Thought experimentStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Two model acceleration thermometers

Physical scope

A hypothetical comparison of two proper accelerations in the selected massless Minkowski-vacuum two-state model, with no apparatus construction, measured outcome, or universal acceleration-temperature claim.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Fix $E > 0$ joules, $c > 0$ metres per second, $ℏ > 0$ joule-seconds, $k_{B} > 0$ joules per kelvin, a coupling $γ_{a c c} > 0$ in square metres per square second, and a Rindler normalization $κ > 0$ inverse metres. In the selected acceleration model of Free-vacuum effective acceleration thermometer, compare the two proper-time hyperbolae with accelerations $a$ and $2 a$ for any $a > 0$ . Write $T_{e f f} (b) : = ℏ b / (2 π c k_{B})$ for the model's effective temperature at proper acceleration $b > 0$ . Then $T_{e f f} (2 a) = 2 T_{e f f} (a)$ . On the Minkowski right wedge and boost field defined in Free-vacuum effective acceleration thermometer, use the normalized boost observer congruence $u_{κ} : = χ_{κ} / N = χ_{κ} / (κ ρ)$ . Its integral curves at fixed $ρ, x^{2}, x^{3}$ are the boost orbits; the two compared orbits have radii $ρ_{a} = c^{2} / a$ and $ρ_{2 a} = c^{2} / (2 a)$ . Their lapses are $N_{a} = κ ρ_{a}$ and $N_{2 a} = N_{a} / 2$ , and their local effective temperatures satisfy $T_{U} (ρ_{a}) = T_{K}^{a c c} / N_{a}$ and $T_{U} (ρ_{2 a}) = T_{K}^{a c c} / N_{2 a}$ for $T_{K}^{a c c} = ℏ c κ / (2 π k_{B})$ . Rescaling the boost generator by any $b > 0$ changes $N$ and $T_{K}^{a c c}$ by $b$ but leaves both local effective temperatures invariant. These are conditional outputs of the specified effective rate model, not observed detector readings.

Facts & Assumptions

Given: Countable Choice and the two-state acceleration model, Rindler lapse, reference normalization, and temperature formula of Free-vacuum effective acceleration thermometer, with $a, κ, E, c, ℏ, k_{B} > 0$ .

[F1]

Free-vacuum effective acceleration thermometer gives the effective acceleration temperature $T_{e f f} = ℏ a / (2 π c k_{B})$ , the radius-dependent Rindler temperature $T_{U} (ρ) = ℏ c / (2 π k_{B} ρ)$ , and $N = κ ρ$ with $T_{U} (ρ) = T_{K}^{a c c} / N$ . This assigned supplier is currently a draft; its accelerated spectral input remains conditional on the batch-74 draft suppliers named in its facts.

[A1]

The comparison uses the same $E, c, ℏ, k_{B}, γ_{a c c}$ , boost congruence, and reference normalization for both hypothetical systems; it reports no empirical event.

[A2]

Assume Countable Choice The Axiom of Countable Choice ( ${A C}_{ω}$ ), retaining the hypothesis of the mathematical spectral and tested-field suppliers used here.

Proof

1.1A2A1F1givenalgebra

Under the Countable Choice assumption [A2], apply [F1] first at acceleration $a$ and then at $2 a$ . The fixed positive constants cancel in the ratio, so $T_{e f f} (2 a) / T_{e f f} (a) = 2$ .

2.1A1F1step 1.1algebra

On $W_{R}$ , [F1] gives $η (χ_{κ}, χ_{κ}) = - κ^{2} ρ^{2}$ and $χ_{κ}^{0} = κ x^{1} > 0$ , so $u_{κ} = χ_{κ} / (κ ρ)$ is a smooth future-directed unit timelike field. Also $χ_{κ} (ρ^{2}) = 0$ and $χ_{κ} (x^{2}) = χ_{κ} (x^{3}) = 0$ , so its integral curves have constant $ρ, x^{2}, x^{3}$ and define the boost congruence used here. The two selected radii satisfy $ρ_{a} = c^{2} / a$ and $ρ_{2 a} = c^{2} / (2 a) = ρ_{a} / 2$ , hence $N_{2 a} = κ ρ_{2 a} = N_{a} / 2$ . By [F1], $T_{e f f} (a) = T_{U} (ρ_{a})$ and $T_{e f f} (2 a) = T_{U} (ρ_{2 a})$ , each with the common reference $T_{K}^{a c c}$ .

3.1F1step 2.1algebra∎

Under $χ_{κ} \mapsto b χ_{κ}$ with $b > 0$ , [F1] gives $N \mapsto b N$ and $T_{K}^{a c c} \mapsto b T_{K}^{a c c}$ at each orbit; therefore $T_{K}^{a c c} / N$ and each effective local temperature are unchanged. The comparison is entirely conditional on [F1]'s model assumptions.

Two model acceleration thermometers

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources