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Reduced exponential current and defined spectral detector adoption — Examples

1 · Prerequisites

2 · Summary

This companion compares two proper accelerations in the specified massless Minkowski-vacuum rate model. The factor-of-two temperature relation and the common reference normalization are conditional model calculations; no detector apparatus or measured outcome is asserted.

The sketch uses length coordinates with $x^{1}$ horizontal and $x^{0} = c t$ vertical. Its shaded open wedge is $W_{R} = {x^{1} > ∣ x^{0} ∣}$ ; the dashed rays are its null boundaries $x^{0} = \pm x^{1}$ . The plotted branches are the right-wedge hyperbolae $ρ = \sqrt{(x^{1})^{2} - (x^{0})^{2}}$ at two positive radii. They illustrate the coordinate definitions only; the temperature and acceleration calculations use the explicit formulas in the item proof, not the drawing.

\begin{tikzpicture}[scale=0.78]
  \fill[blue!8] (0,0) -- (3.5,3.5) -- (4.2,3.5) -- (4.2,-3.5) -- (3.5,-3.5) -- cycle;
  \draw[->] (-0.4,0) -- (4.35,0) node[right] {$x^1$};
  \draw[->] (0,-3.8) -- (0,3.8) node[above] {$x^0=ct$};
  \draw[dashed] (0,0) -- (3.65,3.65) node[pos=0.82,above left] {$x^0=x^1$};
  \draw[dashed] (0,0) -- (3.65,-3.65) node[pos=0.82,below left] {$x^0=-x^1$};
  \draw[thick,orange!85!black,domain=-1.25:1.25,samples=60,variable=\x]
    plot ({1.05*cosh(\x)},{1.05*sinh(\x)});
  \draw[thick,red!75!black,domain=-1.25:1.25,samples=60,variable=\x]
    plot ({1.75*cosh(\x)},{1.75*sinh(\x)});
  \node[orange!85!black] at (2.25,1.45) {$\rho_1$};
  \node[red!75!black] at (3.0,2.55) {$\rho_2>\rho_1$};
  \node[blue!50!black] at (3.85,0.45) {$W_R$};
\end{tikzpicture}

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

Thought experimentStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-6-luna)Open item page →

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.