783C1583F05 PACS 04.70.Bw04.20.Jb98.80.-k
keywords staticeventhorizoncosmologicalblackholePainlevé-GullstrandcoordinateswhiteinexpandinguniverseSchwarzschild-deSitterapparentcouplinganisotropicpressure
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

The paper aims to overturn a series of results that forbid static event horizons inside an expanding cosmological background. It presents explicit Painlevé–Gullstrand metrics in which the surface $r=l_1$ is a static horizon, curvature invariants stay finite there, and the spacetime reduces to the Schwarzschild–de Sitter solution when the Hubble parameter is constant. The key move is to drop the assumption that radial and tangential pressures are equal near the horizon, and instead to require only that the horizon be static and that the metric approach FLRW at large distances. For white holes in expanding universes this is straightforward; for black holes the velocity function must change sign, and a generalized ansatz is given to realize that transition. If the construction works, black holes embedded in cosmology do not have to be cosmologically coupled, and earlier no-go statements survive only under extra assumptions.

What carries the argument

The machinery is the Painlevé–Gullstrand metric ansatz $ds^2=-(1-h^2(t,r))\,dt^2-2h(t,r)\,dt\,dr+dr^2+r^2\,d\Omega^2$, in which the horizon condition reduces to $h(t,l_1)=1$ and the sign of $h$ distinguishes white holes from black holes relative to the expanding background. A second form, $r=a(t,R)R$ with $h=R\dot a$, turns the metric into a semi-homogeneous form that makes the dust and dust-plus-cosmological-constant solutions tractable. The same function $h$ carries the whole argument: it interpolates between Schwarzschild behavior near $r=l_1$ and FLRW behavior at large $r$, and in the black-hole case it is allowed to pass through zero so the spacetime can switch from an expanding cosmological phase to a contracting black-hole phase.

What would settle it

Trace the paths of radial light rays in the metric (2.8) or in the dust white-hole solution and draw the conformal diagram; if any radial null geodesic from just inside $r=l_1$ reaches arbitrarily large radius, then the identified surface is only an apparent or isolated horizon, not a static event horizon.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that a static horizon can coexist with a cosmological background without a curvature singularity at the horizon, contrary to prior no-go theorems. The authors derive the metric from a minimal set of assumptions—no radial energy flux, a static horizon at $r=l_1$, cosmological energy density, and FLRW asymptotics—and show the resulting solution is unique in that class. For white holes in expanding universes, and black holes in contracting ones, the horizon is simply the surface where $h=1$; for black holes in expanding universes the velocity function $h$ must change sign, and the paper presents a generalized ansatz that allows this. The stress tensor, read through Einstein's equations, has anisotropic pressure, violates the null energy condition for the simplest family, and can be modified to respect that condition by relaxing the assumption that the energy density equals the FLRW value. In the constant-equation-of-state section, exact solutions with dust and with dust plus a cosmological constant show apparent horizons whose physical radius approaches a constant at late times, with the white-hole horizon in the latter case tending to its Schwarzschild–de Sitter value.

Load-bearing premise

The load-bearing premise is that the surface $r=l_1$, where the radial metric component vanishes and the curvature invariants stay finite, is truly a global event horizon; the paper does not construct the full causal structure or prove that light rays starting just inside $r=l_1$ can never escape.

Editorial extensions

If this is right

  • If the central claim is correct, the statement that event horizons in cosmology must be cosmologically coupled is not a general theorem; it fails once the pressure near the horizon is allowed to be anisotropic.
  • The explicit metrics give a concrete starting point for studying accretion, shadows, or tidal effects of black holes embedded in an FLRW background, since they approach known cosmological and Schwarzschild–de Sitter limits.
  • In the constant-equation-of-state models, the physical apparent-horizon radius can become asymptotically constant at late times, so a cosmological white hole can appear static to a late-time observer.
  • The black-hole construction with sign-changing $h$ shows how a region of local gravitational collapse can smoothly emerge from an expanding cosmology, a behavior expected in structure formation but rarely realized in exact solutions.

Reading between the lines

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

  • The paper verifies the horizon by the vanishing of the radial metric component and by finite curvature invariants; whether $r=l_1$ is a global event horizon in the strict causal sense requires a full conformal-diagram analysis, which the paper leaves implicit.
  • The same coordinate construction should extend to black holes with charge or rotation by generalizing the function $h$, a step not taken in this spherically symmetric treatment.
  • Because the escape route is the relaxation of pressure isotropy near the horizon, the construction suggests that the no-go results depend sensitively on that assumption and could be bypassed in modified gravity as well.
  • The sign-change criterion for $h$ offers a geometric way to identify the transition from cosmic expansion to local collapse, potentially useful for classifying apparent-horizon formation in inhomogeneous cosmologies.
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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 / 5 minor

Summary. The paper constructs spherically symmetric metrics in Painlevé–Gullstrand form intended to describe black holes and white holes embedded in FLRW cosmology with a static horizon at r=l1. Section 2 derives a white-hole metric from assumptions of no radial energy flux, vanishing g^{rr} at r=l1, FLRW energy density, and asymptotic FLRW behavior. Later sections specialize to constant equations of state: dust white holes with and without cosmological constant, and black holes with sign-changing velocity function. The paper claims these metrics have static event horizons without curvature singularities, reproduce cosmological behavior at large distances, and have apparent horizons that asymptotically approach constant values at late times.

Significance. If the event-horizon claim were correct, it would contradict established no-go results and the recent claim that black hole event horizons are cosmologically coupled. The paper provides explicit exact solutions and a systematic, in places unique, derivation, and it honestly reports in Section 3.1.2 that a dust apparent horizon cannot be exactly static. However, the central claim rests on a local identification of the horizon, and the paper's own analysis shows the apparent horizon is generally dynamical. The significance is therefore conditional on a successful global causal analysis that is not provided in the manuscript.

major comments (3)
  1. [Sec. 2.2, Eq. (2.13)] The identification of r=l1 as a static event horizon is based solely on g^{rr}=0 (h=1) at fixed r, which is an apparent-horizon condition, not an event-horizon condition. After the coordinate transformation t=γ(τ,r), the metric (2.13) still has coefficients that depend on τ through h(γ(τ,r),r) and ˙γ(τ,r), so no Killing vector ∂_τ is exhibited. No global causal-structure analysis is given to prove that radial null geodesics cannot escape from r<l1 to future null infinity. The abstract's statement that these metrics 'retain static event horizons' is therefore not established by the evidence in the paper.
  2. [Sec. 3.1.2, Eq. (3.35)] The authors prove that for dust with ω=0 the apparent horizon cannot be exactly static: Eq. (3.35) shows ˙r_H=0 is impossible, and the footnote reinforces this point. The late-time constant behavior in Eqs. (3.30)–(3.31) is obtained only after the specific choice b(R)=κl_s^4/R^{3-d}, and a different b(R) would give a different late-time limit. Thus the claim of an 'asymptotically static horizon' is a property of selected free functions, not a generic feature of the solution, and in any case what is shown is an apparent horizon, not an event horizon.
  3. [Sec. 2.2, assumption (b)] A static horizon is assumed as an input to the derivation via g^{rr}|_{r=l1}=0. The uniqueness result shows that the metric (2.8) is the unique solution under assumptions (a)–(d), but because the horizon is assumed, the construction does not by itself demonstrate that a static event horizon can exist in a cosmological setting. To support the paper's central claim, the authors would need to prove that the surface r=l1 is a global event horizon, or else explicitly reframe the result as a family of metrics with static apparent horizons and adjust the abstract and introduction accordingly.
minor comments (5)
  1. [Sec. 2.2] The notation ω(t,r) is introduced for h^2(t,r), but later in Section 3.1.2 the symbols β(R) and b(R) are introduced without a clear statement of their physical interpretation or the residual coordinate freedom; clarifying this would improve readability.
  2. [Sec. 2.1, Fig. 1] The text states that curvature invariants are finite at the horizon, but the caption of Figure 1 does not label which curve corresponds to which invariant, and the dependence of the invariants on the scale factor and its derivatives (as in Eq. (2.14)) is not discussed for general t.
  3. [Sec. 3.1.3, Eq. (3.44)] The statement that the white-hole horizon tends to its Schwarzschild–de Sitter counterpart relies on the approximation b(R) such that R_H becomes small at late times; this condition on the free function should be stated explicitly rather than implied.
  4. [General presentation] The word 'static' is used loosely throughout: in Section 3 the horizons are apparent horizons, and the paper should consistently distinguish apparent, isolated, and event horizons in the terminology.
  5. [Sec. 2.1] There is a typographical error 'the the metric is brought to the form' in the paragraph after Eq. (2.4).

Circularity Check

1 steps flagged · score 6.0 of 10

The claimed static event horizon is imposed as assumption (b) in Sec. 2.2, so its existence is an input rather than an output of the construction.

  1. self definitional [Sec. 2.2, assumption (b); Sec. 2.1 after Eq. (2.13); Abstract]
    "b) There is a static horizon at r =l1, such that grr|r=l1 = 0. ... Therefore with these assumptions the proposed metric (2.8) is unique. ... This metric has a static horizon at r =l1 where h = 1 independent of τ."

    The existence of the 'static event horizon' is not a derived output: assumption (b) already imposes g^{rr}|_{r=l1}=0 on the PG ansatz (2.18), and the Einstein equations are then solved subject to that condition. The later statement that (2.13) 'has a static horizon at r = l1 where h = 1' merely restates assumption (b). Since g^{rr}=0 at fixed r is an apparent-horizon condition, and no timelike Killing vector or null-geodesic escape argument is constructed, the paper's headline claim that it demonstrates the existence of a static event horizon reduces to its own input by construction. The uniqueness statement is conditional on the assumed horizon and therefore does not supply independent evidence.

full rationale

The paper is a self-contained construction: it writes explicit metrics, computes Einstein tensor components, and checks curvature invariants, so much of the technical content is not circular and there is no load-bearing self-citation chain. However, the central existence claim — a static event horizon in a cosmological setting — is built into the derivation. In Sec. 2.2, assumption (b) states 'There is a static horizon at r = l1, such that grr|r=l1 = 0'; the Einstein equations are then solved under this condition, and Sec. 2.1 concludes that the resulting metric 'has a static horizon at r = l1 where h = 1 independent of τ.' This is the same condition, restated. The coordinate transformation (2.12)-(2.13) only removes g_{tr}; it does not demonstrate a timelike Killing vector or prove that null geodesics cannot escape, so the upgrade from local g^{rr}=0 to a global event horizon is not established. For the matter-dominated case, the paper itself proves (Eq. 3.35 and footnote 2) that an exactly static apparent horizon is impossible, and the late-time constant horizon obtained in Eq. (3.31) follows from the specific choice b(R)=κ l_s^4/R^{3-d}; this is an engineered limit, not a generic prediction. Because the headline result reduces to an assumed local condition, a moderate circularity score is warranted, although the exact solutions themselves are non-vacuous.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The construction rests on tailored assumptions: a static horizon position, no radial flux, FLRW energy density, and later choices of free functions b(R) and beta(R). These are model-building inputs rather than independently measured constants, so the ledger counts them honestly. No new particles or forces are introduced.

free parameters (3)
  • l1
    Input Schwarzschild or white-hole radius that fixes the static-horizon location; appears in Eqs. (2.8)-(2.10). It is chosen, not derived.
  • beta0
    Asymptotic constant value of the dust mass function in Section 3.1.2; chosen to reproduce FLRW dust density at large R and controls the late-time horizon radius.
  • b(R) = example: b(R) = kappa l_s^4 / R^3 - d
    Arbitrary bang-time function in the dust and Lambda solutions; the paper chooses it to make the apparent horizon tend to a constant radius for large t.
assumptions (5)
  • domain assumption General relativity: T_mu_nu = G_mu_nu / kappa^2, so any computed Einstein tensor defines a stress-energy tensor.
    Used in Section 2.3 and throughout to interpret the constructed metrics as GR solutions.
  • ad hoc to paper No radial energy flux: T^t_r = 0 (assumption a).
    Imposed in Sections 2.2 and 3.1 to close the system; not required by the no-go theorem being challenged.
  • ad hoc to paper Cosmological energy density: G^t_t = -3 H^2 (assumption c).
    Imposed in Section 2.2 so the metric matches FLRW at large r; later relaxed in Section 2.4 to satisfy the null energy condition.
  • domain assumption Asymptotic FLRW behavior at large r (assumption d).
    Ensures the metric tends to the cosmological metric for large r; a standard matching condition for cosmological black holes.
  • domain assumption Constant equation of state p_r = p_perp = omega rho in Section 3.
    Used to obtain exact solutions; omega=0 for dust and Lambda acts as a cosmological constant.

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Cite this review

Pith. "Pith review of Static horizons in cosmology." pith.science (2026). https://pith.science/paper/DFZBBQWB

@misc{pith2026250420701,
  author       = {Pith},
  title        = {Pith review of: Static horizons in cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFZBBQWB}},
  note         = {Machine review of arXiv:2504.20701}
}
read the original abstract

Although previous results have ruled out the possibility of a static event horizon in cosmology, we present black hole and white hole metrics that retain static event horizons while reproducing cosmological behavior at large distances. Using an appropriate coordinate choice, we demonstrate that a static event horizon can exist in a cosmological setting without introducing curvature invariant singularities at the horizon. The resulting metric reduces to the Schwarzschild de Sitter solution when the Hubble parameter is constant. We find that white hole metrics in an expanding universe, or black holes in a contracting universe, are significantly easier to construct, as a black hole in an expanding cosmology requires the velocity function to change sign. Consequently, this work initially examines white holes in expanding cosmologies as a foundation for subsequent analysis of black holes in expanding universes. In later sections, we investigate scenarios involving a white hole coupled with cosmological matter, as well as a white hole with both matter and a cosmological constant. Assuming the pressure component takes its cosmological value, we show that the physical radius of the apparent horizon can asymptotically approach a constant value at late times. This metric avoids pathologies such as a singular horizon in the limit of a vanishing Hubble parameter. Finally, we analyze the realistic case of a black hole embedded in pressureless cosmological matter with and without a cosmological constant and explore its properties. We specifically show that the velocity function can become zero and change sign in the vicinity of a black hole. This means we can smoothly transition from an expanding cosmological phase with a positive velocity function to a contracting black hole phase with a negative velocity function.

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Reference graph

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