KMS orbit clock rescaling

Let $\mathcal{A}$ be a unital complex $C^{\ast }$-algebra and let $t\mapsto {\alpha }_t$ be a strongly continuous one-parameter group of $\ast$-automorphisms of $\mathcal{A}$. Let $\omega$ be a ${\beta }_K$-KMS state for $\alpha$ in the physical-time convention of Physical-time KMS strip equilibrium condition, where ${\beta }_K>0$ has units ${\mathrm{J}}^{-1}$ and $\mathrm{\hslash }>0$ has units $\mathrm{J}\mathrm{s}$. For a constant dimensionless $\lambda >0$, define ${\alpha }_{\tau }^{\lambda }:={\alpha }_{\tau \mathrm{/}\lambda },$ where $t$ and $\tau$ are real time parameters in seconds. Then $\omega$ is a $\lambda {\beta }_K$-KMS state for ${\alpha }^{\lambda }$, with strip width $\lambda {\beta }_K\mathrm{\hslash }$ seconds. Conversely, if $\omega$ is a $\widehat{\beta }$-KMS state for ${\alpha }^{\lambda }$, then it is a $(\widehat{\beta }\mathrm{/}\lambda )$-KMS state for $\alpha$.

For the geometric interpretation, let $O$ be a connected open subset of a smooth time-oriented Lorentzian four-manifold $(M,g)$ of signature $(-,+,+,+)$, and let $\xi$ be a smooth future-directed timelike Killing field on $O$ with dimensionless components in length coordinates $x^0=ct$, where $c>0$ is the speed of light in $\mathrm{m}{\mathrm{s}}^{-1}$. Define $N_{\xi }(x):=\sqrt{-g_x({\xi }_x,{\xi }_x)}>0$ and $u:=\xi \mathrm{/}{N}_{\xi }$. For a selected point $x_0\in O$, let $I_K\subseteq \mathbb{R}$ be an interval in seconds containing $0$, and let ${\gamma }_{x_0}:I_K\to O$ be the $C^1$ integral curve of $c\xi$ with ${\gamma }_{x_0}(0)=x_0$, so ${\gamma }_{x_0}^{\mathrm{\prime }}(t_K)=c{\xi }_{{\gamma }_{x_0}(t_K)}$; this is the Killing orbit through $x_0$ with Killing-time parameter $t_K$. For a cell assigned to $x_i$, take $x_0=x_i$ and denote this curve by ${\gamma }_i$. Define its proper-time parameter from the metric by $\tau (t_K):=\frac{1}{c}{\int }_0^{t_K}\sqrt{-g_{{\gamma }_{x_0}(s)}({\gamma }_{x_0}^{\mathrm{\prime }}(s),{\gamma }_{x_0}^{\mathrm{\prime }}(s))}ds,\tau (0)=0.$ By Killing temperature normalization and rigidity, $\xi (N_{\xi })=0$, so $N:=N_{\xi }(x_0)>0$ is constant along this orbit. The metric definition then gives $d\tau =Nd{t}_K$ and, with the chosen origins, $\tau =N{t}_K$; equivalently $d{\gamma }_{x_0}\mathrm{/}d\tau =cu$ for $u=\xi \mathrm{/}{N}_{\xi }$. For $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$ and reference temperature $T_K>0$ in kelvins, the inverse temperature ${\beta }_K=(k_B{T}_K{)}^{-1}$ rescales orbitwise to ${\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}:=N{\beta }_K$; defining $T_{\mathrm{l}\mathrm{o}\mathrm{c}}:=(k_B{\beta }_{\mathrm{l}\mathrm{o}\mathrm{c}}{)}^{-1}$ gives $T_{\mathrm{l}\mathrm{o}\mathrm{c}}=T_K\mathrm{/}N$. No spatially varying lapse is treated as one global time-rescaled automorphism group.