Physics theoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Free-vacuum effective acceleration thermometer

Physical scope

Effective temperature in the selected massless Minkowski-vacuum, temporally smeared scalar and two-state Markov model for uniform proper acceleration. The optional Rindler equilibrium comparison requires an explicit reference match; no point-field, all-order detector, empirical, or general Hawking claim is made.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Use the massless acceleration branch of Reduced exponential current and defined spectral detector adoption with proper acceleration $a > 0$ in metres per square second, gap $E > 0$ in joules, $c > 0$ in metres per second, $ℏ > 0$ in joule-seconds, $k_{B} > 0$ in joules per kelvin, and $γ_{a c c} > 0$ in square metres per square second. The adopted rates are $r_{↑} = γ_{a c c} S_{a} (E / ℏ)$ and $r_{↓} = γ_{a c c} S_{a} (- E / ℏ)$ . The unique stationary excited-state probability of the defined two-state model and its effective temperature are

$p_{*} = \frac{1}{1 + e^{2 π c E / (a ℏ)}}, T_{e f f} = \frac{ℏ a}{2 π c k_{B}} .$

For any nonzero real $h \in C_{c}^{\infty} (R)$ and switching duration $T > 0$ seconds, let $Q_{T}^{(a)} (ν) = R_{T}^{(C_{a})} (ν) / (T ∥ h ∥_{2}^{2})$ and $K_{h} = (2 π ∥ h ∥_{2}^{2})^{- 1} \int_{R} ∣ z ∣ ∣ \hat{h} (z) ∣^{2} d z$ , with the definitions in Exact defined Markov thermometer and switched coefficient bound. For every $ν \in R$ , the leading switched coefficient obeys $∣ Q_{T}^{(a)} (ν) - S_{a} (ν) ∣ \leq Lip (S_{a}) K_{h} / T$ . This is a coefficient estimate and does not assert an all-order transition probability.

On the Minkowski right Rindler wedge $W_{R} = {(x^{0}, x^{1}, x^{2}, x^{3}) : x^{1} > ∣ x^{0} ∣}$ , use length coordinates $x^{0} = c t$ , metric $η = diag (- 1, 1, 1, 1)$ , $ρ = \sqrt{(x^{1})^{2} - (x^{0})^{2}} > 0$ , and the boost Killing field $χ_{κ} = κ (x^{1} \partial_{x^{0}} + x^{0} \partial_{x^{1}})$ with $κ > 0$ in inverse metres. Its lapse is $N = \sqrt{- η (χ_{κ}, χ_{κ})} = κ ρ$ . The orbit at fixed $ρ$ has proper acceleration $a (ρ) = c^{2} / ρ$ , so the effective model temperature there is $T_{U} (ρ) = ℏ c / (2 π k_{B} ρ)$ . With reference event on $ρ_{0} = κ^{- 1}$ , $N (ρ_{0}) = 1$ and $T_{K}^{a c c} : = T_{U} (ρ_{0}) = ℏ c κ / (2 π k_{B})$ , hence $T_{U} (ρ) = T_{K}^{a c c} / N (ρ)$ . On the future Rindler horizon $x^{0} = x^{1} > 0$ , direct differentiation gives $\nabla_{χ_{κ}} χ_{κ} = κ χ_{κ}$ , so the generator's $κ_{g e o m}$ is $κ$ under the definition of Thermodynamics-GR fields, currents, reference normalization and units. Rescaling $χ_{κ}$ by $b > 0$ rescales $N$ and $T_{K}^{a c c}$ by $b$ while leaving the proper model temperature unchanged.

If a selected stationary bath of Selected stationary thermal equilibrium is additionally adopted on $W_{R}$ with its selected Killing field identified with this same $χ_{κ}$ , and its reference proper temperature is identified with $T_{U} (ρ_{0})$ , then the bath's local proper temperature agrees with $T_{U} (ρ)$ by Conditional Tolman–Ehrenfest relation. Both identifications are extra model assumptions. No general Unruh-state, KMS, horizon-formation, Hawking-radiation, or empirical thermometer claim follows.

Facts & Assumptions

Given: Countable Choice and the acceleration branch, its coupling and two-state rate equation from Reduced exponential current and defined spectral detector adoption, with $a, E, c, ℏ, k_{B}, γ_{a c c} > 0$ and the stated units.

[A1]

Reduced exponential current and defined spectral detector adoption adopts the massless acceleration branch and rates proportional to the two spectral values. It also states that these are effective rate-model inputs, not exact QFT transition probabilities.

[A2]

Selected stationary thermal equilibrium adopts a smooth positive proper-temperature field on a connected domain with a selected future-timelike Killing field and Killing inverse-temperature vector.

[A3]

For the optional Rindler bath comparison, identify the selected bath's Killing field with $χ_{κ}$ and its reference proper temperature at $ρ_{0} = κ^{- 1}$ with $T_{U} (ρ_{0})$ . These are additional physical model identifications.

[F1]

Constructed timelike smear and massless accelerated spectrum defines the massless field by temporal smearing of a future-directed proper-time hyperbola and gives the tested covariance $C_{a}$ and its conditional spectrum $S_{a} (ν) = ν / [2 π c^{2} (e^{2 π c ν / a} - 1)]$ , its continuous value at zero, and $S_{a} (- ν) = e^{2 π c ν / a} S_{a} (ν)$ for $ν > 0$ . Its actual Statement makes the spectral conclusion conditional on the batch-74 draft suppliers Single-pole thermal contour and exact detailed balance and Positive chiral current and exponential test-coordinate model; the latter also depends on draft Positive smooth damping preserves the double-pole boundary. No proof-acceptance claim is made for this chain.

[F2]

The current mathematical draft Exact defined Markov thermometer and switched coefficient bound proves the exact acceleration-branch two-state solution and temperature, and the finite-switch coefficient estimate above, under the stated Fourier hypotheses. Its accelerated conclusions remain conditional on the draft spectrum input recorded in [F1].

[F3]

Thermodynamics-GR fields, currents, reference normalization and units uses length coordinates and defines $κ_{g e o m}$ only on a supplied smooth null hypersurface and generator satisfying $\nabla_{χ} χ = κ_{g e o m} χ$ .

[F4]

The current draft Killing temperature normalization and rigidity gives $N_{ξ} T = T_{K}$ when the inverse-temperature field of a connected stationary carrier is Killing; this optional use remains conditional on that draft result.

[F5]

The current draft Conditional Tolman–Ehrenfest relation gives the consequent local relation $T = T_{K} / N_{ξ}$ and its constant-rescaling convention; this optional use remains conditional on that draft result.

[F6]

The exponential function is smooth and $(\exp)^{'} = \exp$ gives $(e^{x})^{'} = e^{x}$ , which yields the derivatives of the hyperbolic functions used below.

[A4]

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

1.1A4A1F1F2given

Under the Countable Choice assumption [A4], the acceleration construction in [F1] uses the stated smooth future-directed timelike hyperbola and its actual tested field; the spectrum clause is conditional on the exact current batch-74 contour and exponential-current interfaces in [F1]. Applying [F2] to the rates from [A1] gives the stationary probability $p_{*} = 1 / (1 + e^{2 π c E / (a ℏ)})$ , the Boltzmann ratio $p_{*} / (1 - p_{*}) = e^{- E / (k_{B} T_{e f f})}$ , and $T_{e f f} = ℏ a / (2 π c k_{B})$ . The same supplier gives the stated $K_{h} / T$ bound for every declared switch.

2.1F3F6step 1.1givenalgebra

For $ρ > 0$ define $γ_{ρ} (τ) = (ρ \sinh (c τ / ρ), ρ \cosh (c τ / ρ), 0, 0)$ . Differentiation using [F6] gives $η ({\dot{γ}}_{ρ}, {\dot{γ}}_{ρ}) = - c^{2}$ and $η ({\ddot{γ}}_{ρ}, {\ddot{γ}}_{ρ}) = c^{4} / ρ^{2}$ , so the curve is future timelike, parametrized by proper seconds, with acceleration magnitude $c^{2} / ρ$ . On $W_{R}$ , $x^{1} > ∣ x^{0} ∣$ implies $x^{1} > 0$ , so $χ_{κ}^{0} = κ x^{1} > 0$ ; its norm is $- κ^{2} ρ^{2} < 0$ , hence it is future timelike and $N = κ ρ$ . Since $η_{μ ν}$ is constant in this chart, $(L_{χ_{κ}} η)_{μ ν} = \partial_{μ} χ_{κ ν} + \partial_{ν} χ_{κ μ}$ . Its only nonzero terms are $\partial_{0} χ_{κ 1} = κ$ and $\partial_{1} χ_{κ 0} = - κ$ , which cancel; thus $χ_{κ}$ is Killing by the definition in [F3]. Substituting $a = c^{2} / ρ$ into step 1.1 yields $T_{U} (ρ) = ℏ c / (2 π k_{B} ρ) = T_{K}^{a c c} / N$ with the stated reference normalization.

3.1F3step 2.1algebra

Let $H^{+} : = {x^{0} = x^{1} > 0}$ . The map $ι : (0, \infty) \times R^{2} \to R^{4}$ , $ι (s, y, z) = (s, s, y, z)$ , is a smooth embedding onto $H^{+}$ , and its nonzero conormal $d (x^{0} - x^{1})$ is null since $η^{- 1} (d (x^{0} - x^{1}), d (x^{0} - x^{1})) = - 1 + 1 = 0$ ; thus $H^{+}$ is a smooth embedded null hypersurface. On it $χ_{κ}$ is tangent because $χ_{κ} (x^{0} - x^{1}) = κ (x^{1} - x^{0}) = 0$ , nonzero because $x^{0} = x^{1} > 0$ , and null because $η (χ_{κ}, χ_{κ}) = κ^{2} ((x^{0})^{2} - (x^{1})^{2}) = 0$ . Thus it is a generator in the exact sense of [F3]. Since $η$ has constant components in this chart, its Levi-Civita coefficients vanish; with $χ^{0} = κ x^{1}$ and $χ^{1} = κ x^{0}$ , this gives $(\nabla_{χ} χ)^{0} = κ^{2} x^{0}$ and $(\nabla_{χ} χ)^{1} = κ^{2} x^{1}$ . On $H^{+}$ this equals $κ χ$ , verifying $κ_{g e o m} = κ$ under [F3]. Replacing $χ$ by $b χ$ multiplies $κ, N, T_{K}^{a c c}$ by $b$ , so their ratio at every fixed orbit is unchanged.

4.1A2A3F4F5step 2.1∎

Under the additional bath and reference-match assumptions [A2]–[A3], its lapse is the $N$ of step 2.1 because its Killing field is $χ_{κ}$ . Then [F4]–[F5] give the bath temperature $T_{b a t h} (ρ) = T_{K}^{b a t h} / N (ρ)$ and $T_{K}^{b a t h} = T_{b a t h} (ρ_{0}) = T_{U} (ρ_{0}) = T_{K}^{a c c}$ . Therefore $T_{b a t h} (ρ) = T_{U} (ρ)$ throughout the connected wedge. This is the explicitly assumed comparison of two model constructions.

Free-vacuum effective acceleration thermometer

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources