Reduced exponential current and defined spectral detector adoption

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). This is the mathematical context for the tested-current, Fourier and temporally smeared field constructions used below.

Use the length-coordinate, unit, and reference conventions of Thermodynamics-GR fields, currents, reference normalization and units. Adopt two distinct model branches.

Reduced exponential-current branch. Choose a constant $\kappa >0$ in inverse metres and a smooth Killing field $\xi$ on a declared domain in the TGR framework. The real parameter $u$, measured in metres, is the selected flow parameter for $\xi$; the reference event and normalization of this flow are fixed as part of the model. Set the exponential coordinate $U_{\kappa }(u)=-{\kappa }^{-1}{e}^{-\kappa u}$. Adopt the positive-chiral-current vacuum and tested current $j_{\kappa }$ constructed in Positive chiral current and exponential test-coordinate model. Its mathematical two-point covariance is the tested boundary $C_{\kappa }$ there; use its angular-frequency spectrum $S_{\kappa }$ and detailed-balance convention exactly as stated in Single-pole thermal contour and exact detailed balance. The exponential ray map and the choice of $\kappa$ are model inputs. If $\xi$ and $\kappa$ are identified with a generator and geometric surface gravity of a particular horizon, that identification is an additional physical assumption; no collapse geometry, four-dimensional horizon restriction, scattering factor, or backreaction is derived here.

Massless acceleration branch. Adopt the massless scalar and Minkowski vacuum of Specified bosonic scalar and CAR Dirac model adoption. Use the actually defined temporal smearing along the smooth proper-time hyperbola and its tested covariance from Constructed timelike smear and massless accelerated spectrum. The curve-smeared field is not a sharp-point field. Its accelerated spectrum and finite-switch coefficient are used only with the conditional hypotheses recorded in that lemma and in Exact defined Markov thermometer and switched coefficient bound.

For either branch, select an energy gap $E>0$ joules and define a two-state, continuous-time rate model with ground state $g$, excited state $e$, and model-time parameter $t\ge 0$ measured in seconds. In the reduced branch, no identification of $t$ with proper time along a spacetime curve is assumed. In the acceleration branch, take $t=\tau \ge 0$ along the proper-time hyperbola supplied by Constructed timelike smear and massless accelerated spectrum. In the reduced branch let ${\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}>0$ have units metres per second and set $r_{\uparrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(E\mathrm{/}(\mathrm{\hslash }c))$ and $r_{\downarrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(-E\mathrm{/}(\mathrm{\hslash }c))$. In the acceleration branch let ${\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}>0$ have units square metres per square second and set $r_{\uparrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(E\mathrm{/}\mathrm{\hslash })$ and $r_{\downarrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(-E\mathrm{/}\mathrm{\hslash })$. Here $c>0$ is metres per second, $\mathrm{\hslash }>0$ is joule-seconds, and the frequency arguments have units inverse metres and inverse seconds, respectively. These choices make both rates inverse seconds. The adopted population $p(t)$ of state $e$ is the solution of $p^{\mathrm{\prime }}(t)=r_{\uparrow }(1-p(t))-r_{\downarrow }p(t)$ with supplied $p(0)\in [0,1]$. This differential equation is the entire effective two-state model; the rates are adopted inputs, not exact QFT transition probabilities.

Only when a selected construction also supplies a density state and a normalized finite measurement instrument may an outcome be read using Physical interpretation of a supplied constructed quantum model. No such state, instrument, detector apparatus, or observed outcome is supplied here. The postulate asserts no general thermalization, exact all-order transition probability, KMS state, or experimental temperature reading.

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). Use the reduced branch of Reduced exponential current and defined spectral detector adoption with $\kappa >0$ in inverse metres. Let $E>0$ be an energy in joules, $\mathrm{\hslash }>0$ in joule-seconds, $c>0$ in metres per second, $k_B>0$ in joules per kelvin, and ${\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}>0$ in metres per second. Put $\nu =E\mathrm{/}(\mathrm{\hslash }c)$ and use the adopted rates $r_{\uparrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(\nu )$ and $r_{\downarrow }={\gamma }_{\mathrm{r}\mathrm{e}\mathrm{d}}{S}_{\kappa }(-\nu )$ in that postulate's two-state model. The unique stationary excited-state probability and its ratio are

$p_{\ast }=\frac{1}{1+e^{2\pi E\mathrm{/}(\mathrm{\hslash }c\kappa )}},\frac{p_{\ast }}{1-p_{\ast }}=e^{-2\pi E\mathrm{/}(\mathrm{\hslash }c\kappa )}\mathrm{.}$

Define the model's effective reference temperature $T_{\mathrm{r}\mathrm{e}\mathrm{d}}>0$ by $p_{\ast }\mathrm{/}(1-p_{\ast })=e^{-E\mathrm{/}(k_B{T}_{\mathrm{r}\mathrm{e}\mathrm{d}})}$. Then

$T_{\mathrm{r}\mathrm{e}\mathrm{d}}=\frac{\mathrm{\hslash }c\kappa }{2\pi k_B}\mathrm{.}$

Let $\xi$ 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 $\mathcal{H}$ in the domain of $\xi$, require $\chi :=\xi {\mathrm{\mid }}_{\mathcal{H}}$ to be a nonzero tangent null generator with the same length-flow normalization, and assume ${\mathrm{\nabla }}_{\chi }\chi =\kappa \chi$ on $\mathcal{H}$ in the sense of Thermodynamics-GR fields, currents, reference normalization and units. Under these additional identifications of both generator and constant geometric surface gravity, $T_{\mathrm{r}\mathrm{e}\mathrm{d}}$ is the reference temperature of that reduced horizon model. If, in addition, $\xi$ is future timelike on a connected open domain $O$, a selected stationary equilibrium of Selected stationary thermal equilibrium is adopted there with this same $\xi$, a reference event $x_{\mathrm{r}\mathrm{e}\mathrm{f}}\in O$ has Killing lapse $N_{\xi }(x_{\mathrm{r}\mathrm{e}\mathrm{f}})=1$, and its local reference temperature is identified with $T_{\mathrm{r}\mathrm{e}\mathrm{d}}$, then its local proper-temperature field obeys $T(x)=T_{\mathrm{r}\mathrm{e}\mathrm{d}}\mathrm{/}{N}_{\xi }(x)$ on $O$. These are conditional model deductions; the theorem establishes no horizon formation, general Hawking radiation, KMS state, detector thermalization, or empirical reading.

Assume Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$) as defined in The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$). 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, $\mathrm{\hslash }>0$ in joule-seconds, $k_B>0$ in joules per kelvin, and ${\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}>0$ in square metres per square second. The adopted rates are $r_{\uparrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(E\mathrm{/}\mathrm{\hslash })$ and $r_{\downarrow }={\gamma }_{\mathrm{a}\mathrm{c}\mathrm{c}}{S}_a(-E\mathrm{/}\mathrm{\hslash })$. The unique stationary excited-state probability of the defined two-state model and its effective temperature are

$p_{\ast }=\frac{1}{1+e^{2\pi cE\mathrm{/}(a\mathrm{\hslash })}},T_{\mathrm{e}\mathrm{f}\mathrm{f}}=\frac{\mathrm{\hslash }a}{2\pi c{k}_B}\mathrm{.}$

For any nonzero real $h\in C_c^{\mathrm{\infty }}(\mathbb{R})$ and switching duration $\mathsf{T}>0$ seconds, let $Q_{\mathsf{T}}^{(a)}(\nu )=R_{\mathsf{T}}^{(C_a)}(\nu )\mathrm{/}(\mathsf{T}\mathrm{\parallel }h{\mathrm{\parallel }}_2^2)$ and $K_h=(2\pi \mathrm{\parallel }h{\mathrm{\parallel }}_2^2{)}^{-1}{\int }_{\mathbb{R}}\mathrm{\mid }z\mathrm{\mid }\mathrm{\mid }\widehat{h}(z){\mathrm{\mid }}^{2}dz$, with the definitions in Exact defined Markov thermometer and switched coefficient bound. For every $\nu \in \mathbb{R}$, the leading switched coefficient obeys $\mathrm{\mid }{Q}_{\mathsf{T}}^{(a)}(\nu )-S_a(\nu )\mathrm{\mid }\le \mathrm{Lip}(S_a)K_h\mathrm{/}\mathsf{T}$. This is a coefficient estimate and does not assert an all-order transition probability.

On the Minkowski right Rindler wedge $W_R=\{(x^0,x^1,x^2,x^3):x^1>\mathrm{\mid }{x}^0\mathrm{\mid }\}$, use length coordinates $x^0=ct$, metric $\eta =\mathrm{diag}(-1,1,1,1)$, $\rho =\sqrt{(x^1{)}^2-(x^0{)}^2}>0$, and the boost Killing field ${\chi }_{\kappa }=\kappa (x^1{\mathrm{\partial }}_{x^0}+x^0{\mathrm{\partial }}_{x^1})$ with $\kappa >0$ in inverse metres. Its lapse is $N=\sqrt{-\eta ({\chi }_{\kappa },{\chi }_{\kappa })}=\kappa \rho$. The orbit at fixed $\rho$ has proper acceleration $a(\rho )=c^2\mathrm{/}\rho$, so the effective model temperature there is $T_U(\rho )=\mathrm{\hslash }c\mathrm{/}(2\pi k_B\rho )$. With reference event on ${\rho }_0={\kappa }^{-1}$, $N({\rho }_0)=1$ and $T_K^{\mathrm{a}\mathrm{c}\mathrm{c}}:=T_U({\rho }_0)=\mathrm{\hslash }c\kappa \mathrm{/}(2\pi k_B)$, hence $T_U(\rho )=T_K^{\mathrm{a}\mathrm{c}\mathrm{c}}\mathrm{/}N(\rho )$. On the future Rindler horizon $x^0=x^1>0$, direct differentiation gives ${\mathrm{\nabla }}_{{\chi }_{\kappa }}{\chi }_{\kappa }=\kappa {\chi }_{\kappa }$, so the generator's ${\kappa }_{\mathrm{g}\mathrm{e}\mathrm{o}\mathrm{m}}$ is $\kappa$ under the definition of Thermodynamics-GR fields, currents, reference normalization and units. Rescaling ${\chi }_{\kappa }$ by $b>0$ rescales $N$ and $T_K^{\mathrm{a}\mathrm{c}\mathrm{c}}$ 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 ${\chi }_{\kappa }$, and its reference proper temperature is identified with $T_U({\rho }_0)$, then the bath's local proper temperature agrees with $T_U(\rho )$ 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.