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arxiv: 2510.20207 · v2 · submitted 2025-10-23 · 🌀 gr-qc · hep-th· quant-ph

Emergent time and more from wavefunction collapse in general relativity

Sung-Sik Lee This is my paper

Pith reviewed 2026-05-18 05:12 UTC · model grok-4.3

classification 🌀 gr-qc hep-thquant-ph
keywords emergent timewavefunction collapsegeneral relativitydark mattergravitonsdiffeomorphism invariancecosmological perturbations
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The pith

Wavefunction collapse in general relativity generates emergent time with the scale factor as a clock and leaves long-wavelength scalar modes as a dark matter candidate.

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

This paper develops a theory in which time arises because quantum states that violate the momentum and Hamiltonian constraints of general relativity are interpreted as instances of time. Stochastic fluctuations in the lapse and shift drive a non-unitary evolution that gradually collapses an initial state toward a diffeomorphism-invariant configuration. During this process the scale factor grows monotonically and thereby serves as a clock that sets the arrow of time. Scalar, vector, and tensor gravitons appear as physical excitations; tensor modes recover unitary dynamics at late times while the extra modes are damped. The scalar mode damps at a rate proportional to its wave number, so that long-wavelength excitations persist and can source long-range gravitational effects.

Core claim

Under the wavefunction collapse the scale factor monotonically increases and acts as a clock. The scalar, vector, and tensor gravitons arise as physical excitations whose time arrow is fixed by the initial state. In the long-time limit the tensor gravitons exhibit emergent unitary dynamics, while the extra modes are strongly damped by the non-unitary dynamics that suppress constraint-violating excitations. The vector mode is uniformly suppressed at all scales, but the decay rate of the scalar mode is proportional to its wave vector; large-wavelength scalar excitations therefore survive over long periods and contribute to long-range interactions, whereas short-wavelength modes decay rapidly.

What carries the argument

Stochastic fluctuations of the lapse and shift that generate the time evolution driving gradual collapse toward a diffeomorphism-invariant state.

If this is right

  • The scale factor serves as a monotonic clock that defines the direction of time.
  • Tensor gravitons recover unitary evolution at late times.
  • Vector modes are suppressed uniformly across all wavelengths.
  • Long-wavelength scalar excitations remain active and can mediate long-range interactions.

Where Pith is reading between the lines

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

  • The same damping mechanism could leave observable imprints in the spectrum of primordial gravitational waves or in late-time structure formation.
  • The construction may be extended to other background cosmologies once the large-dimension limit used here is relaxed.
  • If the scalar mode accounts for dark matter, its long-range tail should produce measurable effects in galaxy clustering at the largest scales.

Load-bearing premise

The assumption that quantum states violating the momentum and Hamiltonian constraints represent instances of time and that stochastic fluctuations of the lapse and shift produce the evolution toward a constraint-satisfying state.

What would settle it

A direct measurement showing that scalar gravitational perturbations with longer wavelengths persist for longer cosmic times while shorter-wavelength modes decay faster, or the absence of such a wavelength-dependent lifetime in high-resolution cosmological data.

Figures

Figures reproduced from arXiv: 2510.20207 by Sung-Sik Lee.

Figure 1
Figure 1. Figure 1: FIG. 1: In this theory, an initial state [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: The original contour of [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3: The dashed lines from left to right represent [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: The complex energies of the excitations for [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: The cubic vertex that produces a scalar of momentum [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
read the original abstract

In this paper, we further develop a recently proposed theory of time based on wavefunction collapse in general relativity. It is based on the postulations that quantum states, which violate the momentum and Hamiltonian constraints, represent instances of time, and stochastic fluctuations of the lapse and shift generate the time evolution under which an initial state gradually collapses toward a diffeomorphism-invariant state. Under the wavefunction collapse, the scale factor monotonically increases, thus acting as a clock. The scalar, vector, and tensor gravitons arise as physical excitations, and the arrow of time for their evolution is set by the initial state. In the long-time limit, the tensor gravitons exhibit emergent unitary dynamics. However, the extra modes are strongly damped due to the non-unitary dynamics that suppress the constraint-violating excitations. The vector mode is uniformly suppressed over all length scales, but the decay rate of the scalar is proportional to its wave vector. This makes the latter a viable candidate for dark matter; excitations with large wavelengths survive over long periods, contributing to long-range interactions, while the fast decay of short-wavelength modes renders them undetectable without sufficient temporal resolution. These are demonstrated for the cosmological constant-dominated universe through semi-classical and adiabatic approximations, which are controlled in the limit of large space dimension.

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

2 major / 2 minor

Summary. The paper proposes a theory of emergent time in general relativity arising from wavefunction collapse, where states violating the momentum and Hamiltonian constraints represent time instances and stochastic fluctuations of the lapse and shift drive non-unitary evolution toward a diffeomorphism-invariant state. The scale factor increases monotonically and serves as a clock. Scalar, vector, and tensor graviton modes emerge as physical excitations; tensor modes recover unitary dynamics at late times while extra modes are damped. The scalar mode decay rate scales with wave vector k, making long-wavelength excitations long-lived candidates for dark matter while short-wavelength modes are rapidly suppressed. These results are obtained for a cosmological-constant-dominated universe via semi-classical and adiabatic approximations asserted to be controlled in the large space-dimension limit.

Significance. If the k-dependent damping and dark-matter interpretation survive beyond the stated approximations, the work supplies a concrete mechanism linking constraint violation, stochastic lapse-shift fluctuations, and observable cosmology without additional free parameters. The derivation of emergent unitary dynamics for tensor modes and the explicit wavelength-dependent suppression for the scalar mode constitute falsifiable predictions that could be tested against structure-formation data once the large-D control is relaxed or quantified.

major comments (2)
  1. Abstract and the section deriving the mode equations: the proportionality of the scalar decay rate to wave vector k (central to the dark-matter candidacy claim) is obtained only under semi-classical and adiabatic approximations whose validity is stated to hold in the large space-dimension limit. No explicit error estimates, higher-order corrections, or comparison with exact solutions in D=3 are supplied, so it remains possible that the k-dependence is an artifact of the limit and does not persist in 3+1 dimensions.
  2. Section introducing the foundational postulates: the identification of constraint-violating states with time and the assumption that stochastic lapse-shift fluctuations generate the collapse dynamics are introduced as axioms. Because the damping rates and the arrow of time for the graviton modes follow directly from these postulates, the manuscript should provide a quantitative check that the resulting non-unitary evolution is consistent with the classical limit of general relativity when the stochastic terms are taken to zero.
minor comments (2)
  1. Ensure uniform notation for the scalar, vector, and tensor modes between the abstract and the main text; the abstract states the vector mode is uniformly suppressed while the scalar decay depends on k, but the precise definitions of these modes should be cross-referenced to the linearized constraint equations.
  2. Add a brief discussion of how the large-D limit relates to the physical 3+1-dimensional case, perhaps via a short appendix sketching the leading 1/D corrections to the decay rates.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. We address each major comment below and indicate the revisions made to strengthen the presentation of the approximations and consistency with the classical limit.

read point-by-point responses

  1. Referee: Abstract and the section deriving the mode equations: the proportionality of the scalar decay rate to wave vector k (central to the dark-matter candidacy claim) is obtained only under semi-classical and adiabatic approximations whose validity is stated to hold in the large space-dimension limit. No explicit error estimates, higher-order corrections, or comparison with exact solutions in D=3 are supplied, so it remains possible that the k-dependence is an artifact of the limit and does not persist in 3+1 dimensions.

    Authors: We agree that the k-proportional decay is derived within the controlled semi-classical and adiabatic approximations of the large-D limit, and that the absence of explicit error estimates leaves open the question of robustness in D=3. The leading k-dependence arises directly from the form of the stochastic damping term in the mode equations, which is independent of the dimension at this order. In the revised manuscript we have added a dedicated paragraph outlining the structure of higher-order 1/D corrections and arguing that they do not modify the linear k-scaling at leading order. A full numerical comparison with exact D=3 solutions lies outside the present analytical treatment but is noted as a natural direction for follow-up work. revision: partial

  2. Referee: Section introducing the foundational postulates: the identification of constraint-violating states with time and the assumption that stochastic lapse-shift fluctuations generate the collapse dynamics are introduced as axioms. Because the damping rates and the arrow of time for the graviton modes follow directly from these postulates, the manuscript should provide a quantitative check that the resulting non-unitary evolution is consistent with the classical limit of general relativity when the stochastic terms are taken to zero.

    Authors: The postulates define the framework and are introduced as such. To address consistency with the classical limit, we have inserted a new subsection that explicitly takes the stochastic amplitude to zero. In this limit the non-unitary terms vanish identically, the evolution equations reduce to the standard constrained Hamiltonian dynamics of classical GR, and both the background scale-factor evolution and the perturbation equations recover their classical forms without additional assumptions. This quantitative reduction is now shown in detail. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from explicit postulates via controlled approximations

full rationale

The paper explicitly postulates that constraint-violating quantum states represent time and that stochastic lapse/shift fluctuations drive non-unitary collapse toward diffeomorphism-invariant states. From these inputs it derives the monotonic increase of the scale factor, the emergence of scalar/vector/tensor modes, and the damping rates (including scalar decay proportional to wave vector) under semi-classical and adiabatic approximations valid in the large-D limit for a cosmological-constant-dominated universe. These steps are not reductions by construction, nor do they rely on fitted parameters renamed as predictions, self-citation chains, or smuggled ansatze; the wavelength-dependent survival of long modes follows from the non-unitary dynamics rather than being presupposed. The large-D control parameter is an external approximation whose validity can be assessed independently, leaving the central claim self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The central claims rest on two explicit postulations about constraint-violating states and stochastic lapse/shift fluctuations; the large-space-dimension limit and semi-classical/adiabatic approximations are additional controlling assumptions rather than free parameters fitted to data.

axioms (2)
  • ad hoc to paper Quantum states violating the momentum and Hamiltonian constraints represent instances of time
    Stated as the foundational postulate in the abstract for the emergence of time.
  • ad hoc to paper Stochastic fluctuations of the lapse and shift generate time evolution under which an initial state gradually collapses toward a diffeomorphism-invariant state
    Postulated mechanism that produces the collapse dynamics and monotonic scale-factor growth.
invented entities (1)
  • Scalar graviton mode as dark matter candidate no independent evidence
    purpose: To account for long-range interactions via surviving long-wavelength excitations while short modes decay rapidly
    Derived from the wave-vector-dependent decay rate under the non-unitary dynamics; no independent falsifiable prediction outside the model is provided in the abstract.

pith-pipeline@v0.9.0 · 5751 in / 1595 out tokens · 39324 ms · 2026-05-18T05:12:52.013339+00:00 · methodology

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  • Foundation/AlexanderDuality.lean alexander_duality_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    These are demonstrated for the cosmological constant-dominated universe through semi-classical and adiabatic approximations, which are controlled in the limit of large space dimension.

  • Foundation/AlexanderDuality.lean D3_admits_circle_linking contradicts
    ?
    contradicts

    CONTRADICTS: the theorem conflicts with this paper passage, or marks a claim that would need revision before publication.

    the decay rate of the scalar is proportional to its wave vector... excitations with large wavelengths survive over long periods

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