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Reduced quantum horizon temperature calibration

Physical scope

Reference temperature calibration in the selected reduced exponential-current and two-state Markov model. A local stationary temperature formula applies only after the stated equilibrium assumption and reference match; no general Hawking or KMS theorem is asserted.

Statement

Assume Countable Choice ( ${A C}_{ω}$ ) as defined in The Axiom of Countable Choice ( ${A C}_{ω}$ ). Use the reduced branch of Reduced exponential current and defined spectral detector adoption with $κ > 0$ in inverse metres. Let $E > 0$ be an energy in joules, $ℏ > 0$ in joule-seconds, $c > 0$ in metres per second, $k_{B} > 0$ in joules per kelvin, and $γ_{r e d} > 0$ in metres per second. Put $ν = E / (ℏ c)$ and use the adopted rates $r_{↑} = γ_{r e d} S_{κ} (ν)$ and $r_{↓} = γ_{r e d} S_{κ} (- ν)$ in that postulate's two-state model. The unique stationary excited-state probability and its ratio are

$p_{*} = \frac{1}{1 + e^{2 π E / (ℏ c κ)}}, \frac{p_{*}}{1 - p_{*}} = e^{- 2 π E / (ℏ c κ)} .$

Define the model's effective reference temperature $T_{r e d} > 0$ by $p_{*} / (1 - p_{*}) = e^{- E / (k_{B} T_{r e d})}$ . Then

$T_{r e d} = \frac{ℏ c κ}{2 π k_{B}} .$

Let $ξ$ be the same selected reference Killing field and $u$ its length flow parameter from Reduced exponential current and defined spectral detector adoption. For the horizon interpretation, additionally supply a smooth embedded null hypersurface $H$ in the domain of $ξ$ , require $χ : = ξ ∣_{H}$ to be a nonzero tangent null generator with the same length-flow normalization, and assume $\nabla_{χ} χ = κ χ$ on $H$ in the sense of Thermodynamics-GR fields, currents, reference normalization and units. Under these additional identifications of both generator and constant geometric surface gravity, $T_{r e d}$ is the reference temperature of that reduced horizon model. If, in addition, $ξ$ is future timelike on a connected open domain $O$ , a selected stationary equilibrium of Selected stationary thermal equilibrium is adopted there with this same $ξ$ , a reference event $x_{r e f} \in O$ has Killing lapse $N_{ξ} (x_{r e f}) = 1$ , and its local reference temperature is identified with $T_{r e d}$ , then its local proper-temperature field obeys $T (x) = T_{r e d} / N_{ξ} (x)$ on $O$ . These are conditional model deductions; the theorem establishes no horizon formation, general Hawking radiation, KMS state, detector thermalization, or empirical reading.

Facts & Assumptions

Given: Countable Choice and the reduced branch and rate equation of Reduced exponential current and defined spectral detector adoption, with $κ, E, ℏ, c, k_{B}, γ_{r e d} > 0$ and the units stated above.

[A1]

Reduced exponential current and defined spectral detector adoption adopts $r_{↑} = γ_{r e d} S_{κ} (E / (ℏ c))$ , $r_{↓} = γ_{r e d} S_{κ} (- E / (ℏ c))$ , and the differential equation for $p (t)$ ; the rates are model inputs.

[A2]

Selected stationary thermal equilibrium adopts a smooth positive proper-temperature field on a connected domain with a selected future-timelike Killing field $ξ$ and Killing inverse-temperature vector $β$ . The optional use here additionally identifies that selected field with the reduced branch's reference field.

[A3]

For the optional local-temperature clause, additionally take $x_{r e f} \in O$ , identify the selected bath's proper temperature there with $T_{r e d}$ , and normalize $N_{ξ} (x_{r e f}) = 1$ . This is an explicit model identification, not a deduction from the rate equation.

[F1]

Single-pole thermal contour and exact detailed balance states that for $ν > 0$ , $S_{κ} (ν) > 0$ and $S_{κ} (- ν) = e^{2 π ν / κ} S_{κ} (ν)$ , with $S_{κ} (0) = κ / (4 π)$ as a continuous value. This mathematical supplier, its prerequisite Positive chiral current and exponential test-coordinate model, and that item's prerequisite Positive smooth damping preserves the double-pole boundary currently have status draft; the exact chiral spectrum input is conditional on these statements and their proofs, with no independent proof acceptance claimed.

[F2]

Thermodynamics-GR fields, currents, reference normalization and units fixes $x^{0} = c t$ length coordinates, $k_{B}$ in joules per kelvin, $ℏ$ in joule-seconds, $N_{ξ} = \sqrt{- g (ξ, ξ)}$ , and the conditional definition of $κ_{g e o m}$ by $\nabla_{χ} χ = κ_{g e o m} χ$ on a supplied smooth null hypersurface.

[F3]

The current draft Killing temperature normalization and rigidity states and proves that on the connected stationary carrier, the Killing condition on $β$ gives $N_{ξ} T = T_{K}$ for one positive constant; this optional use remains conditional on that draft result.

[F4]

The current draft Conditional Tolman–Ehrenfest relation applies that carrier and states $T (x) = T_{K} / N_{ξ} (x)$ , with the reference value $T_{K} = N_{ξ} (x_{r e f}) T (x_{r e f})$ for $x_{r e f} \in O$ . This optional use remains conditional on that draft result.

[F5]

The exponential function is smooth and $(\exp)^{'} = \exp$ gives $(e^{x})^{'} = e^{x}$ .

[F6]

The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ implies that a differentiable function with zero derivative on an interval is constant there.

[F7]

The exponential is positive and satisfies $\exp (- x) = 1 / \exp (x)$ gives $e^{x} > 0$ for every real $x$ . Together with [F5], this makes the derivative of $x \mapsto e^{x}$ positive; [F6] then implies that the exponential is strictly increasing.

[A5]

For the horizon interpretation only, supply $H$ and identify its generator with $χ = ξ ∣_{H}$ , with the same length-flow normalization and $\nabla_{χ} χ = κ χ$ . These are additional model assumptions; equality of surface-gravity values alone does not identify the generators.

[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

1.1A4A1F1givenalgebra

Under the Countable Choice assumption [A4], since $E, ℏ, c, κ > 0$ , $ν = E / (ℏ c) > 0$ . By [F1], both $S_{κ} (ν)$ and $S_{κ} (- ν)$ are positive and their ratio is $S_{κ} (- ν) / S_{κ} (ν) = e^{2 π ν / κ}$ . Multiplying by the same $γ_{r e d} > 0$ gives $r_{↑}, r_{↓} > 0$ and $r_{↓} / r_{↑} = e^{2 π E / (ℏ c κ)}$ .

2.1A1F5F6F7step 1.1algebra

Put $R = r_{↑} + r_{↓}$ and $p_{*} = r_{↑} / R$ . Then $0 < p_{*} < 1$ , and $p (t) = p_{*} + (p (0) - p_{*}) e^{- R t}$ satisfies the adopted equation and initial condition by [F5]. Since $R > 0$ , [F5] and [F7] give $0 < e^{- R t} \leq 1$ for $t \geq 0$ ; hence $p (t) = e^{- R t} p (0) + (1 - e^{- R t}) p_{*}$ is a probability. For any other differentiable solution, $q (t) = e^{R t} (p (t) - p_{*})$ has zero derivative by the rate equation and [F5]; [F6] makes $q$ constant on every compact time interval, so the displayed solution is unique. Its stationary value is therefore $p_{*}$ .

3.1A5F2F1step 1.1step 2.1F5F6F7algebra

From $p_{*} = r_{↑} / R$ and $1 - p_{*} = r_{↓} / R$ , we have $p_{*} / (1 - p_{*}) = r_{↑} / r_{↓} = e^{- 2 π E / (ℏ c κ)}$ by step 1.1. By [F5] and [F7], the exponential has positive derivative, so [F6] makes it strictly increasing; comparison with the defining Boltzmann ratio gives $E / (k_{B} T_{r e d}) = 2 π E / (ℏ c κ)$ . Since $E > 0$ , solving yields $T_{r e d} = ℏ c κ / (2 π k_{B})$ , which has kelvin units. Under [A5], the generator and its constant geometric surface gravity are exactly those of the selected reduced branch, so this reference scale has the stated reduced-horizon interpretation.

4.1A2A3F2F3F4step 3.1algebra∎

Under the optional assumptions [A2]–[A3], [F3]–[F4] give $T (x) = T_{K} / N_{ξ} (x)$ on $O$ . Evaluating at $x_{r e f}$ and using $N_{ξ} (x_{r e f}) = 1$ and [A3] sets $T_{K} = T_{r e d}$ . Substitution gives $T (x) = T_{r e d} / N_{ξ} (x)$ . The result is conditional on the separately adopted bath and reference match.

Reduced quantum horizon temperature calibration

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources