Black hole apparent horizons become comoving with the universe

Massimiliano Rinaldi; Valerio Faraoni

REVIEW

Black hole apparent horizons become comoving with the universe

Valerio Faraoni, Massimiliano Rinaldi

General Relativity and Quantum Cosmology · Submitted 21 Aug 2026

Pith's verdict: A compact and mostly clean argument that apparent horizons tend to become comoving, but the 'full generality' claim rests on an unstated bound on the lapse, so the central theorem is not proven as stated.

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arXiv ID2608.21222 (version 2)

Pith ID7RAAOJVK, Pith's permanent record of this paper and its review

CitationMassimiliano Rinaldi, Valerio Faraoni. "Black hole apparent horizons become comoving with the universe." arXiv:2608.21222 (2026). https://pith.science/paper/7RAAOJVK

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Editor's note

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Short version: this is a neat little paper that extends the existing no-go result for static event horizons to apparent horizons. The main equation (9) is right, and the observation that 2M=R at the horizon turns mass scaling into horizon-radius scaling is clean. The paper is honest about using no field equations and about the foliation-dependence of apparent horizons. If the conclusion were established, it would matter for the cosmological coupling debate.

What is actually new: the specific statement that, under the authors' exclusions, the black hole apparent horizon radius R_AH asymptotically grows like a(t), so M ∝ a(t), is not in the cited literature. The proof is a short exercise in the standard formalism, but that does not make it wrong.

Soft spots, roughly in order of importance.

1. The dichotomy before Eq. (18) is not exhaustive. To rule out the case \dot{r}_AH > 0 with R_AH growing forever without \dot{r}_AH → 0, the authors need a_AH \dot{r}_AH/T_AH → ∞. They assume this without stating it. With no field equations and T just a lapse with T→1 at infinity, nothing stops T_AH from growing like a_AH; then a_AH \dot{r}_AH/T_AH can tend to a constant β/λ < 1, and R_AH grows forever at rate H(1−β/λ), which is not comoving. This is not gauge: the combination is invariant under time reparametrization. It is an internal gap, not a question of physical taste.

2. The exclusions of the other branches are physical reasonableness arguments, not consequences of the equations: decreasing horizon entropy, naked null singularities, 'disconnected regions.' They might be right, but they are asserted. The paper should say so.

3. There are typos in the intermediate algebra, including the line before Eq. (8) where a dot is missing or misprinted, and Eq. (7) as printed does not obviously reduce to Eq. (8). These are minor and fixable, but they make a compact paper harder to check.

The 'full generality' in the abstract is an overstatement given point 1. The paper needs either a bound on T_AH (some energy condition or fall-off condition on the lapse) or a revised conclusion with restricted scope.

Who it is for: people working on cosmological coupling, primordial black hole growth, and quasi-local horizons. It deserves a serious referee, because the core idea is useful and the gap is repairable. I would not cite the theorem as established; I might cite it as a suggestive argument.

Referee report

Significance

If the central claim were established, the paper would provide a simple, parameter-free kinematical argument for cosmological coupling of black holes and would give a new perspective on the early growth of supermassive and primordial black holes. The derivation of Eq. (9) is clean and self-contained, and the authors are transparent about not using field equations and about the foliation dependence of apparent horizons. However, the proof of the main universality claim is incomplete as it stands: the dichotomy preceding Eq. (18) relies on an unstated asymptotic property of the lapse T(t,r), and the elimination of non-comoving branches depends on imported physical assumptions such as the absence of naked null singularities and the non-decrease of horizon entropy. The paper is a thought-provoking contribution to an active debate, but it needs substantial revision before the claim of 'full generality' can be accepted.

Major objections (3)

[Sec. 'Slower than comoving RAH', around Eq. (18)] The dichotomy used to reach Eq. (18) assumes that whenever dot r_AH does not tend to zero, then a_AH dot r_AH / T_AH tends to infinity. This requires T_AH to grow more slowly than a_AH, but no such condition is stated or derived. The only asymptotic condition on the lapse is that T(t,r) -> 1 for large r at fixed t, and this does not exclude a lapse profile whose peak travels with the horizon, for example T(t,r) = 1 + lambda a(t) f(r/a(t)) with f(x) -> 0 as x -> infinity. On a horizon trajectory with dot r_AH = beta < lambda, Eq. (9) gives d ln R_AH/dt = H_AH (1 - beta/lambda) > 0 forever, so R_AH grows without bound, dot r_AH stays positive and finite, and the claimed eventual decrease leading to Eq. (18) never occurs. Since the manuscript explicitly avoids field equations, this is a genuine gap in the central proof rather than a gauge artifact; the authors need either to prove a bound on T_AH from their assumptions or to add an explicit physical assumption excluding such 'lapse tracking' behaviour.
[Sec. 'Slower than comoving RAH', paragraphs discarding branches] The conclusion that the only physically viable scenario is dot r_AH -> 0 depends on two physical axioms that are asserted rather than derived: that a decreasing apparent-horizon area contradicts 'a tenet of black hole physics' because entropy would decrease, and that an apparent horizon asymptoting to a constant areal radius leads to a naked null singularity dividing spacetime into two disconnected regions, which is 'completely unphysical'. These are reasonable principles in many contexts, but they are not consequences of the apparent-horizon definition or of the Einstein equations, and the paper does not cite a proof of the entropy tenet for dynamic apparent horizons. Because the abstract and conclusions claim the result holds 'in full generality', the manuscript should state explicitly that the theorem is conditional on these physical axioms and, if possible, cite or prove the needed entropy-monotonicity result for the relevant class of spacetimes.
[Sec. 'Slower than comoving RAH', asymptotic assumption a_AH -> infinity] The proof also assumes that the universe expands forever with a_AH -> +infinity, but this is not stated among the 'reasonable physical assumptions' at the start. For a spatially flat FLRW background that recollapses or asymptotes to a static configuration, the dichotomy 'either dot r_AH -> 0 or a_AH dot r_AH / T_AH -> infinity' would not follow. This is a smaller point than the lapse issue, but it is still load-bearing for the claimed asymptotic conclusion and should be made explicit at the outset.

Minor objections (5)

[Before Eq. (8)] The sentence 'Using the expressions R'_AH = -T_AH R'_AH / a_AH and dot R_AH = dot a_AH r_AH' contains a typo: the first expression should read dot R_AH = -T_AH R'_AH / a_AH, which is the relation derived as Eq. (5). As printed, the equation is tautological and confusing.
[Eq. (7)] Equation (7) as typeset appears dimensionally inconsistent: dR_AH/dt = dot R_AH + dot r_AH a_AH + r_AH dot r_AH (dot a_AH + a'_AH dot r_AH) contains an extra factor r_AH dot r_AH in the last term. The correct expression is dR_AH/dt = dot R_AH + dot r_AH (a_AH + a'_AH r_AH), which is what the subsequent reduction to Eq. (8) actually uses.
[Eq. (16)] The norm of the normal to the surface f(t,r)=0 is written as N_a N^a = 1/a_AH^2 - dot r_AH^2 r_AH^2, but the factor r_AH^2 is erroneous. From N_c = (-dot r_AH, 1), the correct norm is N_a N^a = 1/a_AH^2 - dot r_AH^2 / T_AH^2, which is the expression needed to obtain Eq. (17).
[Introduction and metric section] The phrase 'at large radial distances r, T(t,r) must reduce to unity and a(t,r) ~ a(t) loses its spatial dependence' should specify whether the limit is pointwise in t for each fixed r, uniform in t, or in some other sense. The proof's asymptotic dichotomy depends on the mode of approach, so a precise statement would improve the rigor of the presentation.
[Figure 1 caption] The caption describes the red dot-dashed lines as 'comoving apparent horizons described by Eq. (11)', but Eq. (11) writes R_AH = a(t,r_0) r_0 + const and then identifies this with a(t) r_0 + const. This identification requires r_0 to be in the asymptotic FLRW region; the caption should clarify that this is a schematic representation.

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The paper

The paper claims

This paper argues that every spherical black hole apparent horizon embedded in an expanding flat FLRW universe is inevitably driven to comove with the cosmic expansion, so that its physical radius grows in proportion to the scale factor…

The reading

This paper aims to settle how a black hole horizon behaves when embedded in an expanding universe. Working only with spherical symmetry, an expanding FLRW background, and the quasi-local idea of an apparent horizon, the authors derive an exact relation between the horizon's growth rate and the cosmic Hubble rate. They then argue, by excluding every non-comoving alternative as unphysical, that any apparent horizon that forms is inevitably driven to a comoving state: its physical radius grows in proportion to the scale factor, and its mass grows as $M(t)\propto a(t)$. If correct, this makes cosmological coupling of black holes a geometric necessity rather than an optional astrophysical effect, and it opens a fundamental, accretion-independent channel for early black hole growth.

What carries the argument

The load-bearing object is the apparent horizon radius $R_{\rm AH}(t)=a(t,r_{\rm AH})r_{\rm AH}$ in a general spherically symmetric metric, determined by $\nabla_c R\nabla^c R=0$, together with the Misner-Sharp-Hernandez mass $M$, which satisfies $2M_{\rm AH}=R_{\rm AH}$ at the horizon. The key identity is Eq. (9), connecting the fractional growth of $R_{\rm AH}$ and $M$ to the horizon's Hubble rate $H_{\rm AH}$ and the drift $\dot r_{\rm AH}$ of the comoving radius. The argument's machinery is the tripartite classification of possible dynamics—comoving, slower, faster—and the exclusion of all non-comoving branches by appeal to black-hole entropy monotonicity, absence of naked null singularities, and the impossibility of horizon disappearance.

Extended reading notes

Core claim

The central claim is Eq. (9): $\frac{1}{R_{\rm AH}}\frac{dR_{\rm AH}}{dt} = H_{\rm AH}\left(1 - \frac{a_{\rm AH}\dot r_{\rm AH}}{T_{\rm AH}}\right) = \frac{1}{M}\frac{dM}{dt}$. The factor in parentheses measures how much the apparent horizon's comoving radius $r_{\rm AH}$ drifts relative to the cosmic substratum. The paper classifies $\dot r_{\rm AH}=0$ as exactly comoving, $\dot r_{\rm AH}>0$ as slower than comoving, and $\dot r_{\rm AH}<0$ as faster than comoving. It then argues that every branch with $\dot r_{\rm AH}$ not tending to zero is pathological: shrinking horizons violate the second law via decreasing entropy; horizons approaching constant $R_{\rm AH}$ become naked null singularities that split spacetime; horizons reaching $r_{\rm AH}=0$ disappear. Therefore the only physically viable asymptotic state is $\dot r_{\rm AH}\to 0$, which makes $R_{\rm AH}\propto a(t)$ and $M(t)\propto a(t)$. The argument deliberately uses no field equations, only geometry and the physical-principle exclusions.

Load-bearing premise

The conclusion stands or falls on treating as unphysical every non-comoving outcome—decreasing horizon entropy, a naked null singularity splitting spacetime, and a vanishing apparent horizon—appeals that are asserted principles rather than consequences of the Einstein equations; the argument also assumes a globally expanding background with $a(t)\to\infty$ and that $T_{\rm AH}$ does not grow in the dichotomy leading to Eq. (18).

Editorial extensions

If this is right

If the claim is correct, any spherical black hole apparent horizon in a flat expanding FLRW universe asymptotically satisfies $R_{\rm AH}\propto a(t)$ and $M(t)=M_0 a(t)$, making the mass proportional to the cosmic scale factor.
Cosmological coupling of black hole masses is therefore not a conjecture fitted to data but a kinematic consequence of horizon consistency in an expanding background.
During inflation, when $a(t)$ grows exponentially, the comoving requirement forces $\dot r_{\rm AH}$ to vanish rapidly and the inferred mass growth rate can be very large, providing a fundamental mechanism for early black hole growth.
On timescales short compared with the Hubble time the effect is negligible, so standard astrophysical predictions for isolated black holes remain intact; the new behavior accumulates only over cosmic times.

Reading between the lines

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

One can test the sharpest prediction—$M\propto a(t)$ for every black hole—against redshift-resolved black-hole mass measurements in quiescent galaxies; the paper itself cites tentative observational support, but the geometric argument would make such scaling universal, not environment-dependent.
The same dichotomy might be adapted to non-spherical or rotating apparent horizons: if an area law similar to Eq. (9) holds, the comoving conclusion could extend to Kerr-like holes, though frame dragging and multipole falloff could introduce additional branches.
Relaxing any one of the physical exclusions (allowing transient entropy decrease, or a horizon that disappears) would admit non-comoving histories, so the theorem is only as strong as those principles; a more constructive proof might replace the exclusions with a direct dynamical argument.

Original abstract

It has been shown that static black hole event horizons cannot exist in time-dependent backgrounds without becoming naked singularities. In light of this, we are left with the question ``how does a black hole horizon evolve in an expanding universe?'' Here we offer a solution to this puzzle by showing in full generality that regular black hole apparent horizons tend to become comoving, thus confirming the necessity of a cosmological coupling and raising intriguing possibilities for the early growth of black holes.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,358 input tokens.

What the abstract itself states, as probabilities: claims first 0.08 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.04 · compares with prior results 0.59 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

It has been shown that static black hole event horizons cannot exist in time-dependent backgrounds without becoming naked singularities. In light of this, we are left with the question ``how does a black hole horizon evolve in an expanding universe?'' Here we offer a solution to this puzzle by showing in full generality that regular black hole apparent horizons tend to become comoving, thus confirming the necessity of a cosmological coupling and raising intriguing possibilities for the early growth of black holes.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	It has been shown that static black hole event horizons cannot exist in time-dependent backgrounds without becoming naked singularities.	background 1.00	0.05	0.03	asserted 1.00
s1	In light of this, we are left with the question ``how does a black hole horizon evolve in an expanding universe?'' Here we offer a solution to this puzzle by showing in full generality that regular black hole apparent horizons tend to become comoving, thus confirming the necessity of a cosmological coupling and raising intriguing possibilities for the early growth of black holes.	contribution 0.52	0.93	0.96	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Qualitative evolution of slower than comoving ap

Figures reproduced from arXiv:2608.21222 by the authors.

How the literature uses it

What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

What it cites · 29 sentences without a confident label

Cites as support 1 paper

However, black holes exist for substantial fractions of the age of the universe and tiny fundamental effects could accumulate and add to astrophysical ones (as suggested in [ref] ).

Observational evidence for cosmological coupling of black holes and its implications for an astrophysical source of dark energy · Introduction · 0.98

Cites as background 3 papers

At very early epochs, and especially during inflation, the Hubble scale is much smaller than in the present universe and the separation between black hole and cosmological scales is not as sharp [ref] .

Eliminating the LIGO bounds on primordial black hole dark matter · Introduction · 0.80
At very early epochs, and especially during inflation, the Hubble scale is much smaller than in the present universe and the separation between black hole and cosmological scales is not as sharp [ref] .

Cosmological black holes are not described by the Thakurta metric: LIGO-Virgo bounds on PBHs remain unchanged · Introduction · 0.80
At very early epochs, and especially during inflation, the Hubble scale is much smaller than in the present universe and the separation between black hole and cosmological scales is not as sharp [ref] .

Thakurta metric does not describe a cosmological black hole · Introduction · 0.80

Discussion

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