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Salecker-Wigner-Karolyhazy Gedankenexperiment in light of the self-gravity

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Self-gravity of a quantum clock cuts length-measurement error to the Planck length.

desk verdict Worth reading and worth refereeing: the self-gravity equilibrium argument is new and the arithmetic is right, but the Planck-length conclusion is only as good as Eq. (6), an unverified ansatz at the radius where Newtonian gravity is not valid. read the letter →

arxiv 1908.09582 v1 pith:EB5R3PTH submitted 2019-08-26 gr-qc hep-phhep-th

Michael Maziashvili This is my paper · ORCID
classification gr-qchep-phhep-th
keywords gedankenexperimentlengthmeasurementPlanckself-gravitywave-packetspreadingSchrödinger-NewtonequationquantumclockSchwarzschildbound
topics The Measurement Problem
open problems The Measurement Problem
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 when the gravitational pull a clock's own mass distribution exerts on itself is included in the standard quantum thought experiment for measuring a distance, the wave packet describing the clock stops spreading and settles into a stable size. The balance between quantum spreading and this self-gravity makes the minimum measurable length the Planck length, $\delta l \sim l_P$, instead of the larger distance-dependent value $\delta l \sim l^{1/3} l_P^{2/3}$. If this is right, space-time distances could in principle be pinned down to a single Planck unit regardless of how large the measured distance is. The conclusion follows from a one-dimensional differential equation for the wave-packet radius, and the paper suggests a numerical study of the full Schrödinger-Newton system as the next step.

What carries the argument

The central mechanism is a balance between two opposite accelerations acting on the clock's wave-packet radius $r_{wp}$: the quantum spreading acceleration $1/(4m^2 r_{wp}^3)$ and the Newtonian self-gravitational acceleration $l_P^2 m / r_{wp}^2$. Setting their sum to zero gives the stable equilibrium radius $r_c = 1/(4 l_P^2 m^3)$, which is the minimum of the effective potential $V(r_{wp}) = 1/(8m^2 r_{wp}^2) - l_P^2 m / r_{wp}$. Imposing the black-hole bound $m \lesssim r_c/l_P^2$ on the mass then yields $r_c \sim l_P$ and hence $\delta l \sim l_P$. The paper treats the packet radius as a classical variable obeying this ordinary differential equation, and notes that solving the full Schrödinger-Newton system should give the same equilibrium.

What would settle it

Run a numerical solution of the Schrödinger-Newton equation for a Gaussian initial packet of mass $m$ in otherwise empty space and track $r_{wp}(t)$ over long times. If the width does not stabilize at the predicted $1/(4 l_P^2 m^3)$ and instead spreads or collapses, the claimed reduction of $\delta l$ to $l_P$ fails.

Watch

Extended reading notes

Core claim

Adding the clock's own Newtonian gravity to the measurement problem changes the answer. Without gravity, a Gaussian wave packet broadens so that after time $t$ its width is roughly $r_{wp}(t) \simeq \sqrt{r_c^2 + t^2/(4m^2 r_c^2)}$, and optimizing the initial width against the Schwarzschild constraint gives the known cubic-root floor $\delta l \sim l^{1/3} l_P^{2/3}$. With self-gravity, the radial acceleration is $\ddot{r}_{wp} = 1/(4m^2 r_{wp}^3) - l_P^2 m / r_{wp}^2$, which has a stable equilibrium at $r_c = 1/(4 l_P^2 m^3)$. Requiring the clock's radius to stay above its Schwarzschild radius then forces $r_c \sim l_P$, so the length-measurement uncertainty is reduced to $\delta l \sim l_P$. The paper also checks whether the self-gravitational reduction of the clock's mass changes this conclusion, using two different formulas for the reduced mass, and finds the Planck-length result survives in both cases.

Load-bearing premise

The whole argument rests on treating the wave-packet radius $r_{wp}$ as a classical variable whose expansion is precisely counteracted by Newtonian self-gravity, even at radii as small as the packet's own Schwarzschild radius, where the Newtonian formula is no longer trustworthy.

Editorial extensions

If this is right

  • If the central claim holds, the uncertainty in measuring any distance $l$ is bounded below by about one Planck length, independent of $l$.
  • The clock that achieves this limit is close to a black hole: its wave-packet radius is held at the equilibrium value $r_c = 1/(4 l_P^2 m^3)$, which the Schwarzschild bound forces down to $l_P$.
  • Including the reduction of the clock's mass by self-gravity does not alter the outcome; both candidate mass formulas lead to $r_c \simeq l_P$ for $m \simeq 1/l_P$.
  • The argument predicts that the full Schrödinger-Newton dynamics, not just the classical-radius equation, should exhibit a stationary Gaussian wave packet at the same equilibrium radius.

Reading between the lines

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

  • Extending beyond the paper: if the Planck-length floor is independent of measured distance, searches for space-time fluctuations in interferometers should look for a distance-independent Planck-scale noise rather than the larger $l^{1/3} l_P^{2/3}$ signature.
  • Extending beyond the paper: the stable equilibrium relies on Newtonian gravity acting at radii close to the Schwarzschild radius; a general-relativistic treatment could either shift $r_c$ or destroy the equilibrium, so a relativistic version of the Schrödinger-Newton equation is a sharper test.
  • Extending beyond the paper: a direct numerical experiment with the Schrödinger-Newton equation, starting from a Gaussian packet of mass $m$, would either confirm the packet width stabilizes at $1/(4 l_P^2 m^3)$ or falsify the Planck-length conclusion.
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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 / 3 minor

Summary. The manuscript revisits the Salecker-Wigner-Károlyházy clock argument and claims that once the self-gravity of the clock's Gaussian wave packet is included, the wave-packet spreading is halted at a stable equilibrium radius r_c = 1/(4 l_P^2 m^3) (Eq. (8)). Combining this equilibrium with the Schwarzschild-radius bound m ≲ r_c/l_P^2 yields r_c ~ l_P and hence a length-measurement uncertainty δl ~ l_P, independent of the measured distance l. Section IV considers two heuristic modifications of the gravitating mass m -> m_c and argues that they leave the conclusion unchanged. Section V suggests that the Schrödinger-Newton system with source m_c|ψ|^2 should be studied numerically and closes with a speculative remark about width-dependent incorporation of l_P into quantum mechanics.

Significance. If the central claim holds, the paper would replace the Károlyházy scaling δl ~ l^{1/3} l_P^{2/3} with a Planck-length floor δl ~ l_P, which is significant for phenomenological bounds on space-time fluctuations and for the interpretation of Salecker-Wigner-Károlyházy-type gedanken experiments. The manuscript is transparent: it contains no fitted parameters, the dimensional analysis is clear, and the main calculation is straightforward to check. However, the claimed Planck-length result rests on an effective equation of motion for the wave-packet radius that is not derived from the Schrödinger-Newton system and is evaluated at a radius where the Newtonian gravitational form is not justified. These gaps make the conclusion plausible but not established.

major comments (3)
  1. [§III, Eq. (6)] The central equation of motion for r_wp is not derived from the Schrödinger-Newton system (Eq. (9)). The spreading term is taken from the free Gaussian packet and the gravitational term is the Newtonian acceleration evaluated at r_wp; the two are assumed to be additive and r_wp is treated as a classical variable. The author explicitly writes that it 'seems likely' that solving Eq. (9) gives the same result, and Section V calls for a numerical study as future work. Because Eq. (8) and the subsequent δl ~ l_P conclusion are direct consequences of Eq. (6), the central claim is conditional on an unverified ansatz. A derivation of Eq. (6) from Eq. (9) in a controlled limit, or a numerical solution of Eq. (9), is needed before the claim can be stated as established.
  2. [§III, after Eq. (8)] The equilibrium r_c ~ l_P is evaluated in a regime where the Newtonian expression l_P^2 m/r_wp^2 is not justified. The Schwarzschild bound used to fix the mass places the equilibrium at r_wp ~ r_g ~ l_P, where general-relativistic corrections are of order unity. If the actual self-gravitational force differs from the Newtonian form at this radius, the location and even the existence of the stable equilibrium in Eq. (8) can change. The manuscript should either justify the Newtonian extrapolation to r_wp ~ l_P or show that the δl ~ l_P conclusion is robust to order-one general-relativistic corrections.
  3. [§IV, Eq. (14)] The mass-reduction variants are introduced through Eqs. (10)-(13) on the basis of heuristic identifications of m_c, but the resulting potentials are not derived from a single Hamiltonian or from the Schrödinger-Newton system. The statement that the potential has 'the same qualitative behavior' and the derivation of the equilibrium condition for r_c are asserted rather than demonstrated in detail. Since Section IV is used to argue that mass reduction does not alter the δl ~ l_P result, the argument should be made explicit, or the section should be clearly labeled as heuristic and non-essential to the main claim.
minor comments (3)
  1. [§II, Eq. (3)] The displayed inequality sqrt(r_c^2 + t^2/(4m^2 r_c^2)) ≳ r_c + t/(4 m r_c) has the wrong direction; for nonnegative quantities the sum is an upper bound, not a lower bound. Since only orders of magnitude are involved this does not affect the conclusions, but the notation should be corrected.
  2. [General] The manuscript contains numerous typographical and OCR-style artifacts (for example, '/greaterorsimilar' for ≳, 'Wellenpaket', and inconsistent spacing in equations). The published version should be carefully proofread.
  3. [§V, closing paragraph] The closing claim that incorporating l_P as a function of l_P/r_wp rather than l_P⟨p⟩ is 'physically meaningful' and that the alternative gives 'evidently misleading results' is not substantiated in the text; either a short derivation or a more detailed reference to [33] should be provided.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the δl~l_P conclusion follows from standard quantum spreading, Newtonian self-gravity, and the Schwarzschild bound; the unsolved Schrödinger–Newton system is a rigor gap, not a self-referential input.

full rationale

The derivation chain is not circular. The free wave-packet spreading gives r_wp(t)=sqrt(r_c^2+t^2/(4m^2r_c^2)) and the acceleration (5); the paper then adds the standard Newtonian self-gravitational acceleration l_P^2 m / r_wp^2, attributed to [25], to form Eq. (6). The equilibrium radius (8), r_c=1/(4l_P^2m^3), is combined with the Schwarzschild bound m≲r_c/l_P^2, yielding r_c∼l_P and δl∼l_P. None of these inputs contains δl∼l_P as a free parameter: no quantity is fitted to data and no prediction is a renamed input; l_P is the fixed gravitational coupling. The main weakness is that Eq. (6) is an effective ansatz rather than a solution of the Schrödinger–Newton system (9); the author explicitly says it 'seems likely' that solving (9) gives the same result and Section V calls for a numerical study. That is a correctness/derivation gap, not circularity, because the target result is not assumed in constructing Eq. (6). The two self-citations [33,34] appear only in the closing analogy about modifying the Schrödinger equation by functions of l_P/r_wp and are not load-bearing for the δl∼l_P claim; they do not supply a uniqueness theorem or forbid alternatives. No step reduces by construction to its own input, so the circularity score is 0.

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

The derivation introduces no new constants or fitted parameters: m and r_c are variables of the gedankenexperiment, minimized over, and l_P enters as the standard gravitational coupling. The load-bearing input is the asserted equation of motion (6) for the wave-packet radius, a Gaussian ansatz with Newtonian self-gravity, plus its validity at the black-hole threshold, together with the Schwarzschild constraint. These are stated in the paper but the first is only heuristic.

assumptions (5)
  • ad hoc to paper The wave-packet radius r_wp obeys the classical ODE r̈_wp = 1/(4m^2 r_wp^3) - l_P^2 m/r_wp^2 (Eq. 6).
    Posited in Section III without a derivation from the Schrödinger or Schrödinger-Newton equations. It is a Gaussian-ansatz approximation; the author suggests solving the SN system (Eq. 9) to confirm, so this is the main structural assumption of the paper.
  • ad hoc to paper Newtonian self-gravity, a_g = l_P^2 m/r_wp^2, remains valid at the optimum where r_wp ~ r_g ~ l_P.
    The equilibrium that yields δl ~ l_P sits at the Schwarzschild radius. No relativistic justification is provided for the Newtonian force law in that regime.
  • domain assumption The clock radius must exceed its Schwarzschild radius, r_c > l_P^2 m (Section II).
    Standard general-relativity input from Karolyhazy [16], imported as an inequality constraint on the clock's mass.
  • domain assumption The gravitating mass m_c is given by either Eq. (10) (ADM-based) or Eq. (11) (Duff).
    Section IV imports both expressions from [28-32] and notes that Duff calls Eq. (10) a mistake, then declines to adjudicate. The l_P conclusion is checked against both, which limits the risk.
  • ad hoc to paper The Schrödinger-Newton source may be modified from m|ψ|^2 to m_c|ψ|^2 (Section V).
    Proposed in Section V as a guide for future numerical work, not derived. It is used only to argue that mass reduction does not change the l_P conclusion.

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Cite this review

Pith. "Pith review of Salecker-Wigner-Karolyhazy Gedankenexperiment in light of the self-gravity." pith.science (2026). https://pith.science/paper/EB5R3PTH

@misc{pith2026190809582,
  author       = {Pith},
  title        = {Pith review of: Salecker-Wigner-Karolyhazy Gedankenexperiment in light of the self-gravity},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EB5R3PTH}},
  note         = {Machine review of arXiv:1908.09582}
}
read the original abstract

In Gedankenexperiment mentioned in the title, the imprecision in space-time measurement is related to the spreading of clock's wave-function with the passage of time required for the measurement. Special relativity puts a bound on the measurement time, it cannot be reduced arbitrarily as the signal used for the measurement cannot propagate with speed greater than that of light. In view of this reasoning, one is led to conclude that the clock should be heavy enough to slow down its wave-function from spreading with time. However, the general relativity puts an upper bound on clock's mass, since its size must remain greater then the Schwarzschild radius associated to it. This way one reaches a limit in length measurement. However, as is discussed below, an additional insight into the question comes by taking into account self-gravitational effects. As a result, the uncertainty in length measurement is reduced to the Planck length.

Figures

Figures reproduced from arXiv: 1908.09582 by the authors.

Figure 1
Figure 1. FIG. 1. The potential: 1 [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

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