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arxiv: 2501.09676 · v2 · submitted 2025-01-16 · 🌀 gr-qc · hep-th· quant-ph

Finite-time Unruh effect: Waiting for the transient effects to fade off

D. Jaffino Stargen This is my paper

Pith reviewed 2026-05-23 05:03 UTC · model grok-4.3

classification 🌀 gr-qc hep-thquant-ph
keywords Unruh effectUnruh-DeWitt detectorfinite timethermalization timetransient effectsaccelerating detectorquantum field in accelerated frames
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The pith

A uniformly accelerating detector reaches the Unruh thermal spectrum only after an exponentially long thermalization time when acceleration is small compared to its energy gap.

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

The paper decomposes the finite-time transition probability rate of a Unruh-DeWitt detector into time-independent thermal terms and oscillatory non-thermal transient terms that depend on the product of energy gap and interaction time. These transients can be averaged away once the interaction time T satisfies ΔE T ≫ 1, independent of acceleration. The authors introduce a non-thermal parameter ε_nt to quantify the size of the transients and solve for the thermalization time τ_th at which ε_nt drops below a small threshold δ. This yields an exponentially large τ_th for small accelerations a ≪ ΔE and a much shorter τ_th inversely proportional to ΔE for large accelerations.

Core claim

For a uniformly accelerating Unruh-DeWitt detector coupled to a massless scalar field over finite proper time T, the transition probability rate is the sum of purely thermal terms independent of T and non-thermal transient terms that oscillate with ΔE T. The transients become negligible after averaging when ΔE T ≫ 1 for any a T and ΔE/a. Defining the non-thermal parameter ε_nt, the time τ_th needed to reach ε_nt = δ is τ_th ∼ (ΔE)^{-1} exp(2π |ΔE|/a)/δ when a ≪ ΔE and τ_th ∼ (ΔE)^{-1}/δ when a ≫ ΔE.

What carries the argument

Decomposition of the finite-time transition probability rate into T-independent thermal terms plus oscillatory non-thermal transient terms controlled by the parameter ε_nt.

If this is right

  • The detector appears thermalized to the Unruh temperature only after waiting time τ_th that grows exponentially with |ΔE|/a at small accelerations.
  • At large accelerations the required interaction time shrinks to order 1/ΔE divided by the allowed non-thermality δ.
  • The non-thermal transients vanish in the long-time limit irrespective of acceleration strength.
  • The paper notes possible ways to reduce the exponentially large τ_th at small accelerations.

Where Pith is reading between the lines

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

  • Laboratory tests of the Unruh effect at modest accelerations would face exponentially long required observation windows.
  • Repeated short-time measurements would need statistical averaging over the oscillatory transients to recover the thermal spectrum.
  • The same decomposition may apply to detectors in other trajectories or in curved backgrounds where exact thermality is not guaranteed.

Load-bearing premise

The non-thermal transient terms oscillate with respect to ΔE T and can be averaged to insignificance once ΔE T is large, regardless of the values of a T and ΔE/a.

What would settle it

Measure the detector transition rate after an interaction time T much smaller than the predicted τ_th and check whether the fractional deviation from the purely thermal rate remains larger than the chosen δ.

read the original abstract

We investigate the transition probability rate of a Unruh-DeWitt (UD) detector interacting with massless scalar field for a finite duration of proper time, $T$, of the detector. For a UD detector moving at a uniform acceleration, $a$, we explicitly show that the finite-time transition probability rate can be written as a sum of purely thermal terms, and non-thermal transient terms. While the thermal terms are independent of time, $T$, the non-thermal transient terms depend on $(\Delta ET)$, $(aT)$, and $(\Delta E/a)$, where $\Delta E$ is the energy gap of the detector. Particularly, the non-thermal terms are oscillatory with respect to the variable $(\Delta ET)$, so that they may be averaged out to be insignificant in the limit $\Delta ET \gg 1$, irrespective of the values of $(aT)$ and $(\Delta E/a)$. To quantify the contribution of non-thermal transient terms to the transition probability rate of a uniformly accelerating detector, we introduce a parameter, $\varepsilon_{\rm nt}$, called non-thermal parameter. Demanding the contribution of non-thermal terms in the finite-time transition probability rate to be negligibly small, \ie, $\varepsilon_{\rm nt}=\delta\ll1$, we calculate the thermalization time -- the time required for the detector to interact with the field to arrive at the required non-thermality, $\varepsilon_{\rm nt}=\delta$, and the detector to be (almost) thermalized with the Unruh bath in its comoving frame. Specifically, for small accelerations, $a\ll\Delta E$, we find the thermalization time, $\tau_{\rm th}$, to be $\tau_{\rm th} \sim (\Delta E)^{-1} \times {\rm e}^{2\pi|\Delta E|/a}/\delta$; and for large accelerations, $a\gg \Delta E$, we find the thermalization time to be $\tau_{\rm th} \sim (\Delta E)^{-1}/\delta$. We comment on the possibilities of bringing down the exponentially large thermalization time at small accelerations, $a\ll\Delta E$.

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 manuscript examines the finite-time transition probability rate of a uniformly accelerated Unruh-DeWitt detector coupled to a massless scalar field over proper time T. It decomposes the rate into T-independent thermal terms plus non-thermal transient terms depending on ΔET, aT and ΔE/a. The transients are stated to be oscillatory in ΔET and therefore average to negligible size for ΔET ≫ 1 irrespective of aT and ΔE/a. A non-thermal parameter ε_nt is introduced; setting ε_nt = δ ≪ 1 yields explicit thermalization times τ_th ∼ (ΔE)^{-1} exp(2π|ΔE|/a)/δ for a ≪ ΔE and τ_th ∼ (ΔE)^{-1}/δ for a ≫ ΔE.

Significance. If the decomposition and the uniform averaging of transients are rigorously established, the results supply concrete, regime-dependent estimates for the proper time needed for an accelerated detector to reach the Unruh thermal state. This is directly relevant to finite-time proposals for detecting the Unruh effect and clarifies the approach to equilibrium in accelerated frames. The explicit scaling with the usual Unruh factor e^{2πΔE/a} for small a is a useful quantitative output.

major comments (2)
  1. [Sections deriving the decomposition and the averaging argument (likely §3–4)] The central claim that non-thermal transients average to insignificant values for ΔET ≫ 1 irrespective of aT and ΔE/a is load-bearing for both quoted τ_th expressions. The amplitude of the oscillatory integrals is set by the accelerated Wightman function integrated over the finite interval; this amplitude carries explicit dependence on aT and ΔE/a. Without explicit bounds or estimates demonstrating uniform suppression across all regimes, the averaging step does not automatically guarantee ε_nt ≪ 1, undermining the derived thermalization times.
  2. [Section introducing ε_nt and deriving τ_th] The definition and evaluation of the non-thermal parameter ε_nt (used to set δ and extract τ_th) must be shown to produce the stated exponential factor for a ≪ ΔE. The manuscript states the final expressions but the intermediate steps connecting the oscillatory integrals to the quoted scaling of τ_th require explicit verification.
minor comments (2)
  1. [Abstract and §2] Notation for the energy gap (ΔE vs. |ΔE|) and the precise definition of the transition rate (probability per unit time or integrated probability) should be stated consistently in the abstract and main text.
  2. [Discussion section] The manuscript would benefit from a brief comparison of the derived τ_th with existing finite-time Unruh results in the literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the potential relevance of our finite-time results. We address each major comment below and will revise the manuscript to strengthen the rigor of the claims.

read point-by-point responses

  1. Referee: [Sections deriving the decomposition and the averaging argument (likely §3–4)] The central claim that non-thermal transients average to insignificant values for ΔET ≫ 1 irrespective of aT and ΔE/a is load-bearing for both quoted τ_th expressions. The amplitude of the oscillatory integrals is set by the accelerated Wightman function integrated over the finite interval; this amplitude carries explicit dependence on aT and ΔE/a. Without explicit bounds or estimates demonstrating uniform suppression across all regimes, the averaging step does not automatically guarantee ε_nt ≪ 1, undermining the derived thermalization times.

    Authors: We agree that the manuscript would be strengthened by explicit bounds demonstrating that the amplitude of the transients remains controlled for large ΔET independently of aT and ΔE/a. While the decomposition into T-independent thermal terms and oscillatory transients in ΔET is derived explicitly, the original text relies on the rapid oscillation to argue for averaging without supplying uniform estimates. In the revision we will add an appendix that applies integration by parts to the finite-time integrals of the accelerated Wightman function, yielding explicit O(1/ΔET) bounds whose prefactors are shown to be independent of aT and ΔE/a in the regimes relevant to the thermalization times. revision: yes

  2. Referee: [Section introducing ε_nt and deriving τ_th] The definition and evaluation of the non-thermal parameter ε_nt (used to set δ and extract τ_th) must be shown to produce the stated exponential factor for a ≪ ΔE. The manuscript states the final expressions but the intermediate steps connecting the oscillatory integrals to the quoted scaling of τ_th require explicit verification.

    Authors: The exponential scaling for a ≪ ΔE follows because the thermal contribution itself is suppressed by the Unruh factor e^{-2π|ΔE|/a}; the non-thermal transients must therefore be suppressed below this exponentially small level, which requires T to be exponentially large. We will expand the section defining and evaluating ε_nt to include the intermediate steps: the explicit form of the integrated transient terms in the small-a limit, the ratio that defines ε_nt, and the inversion that produces τ_th ∼ (ΔE)^{-1} exp(2π|ΔE|/a)/δ. These steps will be written out in full in the revised manuscript. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation follows from explicit integration of finite-time detector response

full rationale

The paper computes the finite-time transition rate via direct integration of the Unruh-DeWitt detector coupling to the field over proper time interval T, decomposes the result into T-independent thermal pieces plus oscillatory transients in (ΔET) whose amplitudes depend on (aT) and (ΔE/a), and defines τ_th by the condition ε_nt=δ on the size of those transients. This chain is self-contained in the model definitions and standard QFT integrals; no parameter is fitted to data and then relabeled as a prediction, no self-citation supplies a load-bearing uniqueness theorem, and the thermalization time is not smuggled in by prior ansatz. The skeptic concern about amplitude dependence affects correctness of the 'irrespective' claim but does not create a definitional or self-referential reduction.

Axiom & Free-Parameter Ledger

3 free parameters · 2 axioms · 0 invented entities

The central claim rests on the standard Unruh-DeWitt detector model coupled to a massless scalar field in Minkowski spacetime plus the assumption that the finite-time rate admits a clean separation into T-independent thermal and oscillatory transient pieces. No new entities are postulated.

free parameters (3)
  • δ
    Threshold value chosen by hand to define when non-thermality ε_nt is acceptably small; directly sets the numerical value of τ_th.
  • a
    Uniform proper acceleration; physical input parameter of the trajectory.
  • ΔE
    Detector energy gap; physical input parameter of the two-level system.
axioms (2)
  • domain assumption The finite-time transition probability rate decomposes into purely thermal terms independent of T and non-thermal transient terms that depend on (ΔET), (aT) and (ΔE/a).
    Invoked to obtain the separation used for all subsequent results.
  • domain assumption Non-thermal transient terms are oscillatory in (ΔET) and become insignificant after averaging when ΔET ≫ 1 regardless of aT and ΔE/a.
    Used to justify the definition of thermalization time via ε_nt = δ.

pith-pipeline@v0.9.0 · 5936 in / 1713 out tokens · 58744 ms · 2026-05-23T05:03:23.559349+00:00 · methodology

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Reference graph

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