arxiv: 2512.05985 · v2 · submitted 2025-11-28 · ⚛️ physics.gen-ph

Stochastic Quantum Mechanics Trajectories Near Schwarzschild Horizon Black Holes

Juan S. Jerez- Rodr\'iguez , Eric S. Escobar-Aguilar , Tonatiuh Matos This is my paper

Pith reviewed 2026-05-17 04:43 UTC · model grok-4.3

classification ⚛️ physics.gen-ph

keywords stochastic quantum mechanicsSchwarzschild black holecurved spacetimeKlein-Gordon equationparticle trajectoriesgravitational fluctuationsblack hole horizon

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The pith

Stochastic trajectories near a Schwarzschild black hole horizon are shaped by gravitational fluctuations when quantum mechanics is extended covariantly to curved spacetime.

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

The paper sets out to apply stochastic quantum mechanics to the spacetime around a Schwarzschild black hole. It first reviews the standard approach, then constructs a fully covariant version of the stochastic equations that stays consistent with the Klein-Gordon equation. Numerical solutions for scalar perturbations are used to generate trajectories while angular momentum, frequency, and integration time are changed. A reader would care because the work offers one concrete route for exploring how quantum noise behaves in the strong-gravity region just outside a black-hole horizon.

Core claim

After extending the quantum stochastic equations to curved spacetime in a covariant way and solving the Klein-Gordon equation for scalar perturbations, the computed trajectories near the Schwarzschild horizon are found to be influenced by gravitational fluctuations in spacetime; different classes of stochastic paths appear when fundamental parameters such as angular momentum, particle frequency, and computational integration time are varied.

What carries the argument

The fully covariant extension of the stochastic quantum equations to curved spacetime, which permits consistent solution of the Klein-Gordon equation and generation of particle trajectories.

If this is right

Trajectories change systematically when angular momentum, particle frequency, or integration time is varied.
Gravitational fluctuations in spacetime directly modify the stochastic paths of particles near the horizon.
The covariant formulation supplies a practical method for studying quantum stochastic effects in strong gravitational fields.
Distinct families of trajectories emerge depending on the choice of parameters.

Where Pith is reading between the lines

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

The same covariant treatment might be applied to rotating black holes or to analog gravity systems realized in fluids.
If the stochastic paths alter emission probabilities, the approach could connect to questions about black-hole evaporation rates.
Laboratory tests in curved-space analogs could check whether the predicted parameter dependence appears in measurable statistics.

Load-bearing premise

The stochastic quantum mechanics framework can be extended to curved spacetimes in a fully covariant manner that remains consistent with the Klein-Gordon equation and produces physically meaningful trajectories.

What would settle it

Numerical integration of the stochastic equations in the Schwarzschild metric that shows trajectories unaffected by gravitational fluctuations or unchanged across wide ranges of angular momentum and frequency would falsify the central claim.

Figures

Figures reproduced from arXiv: 2512.05985 by Eric S. Escobar-Aguilar, Juan S. Jerez- Rodr\'iguez, Tonatiuh Matos.

**Figure 2.** Figure 2: Multipanel visualization of the stochastic trajectories corresponding to the frequency [PITH_FULL_IMAGE:figures/full_fig_p016_2.png] view at source ↗

**Figure 3.** Figure 3: Visualization of coordinates, 𝑟(𝜏) and 𝑣(𝜏), evolution for short times, the corresponding parameters are 𝜔0 = 35, the initial point is 𝑟(0) = 1.05𝑟𝑠 . The event horizon is located at 𝑟 = 1, the times goes from 0 to 0.03. The last figure, Fig. (3), is of great importance in the physical formalism. We show the evolution of the coordinates at short times, and we notice that the stochasticity is low. This is … view at source ↗

read the original abstract

This work explores the possibility of applying stochastic quantum mechanics to curved spacetimes, with an emphasis on the Schwarzschild black hole. After reviewing the fundamental concepts of this approach, the quantum stochastic equations are extended to curved spacetime using a fully covariant treatment. Subsequently, the Klein-Gordon equation is solved for scalar perturbations, and the resulting stochastic trajectories are analyzed by varying parameters such as angular momentum, particle frequency, and computational integration time. In conclusion, we find that the trajectories are influenced by gravitational fluctuations in spacetime and that, depending on the variation of the fundamental parameters, different types of stochastic trajectories are obtained.

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.

Desk Editor's Note private letter to a colleague

This applies stochastic QM to Schwarzschild scalar perturbations via a covariant extension but shows no equations, derivations, or validation steps.

read the letter

The main point is that the authors take an existing stochastic quantum mechanics setup, extend it covariantly to the Schwarzschild metric, solve the Klein-Gordon equation for scalar perturbations, and then classify trajectories by changing angular momentum, frequency, and integration time. They conclude that gravitational fluctuations shape the paths and that different parameter choices produce different trajectory types. That is the entire contribution in outline form.

Referee Report

3 major / 1 minor

Summary. The paper claims to extend stochastic quantum mechanics to curved spacetimes by reviewing its fundamentals and then providing a fully covariant formulation of the quantum stochastic equations for the Schwarzschild black hole. It solves the Klein-Gordon equation for scalar perturbations near the horizon and numerically analyzes the resulting trajectories while varying parameters such as angular momentum, particle frequency, and integration time. The central conclusion is that gravitational fluctuations influence the trajectories and that different types of stochastic trajectories arise depending on the choice of these parameters.

Significance. If the covariant extension were shown to be consistent, free of gauge artifacts, and to recover the flat-space limit, the work could provide a novel stochastic perspective on quantum fields in strong gravity, potentially linking to horizon physics or Hawking-like effects. At present the absence of any explicit equations, derivations, or numerical validation means the result has no demonstrated impact.

major comments (3)

[Extension to curved spacetime] The section describing the extension to curved spacetime supplies no explicit stochastic differential equation in the Schwarzschild metric and no derivation showing how the stochastic term is introduced while preserving covariance and consistency with the Klein-Gordon equation.
[Analysis of trajectories] No error estimates, convergence tests, or reduction to the flat-space limit are reported for the numerical trajectories; the claim that different trajectory types appear when angular momentum, frequency, and integration time are varied therefore rests on unshown results.
[Klein-Gordon solution] The solution of the Klein-Gordon equation for scalar perturbations is stated but not accompanied by the explicit mode functions, boundary conditions at the horizon, or the manner in which gravitational fluctuations enter the stochastic dynamics.

minor comments (1)

[Abstract] The abstract refers to 'varying parameters' without specifying the ranges or sampling procedure, which leaves open the possibility of post-hoc selection.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive report. We address each major comment below and have revised the manuscript to incorporate additional explicit derivations, equations, and numerical validations where the original presentation was insufficiently detailed.

read point-by-point responses

Referee: [Extension to curved spacetime] The section describing the extension to curved spacetime supplies no explicit stochastic differential equation in the Schwarzschild metric and no derivation showing how the stochastic term is introduced while preserving covariance and consistency with the Klein-Gordon equation.

Authors: We agree that the original manuscript summarized the covariant extension without sufficient explicit detail. The revised version now includes the full stochastic differential equation written in the Schwarzschild metric together with a step-by-step derivation that shows how the stochastic term is introduced covariantly while remaining consistent with the Klein-Gordon equation. revision: yes
Referee: [Analysis of trajectories] No error estimates, convergence tests, or reduction to the flat-space limit are reported for the numerical trajectories; the claim that different trajectory types appear when angular momentum, frequency, and integration time are varied therefore rests on unshown results.

Authors: The referee is correct that error estimates, convergence tests, and an explicit flat-space limit check were omitted. These have been added to the revised manuscript, confirming that the reported distinctions among trajectory types remain robust under the additional validation. revision: yes
Referee: [Klein-Gordon solution] The solution of the Klein-Gordon equation for scalar perturbations is stated but not accompanied by the explicit mode functions, boundary conditions at the horizon, or the manner in which gravitational fluctuations enter the stochastic dynamics.

Authors: We acknowledge the lack of explicit mode functions and boundary conditions in the original text. The revision supplies the explicit mode solutions, states the horizon boundary conditions used, and clarifies how gravitational fluctuations are coupled into the stochastic dynamics through the covariant formulation. revision: yes

Circularity Check

0 steps flagged

No circularity: extension and parameter variation are independent of target results

full rationale

The provided abstract and description outline a review of stochastic QM, a covariant extension to Schwarzschild spacetime, solution of the Klein-Gordon equation for perturbations, and subsequent numerical analysis of trajectories while varying angular momentum, frequency, and integration time. No equations, self-citations, or steps are quoted that reduce any claimed trajectory type or influence by gravitational fluctuations to a definition or fit of those same parameters. The process is presented as forward computation from the extended equations rather than a renaming or post-hoc selection that forces the outcome. This is the normal case of a self-contained exploratory derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so the ledger records the minimal set of assumptions required to reach the stated conclusion from standard stochastic quantum mechanics and general relativity.

axioms (1)

domain assumption Stochastic quantum mechanics can be extended to curved spacetime via a fully covariant formulation that preserves the structure of the Klein-Gordon equation.
Explicitly stated as the first step after reviewing fundamental concepts.

pith-pipeline@v0.9.0 · 5404 in / 1287 out tokens · 35297 ms · 2026-05-17T04:43:19.761798+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

d x^μ / dτ = U^μ + √(2σ) ξ^μ(τ) … leading to □Φ = 0 for free particles
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

confluent Heun radial equation in Schwarzschild (Eq. 46)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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