Space cannot stretch too {\it fast}

Samir D. Mathur

REVIEW 5 major objections 6 minor 17 references

Space cannot stretch too {\it fast}

T0 review · 5 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read The paper claims that when spacetime stretches faster than causality lets vacuum entanglement relax, the stored entanglement energy becomes real and turns a collapsing shell into a horizonless fuzzball, resolving the information paradox.

desk verdict A clear speculative essay with a new mechanism, but the central scale postulate is assumed rather than derived, so the cosmological and black hole predictions rest on the same unproved equation. read the letter →

arxiv 2505.10368 v1 pith:7R5FT4AS submitted 2025-05-15 hep-th gr-qc

Samir D. Mathur This is my paper · ORCID

classification hep-thgr-qc PACS 04.70.-s04.60.-m98.80.-k

keywords blackholeinformationparadoxfuzzballsvacuumentanglementfaststretchingdarkenergyHubbletensionsemiclassicalgravitybreakdownnonperturbativefluctuations

topics Dark Energy The Hubble Tension

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

The paper argues that the vacuum of quantum gravity carries virtual imprints of all black-hole microstates, so Planck-scale fluctuations remain entangled across all distance scales with a power-law falloff. When spacetime stretches faster than light can cross the stretched region, these entanglements cannot relax, and the stored entanglement energy becomes real. The paper claims this extra energy is of order $\Delta E \sim R_{\max}/G$, so at the moment a horizon forms it equals the black hole mass $M$ and nucleates a horizonless fuzzball instead of a semiclassical singularity. The same effect at the cosmological horizon gives a dark-energy density of order the closure density and a natural source for early dark energy at the radiation-dust transition. A sympathetic reader would care because this would resolve the information paradox while making a testable connection to cosmology.

What carries the argument

The operative object is a hierarchical spin model of the vacuum: lattice sites carry Planck-scale nonperturbative objects; neighboring pairs form singlets with small triplet admixtures, and triplets at successively larger blocks entangle into higher singlets, producing a hierarchy of entanglements on scales $R_n \sim 2^n l_p$. The identity that does the work is monogamy of entanglement plus the causality constraint that relaxation requires signal exchange, quantified by the scale relation $\Delta E \sim R_{\max}/G$. This scale relation converts a failure to relax into a concrete mass-scale energy and is what turns the toy model into a black-hole and cosmological prediction.

What would settle it

A lattice or nonperturbative computation of the vacuum wavefunctional that shows entanglement between Planck-scale degrees of freedom decays exponentially in separation $R/l_p$ would contradict the power-law postulate behind equation (1). Observationally, a black-hole merger whose post-merger signal is indistinguishable from the standard ringing of a horizon with no prompt near-horizon radiation would count against fuzzball nucleation from fast stretching.

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Extended reading notes

Core claim

The central claim is that the gravitational vacuum is not the empty QFT vacuum: virtual fluctuations of fuzzball microstates imprint correlations between Planck-scale nonperturbative objects at all separations, producing entanglements that fall as a power of the separation rather than exponentially. Because entanglement is monogamous, a fast stretch that inserts new degrees of freedom before signals can pass leaves the old entanglements frozen, and the deficit from the optimal vacuum state carries energy $\Delta E \sim R_{\max}/G$. In gravitational collapse, the light-cone structure inside the horizon prevents relaxation across a region $R_{\max} \sim GM$, so the slice acquires extra energy $\Delta E \sim M$; semiclassical evolution through the good slices is impossible, and the collapse instead spreads over horizonless fuzzball states. The author takes the same mechanism to apply at the cosmological horizon, yielding $\Delta\rho \sim H^2/G \sim \rho_c$ and a burst of early dark energy when the radiation phase gives way to dust.

Load-bearing premise

The argument rests on the relation $\Delta E \sim R_{\max}/G$ being the right energy cost for missing vacuum entanglements; that scale is imported from black-hole thermodynamics rather than derived from the spin model, and if the true vacuum entanglement energy falls off differently the information-paradox resolution and the dark-energy explanation both collapse.

Editorial extensions

If this is right

Black-hole horizons are replaced by fuzzball surfaces, so radiation carries the collapsing shell's information from a normal surface and the information paradox disappears.
Semiclassical gravity fails at low curvature whenever stretching is fast, so Einstein's equations cease to be valid at horizon formation and at the cosmological horizon.
The extra energy at the cosmological horizon has the observed order of magnitude of dark energy, $\Delta\rho \sim H^2/G \sim \rho_c$.
The radiation-to-dust transition changes the entanglement profile and produces early dark energy, offering a mechanism for the Hubble tension.
Slow processes like star formation remain semiclassical because light crosses the system many times during the evolution, so the radical departure is specific to horizon formation.

Reading between the lines

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

The argument suggests a quantitative prediction not spelt out in the essay: the fraction of the closure density contributed by fast stretching should track the ratio $R_p/H^{-1}$, so precision measurements of the dark-energy equation of state could discriminate this mechanism from a cosmological constant.
If the mechanism is right, the transition from semiclassical collapse to fuzzball should be prompt rather than slow, so gravitational-wave ringdowns of black-hole mergers may show horizon-scale deviations or echoes, a testable difference from standard general relativity.
One could test the entanglement-scale relation in analogue systems: a rapidly quenched lattice spin model with long-range power-law couplings should show residual energy set by the quench scale, providing a condensed-matter analogue of equation (1).

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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

Here is my take. The paper has one genuinely new move: it argues that fast stretching of space prevents vacuum entanglement from relaxing, and by monogamy this forces extra energy, ΔE ~ R_max/G. The application to horizon formation and to dark energy/Hubble tension is new relative to Mathur's earlier essays. What it does well: it is clearly written, states the central assumption plainly, and the spin-block picture is a concrete toy for a hard problem.

The soft spot is exactly what the stress-test says. Equation (1) is not derived from the spin model; it is the black hole relation R ~ GM imported as a postulate. The cosmological dark energy density Δρ ~ H^2/G is that same postulate with R_max = H^{-1}, so the headline prediction is the assumption itself. A different coefficient or power law changes both the black hole and cosmological numbers; nothing in the toy model constrains them. The information-paradox conclusion also depends on trusting the fuzzball picture from previous work, which is reasonable within that program but is not a derivation here. I do not think the circularity is fatal for an essay aimed at stimulating thought, but it is fatal for a claim of having resolved the paradox or explained dark energy.

For peer review: I would not desk-reject this. It is a serious speculative essay by someone who knows the subject, and the fast-stretch mechanism is concrete enough to be argued with. A competent referee should be asked to work through whether any independent input fixes the scale in (1). My own verdict would be skeptical unless that missing step appears. If the journal publishes essays, this is appropriate; if it requires derivations, it falls short.

Referee Report

5 major / 6 minor

Summary. This manuscript argues that virtual fluctuations of black hole microstates ('fuzzballs') imprint long-range entanglement on the quantum-gravity vacuum. In the proposed model, Planck-scale spins arranged in hierarchical blocks have entanglement across all scales; when space stretches faster than light can propagate across the relevant scale, the vacuum cannot relax to its optimal entanglement, generating extra energy ΔE ∼ Rmax/G per region of size Rmax. The author applies this to gravitational collapse: inside a horizon, causal disconnection prevents relaxation, so an energy ∼ M appears and the collapse produces a fuzzball rather than a semiclassical black hole, resolving the information paradox. The same mechanism at the cosmological horizon yields Δρ ∼ H^2/G ∼ ρc, proposed as dark energy, with a possible early-dark-energy signal at the radiation-dust transition. The paper is written as an essay and does not present a complete derivation of its central relation.

Significance. If Eq. (1) were a derived consequence of a controlled microscopic model, the proposal would be significant: it would link the black hole information paradox to a concrete failure of semiclassical gravity at low curvature and would connect that failure to the dark energy scale. The essay is clearly written and makes honest contact with the small-corrections theorem and with explicit fuzzball microstate constructions in string theory. It also makes a definite, in-principle falsifiable claim: horizon formation is replaced by fuzzball nucleation, and a horizon-scale energy density of order ρc arises in the dust era. However, the central scale relation is not derived from the spin-lattice model, and no observational constraint is confronted beyond dimensional agreement. The significance is therefore conditional on a missing derivation, and the paper is best read as a speculative proposal rather than a demonstrated result.

major comments (5)

[Modeling the vacuum, Eq. (1)] The load-bearing relation ΔE ∼ Rmax/G is asserted rather than derived. The spin-lattice model specifies only the structural form of the singlet/triplet entanglements; it does not determine the energy of a state in which entanglements above scale Rmax are absent. The text makes this explicit by saying 'We set the extra energy ... to be of the order suggested by the black hole relation R ∼ GM.' Because the later black-hole conclusion ΔE ∼ M uses Rmax ∼ GM and the dark-energy conclusion Δρ ∼ H^2/G uses Rmax ∼ H^{-1}, both results are restatements of Eq. (1) with different choices of Rmax, not consequences of the microscopic model.
[Modeling the vacuum, Eq. (1)] There is a circularity in using R ∼ GM to set the energy cost of imperfect entanglement and then using that cost to infer that a collapsing shell acquires energy ∼ M and forms a fuzzball. The relation R ∼ GM is the very horizon relation that the mechanism is supposed to explain. To avoid circularity, the paper would need an independent derivation of ΔE(Rmax) from the Hamiltonian of the proposed vacuum model, rather than importing the classical black hole relation.
[Cosmology] The dark-energy claim is not checked against quantitative cosmological constraints. The paper states that Δρ ∼ ρc at horizon scales in the dust phase and mentions Big Bang Nucleosynthesis only for the radiation phase, but it does not demonstrate that the extra energy has the required negative-pressure equation of state, that it is compatible with CMB and supernova data, or that the radiation-dust transition yields early dark energy with the correct amplitude. Without such checks, the statement that this is 'of the correct order to account for the dark energy we see today' remains a dimensional coincidence.
[Modeling the vacuum, item (c)] The assertion that the enormous number of fuzzball microstates offsets the mass suppression of virtual fuzzball fluctuations is not quantified. This offset is the justification for replacing exponential falloff of entanglement with a power-law falloff, and it is the physical basis for Eq. (1). The text cites reference [11] for this effect but provides no estimate of the density of states, the coupling of those states to the vacuum wavefunctional, or the resulting power law.
[Fast stretching] No precise criterion for 'too fast' is derived. The paper says that relaxation fails when signals cannot be exchanged between the relevant groups of spins, but the spin model's Hamiltonian and signal speed are unspecified, so the threshold scale Rmax at which relaxation fails is not computed. In the collapse argument, Rmax ∼ GM is assumed rather than derived from the dynamics of the proposed lattice model; this is closely related to the status of Eq. (1) but is a distinct gap in the mechanism.

minor comments (6)

[Abstract] The phrase 'black holes microstates' should be 'black hole microstates' (grammatical correction).
[Page 3, Fig. 2(c) caption] There is a typo in 'fuzzballls'; it should be 'fuzzballs'.
[Page 1] The name is misspelled as 'Galilieo'; it should be 'Galileo'.
[Page 8] The expression 'r /greaterorsimilarH −1' appears to be a rendering artifact; it should read 'r ≳ H^{-1}'.
[Notation] The symbol δ is used both for the triplet admixture in the spin-pair state and for the energy density fluctuation Δρ; using distinct symbols would prevent ambiguity.
[References] Reference [6] is listed as an arXiv preprint with no version number; since it is described as a crucial recent insight, a more complete citation or an expanded presentation of its result would be helpful.

Circularity Check

2 steps flagged · score 8.0 of 10

Central 'predictions' reduce to Eq. (1), a postulate that imports R~GM; the dark-energy and black-hole-energy results are substitutions of the postulate.

self definitional [Section 'Modeling the vacuum', Eq. (1); applied in 'Fast stretching' and 'Cosmology']
"We set the extra energy in the region of size Rmax to be of the order suggested by the black hole relation R ∼ GM: ∆ E ∼ Rmax/G (1)"

The paper later obtains ΔE∼M on the 'good slice' by taking Rmax∼GM and obtains Δρ∼ρc by taking Rmax∼H^{-1}. Both results are obtained by substituting a length into Eq. (1). The relation R∼GM is itself the classical horizon relation that the mechanism is supposed to explain, and the dark-energy density is Eq. (1) with Rmax set to the cosmological horizon length. The spin model supplies a causal story, but not the Rmax/G scale; the 'predictions' are the postulate restated with different Rmax values, not outputs of a first-principles derivation.
uniqueness imported from authors [Introduction, after Fig. 1; see also Refs. [2] and [6]]
"The small corrections theorem has provided a rigorous version of this conﬂict: if usual semiclassical physics holds at leading order around the horizon, then there is no way to prevent this monotonic rise of entanglement [2]. ... a crucial recent insight [6] about the entanglement structure of these ﬂuctuations leads to the complete picture described below."

The no-go theorem that rules out semiclassical evolution and forces a second failure mode is Ref. [2], by the same author, and the 'crucial recent insight' that supplies the new entanglement mechanism is Ref. [6], also by the same author and coauthor. This is a load-bearing self-citation chain: the paper's conclusion that a low-curvature failure is forced and that power-law vacuum entanglement resolves the paradox depends on the author's own prior results. No external proof, machine-checked formalization, or independent benchmark is cited for these key steps.

full rationale

The paper's central quantitative claims are not derived from the hierarchical spin model. Eq. (1) is introduced as 'the order suggested by the black hole relation R ∼ GM' and is explicitly called 'the postulate (1)' later in the text. The two headline quantitative results—ΔE∼M for the collapsing shell and Δρ∼ρ_c for dark energy—are obtained by substituting Rmax∼GM and Rmax∼H^{-1} into this same equation. They are therefore restatements of the input, not outputs of the spin model. The spin model supplies a mechanism for 'extra energy when stretching is fast', but it does not fix the coefficient or the Rmax/G scaling; that scaling is imported from the classical black hole relation that the mechanism is supposed to explain. Load-bearing support is also drawn from the author's own prior work. Ref. [2] supplies the small-corrections theorem that forces a second failure mode, and Ref. [6] supplies the 'crucial recent insight' about entanglement structure; both are self-citations, and no independent verification or external benchmark is cited in the essay for the key step. I am not claiming the fuzzball microstate literature is itself circular; the circularity lies in the specific reductions above. Because the main 'predictions' reduce by construction to Eq. (1) and the remaining mechanism is imported from the author's own no-go theorem and insight, the score is 8.

Assumptions & free parameters 2 free parameters · 5 assumptions · 2 invented entities

All quantitative predictions flow from equation (1), a postulate calibrated to the black hole relation R ∼ GM and then evaluated at the cosmological horizon. The toy spin model supplies a vocabulary but no independent constraint on the energy scale. The paper also leans on the fuzzball program and on the claim that the number of microstates overcomes their mass suppression, both carried by prior self-citations. There are no invented entities with independent falsifiable handles.

free parameters (2)

Scale coefficient in ΔE ∼ Rmax/G (equation 1) = order one, unspecified
Central postulate setting the vacuum entanglement energy; chosen by hand to match the black hole relation R ∼ GM, not derived.
Triplet admixture δ in spin-pair entanglements = small, unspecified
Toy model parameter controlling the hierarchy of correlations; no value or dynamics specified.

assumptions (5)

domain assumption Black hole microstates are horizonless fuzzballs made of strings and branes.
Taken from the string theory constructions cited in [8]; assumed as the starting point for the vacuum-imprint argument.
ad hoc to paper Virtual fuzzball imprints dominate vacuum entanglement despite individual mass suppression.
The text says the number of states offsets the suppression [11] but gives no quantitative comparison; this assumption is needed for the power-law falloff.
ad hoc to paper The energy cost of non-optimal vacuum entanglement is ΔE ∼ Rmax/G.
Equation (1) is the central scale relation; all quantitative black hole and cosmological results derive from it.
standard math Monogamy of entanglement holds for the spin model.
Invoked in the fast-stretching argument: T1 cannot form the required new entanglement with T3 while its entanglement with T2 remains unbroken.
domain assumption Light cones inside a horizon prevent causal contact between neighboring regions on a good slice.
Standard general-relativity causal structure, used to justify that entanglements cannot relax during collapse.

invented entities (2)

Spin-1 nonperturbative objects at every Planck lattice site
purpose: To represent virtual fuzzball imprints in the vacuum wavefunctional
Schematic element of the toy model; no mass, couplings, or observables are specified, and no independent detection handle is proposed.
Hierarchical block entanglements with spin triplets at scales R_n = 2^n l_p
purpose: To produce power-law correlations across length scales
A model construct; its only quantitative output is through the postulated energy scale in equation (1), so it carries no independent evidence.

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Cite this review

Pith. "Pith review of Space cannot stretch too {\it fast}." pith.science (2026). https://pith.science/paper/7R5FT4AS

@misc{pith2026250510368,
  author       = {Pith},
  title        = {Pith review of: Space cannot stretch too \it fast},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7R5FT4AS}},
  note         = {Machine review of arXiv:2505.10368}
}

read the original abstract

We argue that black holes microstates leave an imprint on the gravitational vacuum through their virtual fluctuations. This imprint yields a power law fall off -- rather than an exponential fall off -- for the entanglement of planck scale fluctuations at different points. These entanglements generate an extra energy when space stretches too {\it fast}, since causality prevents a relaxation of these entanglements to their vacuum values. We obtain semiclassical dynamics for slow processes like star formation, but a radical departure from semiclassicality when a black hole horizon forms even though curvatures remain low everywhere. This resolution of the information puzzle also implies an extra energy source at the scale of the cosmological horizon, which may explain the mysteries of dark energy and the Hubble tension.

Figures

Figures reproduced from arXiv: 2505.10368 by the authors.

**Figure 1.** Semiclassical picture of gravitational collapse, in Eddington-Finkelstein coordinates. A shell collapses to a point, generating a singularity. Spacelike slices in this geometry stretch more and more as time evolves; this stretching leads to the creation of entangled pairs and the resulting information paradox. In this essay, we will see how a further fundamental change to our picture of the 1 [PITH_FULL_IMAGE:figu… view at source ↗

**Figure 2.** (a) Small deformations of compact directions give [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

**Figure 3.** A 1-dimensional illustration: (a) Neighboring pa [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: The fate of the collapsing shell, with our new [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

17 extracted references · 3 canonical work pages

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [11]

I. Bena, S. Giusto, R. Russo, M. Shigemori and N. P. Warner, JHEP 05, 110 (2015) [arXiv:1503.01463 [hep-th]]

arXiv 2015

[2] [1]

S. W. Hawking, Commun.\ Math.\ Phys.\ 43 , 199 (1975) [Erratum-ibid.\ 46 , 206 (1976)]; S. W. Hawking, Phys.\ Rev.\ D 14 , 2460 (1976)

1975

[3] [2]

S. D. Mathur, Class.\ Quant.\ Grav.\ 26 , 224001 (2009) [arXiv:0909.1038 [hep-th]]

arXiv 2009

[4] [3]

S. D. Mathur, doi:10.1142/S0218271820300098 [arXiv:2001.11057 [hep-th]]

work page Pith review arXiv 2001

[5] [4]

S. D. Mathur, Int. J. Mod. Phys. D 30, no.14, 2141001 (2021) doi:10.1142/S0218271821410017 [arXiv:2105.06963 [hep-th]]

work page Pith review arXiv 2021

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S. D. Mathur, Int. J. Mod. Phys. D 33, no.15, 2440002 (2024) doi:10.1142/S0218271824400029 [arXiv:2405.08945 [hep-th]]

arXiv 2024

[7] [6]

S. D. Mathur and M. Mehta, [arXiv:2412.09495 [hep-th]]

arXiv

[8] [7]

J. D. Bekenstein, Phys.\ Rev.\ D 7 , 2333 (1973)

work page 1973