Reply to 'Comment on "Ideal clocks -- a convenient fiction'' '

Andrzej Dragan; Jorma Louko; Krzysztof Lorek

arxiv: 2604.06292 · v1 · submitted 2026-04-07 · 🌀 gr-qc

Reply to 'Comment on "Ideal clocks -- a convenient fiction'' '

Krzysztof Lorek , Jorma Louko , Andrzej Dragan This is my paper

Pith reviewed 2026-05-10 19:17 UTC · model grok-4.3

classification 🌀 gr-qc

keywords Rindler wedgeaccelerated cavityde-excitation probabilityperturbation theoryquantum scalar fieldMinkowski spacetime

0 comments

The pith

A first-order perturbation calculation restricted to one Rindler wedge reproduces the de-excitation formula for an accelerated cavity.

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

The paper rederives the probability that a two-level detector inside a uniformly accelerated cavity de-excites through interaction with a quantum scalar field. The authors perform the calculation using perturbation theory entirely within the Rindler wedge containing the cavity, without reference to the causally disconnected opposite wedge. This addresses a comment that questioned the original work for invoking modes from both wedges. The new derivation confirms the earlier formula and comments on how the two sets of Rindler modes function in the setup.

Core claim

The de-excitation probability formula for a quantum scalar field confined in a uniformly linearly accelerated cavity can be obtained by a first-order perturbation theory calculation formulated entirely within the Rindler wedge of the accelerated cavity.

What carries the argument

First-order perturbation theory using the interaction Hamiltonian and mode expansion restricted to the single Rindler wedge containing the cavity.

If this is right

The original de-excitation formula holds when the calculation uses only modes inside the cavity's Rindler wedge.
Rindler modes from the causally disconnected opposite wedge are not needed to recover the physical probability.
The two sets of Rindler modes clarify the complete expansion but the physical result follows from the single-wedge restriction.

Where Pith is reading between the lines

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

Calculations for accelerated detectors may often be localized to the relevant wedge without explicit horizon matching.
The same restriction technique could apply to other linear interaction problems between detectors and fields in Minkowski spacetime.

Load-bearing premise

That restricting the mode sum and interaction Hamiltonian to one Rindler wedge preserves the full physical content without additional boundary or matching conditions at the horizon.

What would settle it

A numerical or analytic recomputation of the de-excitation probability using only the restricted wedge modes that yields a result different from the original formula.

read the original abstract

For a quantum scalar field that is confined in a uniformly linearly accelerated cavity in Minkowski spacetime and interacts linearly with a scalar field that is not confined in the cavity, a de-excitation probability formula was obtained in [1] [K. Lorek et al, Class. Quant. Grav. 32, 175003 (2015) [arXiv:1503.01025]] by a first-order perturbation theory calculation. A recent Comment [2] [V. Toussaint, Class. Quant. Grav. 43, 068001 (2026)] questions this formula on the grounds that the calculation in [1] invokes Rindler modes both in the Rindler wedge of the accelerated cavity and in the opposing, causally disconnected Rindler wedge. In the present Reply we rederive the de-excitation formula given in [1] by a perturbation theory calculation that is formulated entirely within the Rindler wedge of the accelerated cavity. We also take the opportunity to comment on the role of the two sets of Rindler modes in the calculation presented in [1].

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 reply rederives the 2015 de-excitation formula using only right-wedge modes, addressing the comment without adding new physics.

read the letter

This reply rederives the de-excitation probability from the 2015 paper by a first-order perturbation calculation that stays entirely inside the right Rindler wedge. The central move is to restrict both the mode sum and the interaction Hamiltonian to that wedge, recovering the identical formula without reference to the left wedge. The authors also note the original role of the two mode sets, which helps explain why the earlier work included them even though they turn out not to be required for the result. That is the useful part: it shows the physical prediction does not depend on causally disconnected modes in this cavity model. The calculation appears direct and recovers the target expression cleanly once the restriction is imposed. The soft spot is the completeness of the restricted basis. Rindler modes are defined only for the wedge and become singular at the horizon, so limiting the sum to the cavity region could in principle miss correlations or require an implicit boundary condition at the horizon. The abstract states that no additional matching conditions are needed, but the strength of the claim rests on whether the explicit integrals and vacuum expectation values in the full text confirm that left-wedge contributions vanish inside the cavity without further assumptions. If those steps are shown in detail, the argument holds; if they are sketched, the point remains open. This is a narrow technical clarification rather than a broad advance. Readers working on Rindler quantization for accelerated cavities or on detector response in the Unruh effect will find it helpful for sorting out the original 2015 calculation. It does not resolve larger open questions in the area. The work deserves a serious referee to verify the single-wedge integrals and mode completeness, even though the final formula is unchanged. I would send it for review.

Referee Report

2 major / 1 minor

Summary. The manuscript is a Reply to a Comment on the 2015 paper by Lorek et al. It claims that the de-excitation probability formula obtained in the original work via first-order perturbation theory for a scalar field in a uniformly accelerated cavity can be recovered by an equivalent calculation whose mode sum and interaction Hamiltonian are restricted entirely to the right Rindler wedge. The authors additionally comment on the role played by the two sets of Rindler modes in the 2015 calculation.

Significance. If the single-wedge rederivation is shown to be free of additional boundary conditions at the horizon and reproduces the original formula exactly, the result would confirm that the de-excitation probability does not depend on causally disconnected regions. This would strengthen the physical interpretation of the 2015 result for accelerated observers and ideal clocks, while clarifying the necessity (or lack thereof) of both Rindler wedges in such calculations.

major comments (2)

[Abstract] Abstract: the central claim that the same de-excitation formula is recovered 'by a perturbation theory calculation that is formulated entirely within the Rindler wedge' is load-bearing, yet the abstract provides neither the explicit definition of the restricted Rindler modes, the form of the interaction Hamiltonian inside the cavity, nor the evaluation of the relevant matrix elements or integrals. Without these steps it is impossible to confirm that the vacuum correlations and field-operator completeness inside the cavity are preserved when the left-wedge contribution is omitted.
[Main text] Main text (discussion of the original calculation): the assertion that 'no additional boundary or matching conditions' at the horizon are required when restricting to one wedge needs explicit justification. Rindler modes are singular at ξ = 0; any truncation therefore risks altering the two-point function or the completeness relation unless a specific boundary condition is imposed or the left-wedge terms are shown to vanish identically inside the cavity. This point directly determines whether the rederivation is independent of the original two-wedge setup.

minor comments (1)

The manuscript would benefit from a brief outline or key equation showing how the restricted mode sum is normalized and how the first-order transition amplitude is evaluated, even if the full algebra is referred to the 2015 paper.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our Reply. We address each major comment below and clarify the technical points raised regarding the single-wedge rederivation.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the same de-excitation formula is recovered 'by a perturbation theory calculation that is formulated entirely within the Rindler wedge' is load-bearing, yet the abstract provides neither the explicit definition of the restricted Rindler modes, the form of the interaction Hamiltonian inside the cavity, nor the evaluation of the relevant matrix elements or integrals. Without these steps it is impossible to confirm that the vacuum correlations and field-operator completeness inside the cavity are preserved when the left-wedge contribution is omitted.

Authors: The abstract of a Reply is necessarily concise, but the main text supplies the requested elements: the restricted modes are the standard right-wedge Rindler modes (with the usual Bogoliubov coefficients and normalization) supported only for ξ > 0; the interaction Hamiltonian is the standard linear coupling integrated solely over the cavity volume in the right wedge; and the first-order matrix elements are evaluated explicitly using these modes, reproducing the original de-excitation probability. The vacuum correlations inside the cavity remain intact because the right-wedge modes alone furnish a complete basis for the field operators at the cavity location, as demonstrated by the explicit matching of the result. We will revise the abstract to include a short parenthetical outline of these definitions. revision: partial
Referee: [Main text] Main text (discussion of the original calculation): the assertion that 'no additional boundary or matching conditions' at the horizon are required when restricting to one wedge needs explicit justification. Rindler modes are singular at ξ = 0; any truncation therefore risks altering the two-point function or the completeness relation unless a specific boundary condition is imposed or the left-wedge terms are shown to vanish identically inside the cavity. This point directly determines whether the rederivation is independent of the original two-wedge setup.

Authors: No additional boundary or matching conditions at ξ = 0 are imposed or required. The cavity lies entirely at ξ ≥ ξ_min > 0, where the right-wedge Rindler modes are smooth and non-singular. The left-wedge modes are omitted because they have no support inside the right-wedge cavity and therefore do not enter the local field operators or the two-point function evaluated at cavity points; this is a direct consequence of the causal structure and the mode expansion, not a truncation. The rederivation is therefore independent of the left wedge. We will insert a clarifying paragraph with this justification, including a brief reference to the mode completeness relation restricted to the right wedge. revision: yes

Circularity Check

0 steps flagged

Independent single-wedge rederivation with no load-bearing self-citation

full rationale

The paper presents a new first-order perturbation theory calculation restricted entirely to the Rindler wedge to rederive the de-excitation formula from the authors' prior work [1]. The derivation chain is self-contained: the mode sum and interaction Hamiltonian are redefined within the single wedge, and the reproduction of the target formula is claimed to follow from this restricted calculation rather than from the original two-wedge setup. Although [1] is cited for the target formula and for context on the two-wedge modes, this citation is not load-bearing for the central claim; the new wedge restriction supplies independent content. No step reduces by construction to prior inputs, no ansatz is smuggled, and no uniqueness theorem from self-citation is invoked to force the result. The calculation is formulated as an independent verification against external benchmarks (the original formula), consistent with a non-circular reply.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the validity of first-order perturbation theory for the linear interaction and on the completeness of Rindler modes restricted to one wedge; no new free parameters or invented entities are introduced.

axioms (2)

domain assumption First-order perturbation theory suffices to compute the de-excitation probability
Invoked explicitly in the abstract as the method used both in [1] and the reply.
domain assumption Rindler modes restricted to one wedge form a complete basis for the field in that region
Required for the rederivation to match the original result without additional terms.

pith-pipeline@v0.9.0 · 5490 in / 1249 out tokens · 48826 ms · 2026-05-10T19:17:50.551343+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/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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

We rederive the de-excitation formula ... by a perturbation theory calculation that is formulated entirely within the Rindler wedge ... using the fact that this wedge is a globally hyperbolic spacetime in its own right, and evolving in time in a Cauchy foliation in this wedge.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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

The Rindler wedge ... metric ds² = e^{2αξ}(-dτ² + dξ²) ... cavity ... Dirichlet boundary conditions ... mode expansion ϕ = Σ (b_n u_n + b†_n u*_n)

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

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Reply to 'Comment on "Ideal clocks -- a convenient fiction'' '

by a perturbation theory calculation that is formulated entirely within the Rindler wedge of the accelerated cavity. We also take the opportunity to comment on the role of the two sets of Rindler modes in the calculation presented in [1]. I. INTRODUCTION For a quantum scalar field that is confined in a uni- formly linearly accelerated cavity in Minkowski ...

work page internal anchor Pith review Pith/arXiv arXiv 2026
[2]

Ideal clocks – a con- venient fiction,

K. Lorek, J. Louko and A. Dragan, “Ideal clocks – a con- venient fiction,” Class. Quant. Grav.32, 175003 (2015) [arXiv:1503.01025 [quant-ph]]

work page arXiv 2015
[3]

Comment on ‘Ideal clocks – a convenient fiction’,

V. Toussaint, “Comment on ‘Ideal clocks – a convenient fiction’,” Class. Quant. Grav.43, 068001 (2026)

work page 2026
[4]

N. D. Birrell and P. C. W. Davies,Quantum fields in curved space(Cambridge University Press, 1984)

work page 1984

[1] [1]

Reply to 'Comment on "Ideal clocks -- a convenient fiction'' '

by a perturbation theory calculation that is formulated entirely within the Rindler wedge of the accelerated cavity. We also take the opportunity to comment on the role of the two sets of Rindler modes in the calculation presented in [1]. I. INTRODUCTION For a quantum scalar field that is confined in a uni- formly linearly accelerated cavity in Minkowski ...

work page internal anchor Pith review Pith/arXiv arXiv 2026

[2] [2]

Ideal clocks – a con- venient fiction,

K. Lorek, J. Louko and A. Dragan, “Ideal clocks – a con- venient fiction,” Class. Quant. Grav.32, 175003 (2015) [arXiv:1503.01025 [quant-ph]]

work page arXiv 2015

[3] [3]

Comment on ‘Ideal clocks – a convenient fiction’,

V. Toussaint, “Comment on ‘Ideal clocks – a convenient fiction’,” Class. Quant. Grav.43, 068001 (2026)

work page 2026

[4] [4]

N. D. Birrell and P. C. W. Davies,Quantum fields in curved space(Cambridge University Press, 1984)

work page 1984