Finite Killing energy current and reference normalization — Examples

Use the static subcase, finite coordinate slab, and open chart domain $U$ of Finite Killing energy current and reference normalization. Adopt that lemma's SI convention: $x^0=c{t}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, the spatial coordinates $x^i$ are in metres, $c>0$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, and the metric components and length-normalized Killing field are dimensionless. Let $\xi ={\mathrm{\partial }}_{x^0}$ be future timelike and orthogonal to the coordinate slices, and put ${\mathcal{N}}_{\xi }(q):=\sqrt{-g_q({\xi }_q,{\xi }_q)}>0$ and $u_q:={\xi }_q\mathrm{/}{\mathcal{N}}_{\xi }(q)$ for $q\in U$. Let $W\subset U$ be a specified smooth embedded codimension-one submanifold whose induced metric has signature $(-,+,+)$ (a timelike cavity wall), and choose a reference event $q_{\mathrm{r}\mathrm{e}\mathrm{f}}\in W$ in the static region. Put ${\mathcal{N}}_{\mathrm{r}\mathrm{e}\mathrm{f}}:={\mathcal{N}}_{\xi }(q_{\mathrm{r}\mathrm{e}\mathrm{f}})>0$, which is dimensionless in this convention. No asymptotically flat end is assumed. Supply a function $\vartheta :U\to (0,\mathrm{\infty })$ assigned units kelvins and interpreted as the local proper temperature measured by observer field $u$, and a covector field $p$ on $U$ whose components have units $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$. When a particle-energy interpretation is used, supply a vector field $P$ on $U$ that is future-directed and causal and satisfies $p_q=g_q(P_q,\cdot )$ at each $q\in U$; its components then have the same units as $p$ because the metric is dimensionless. Define, pointwise on $U$, ${\mathcal{T}}_{\xi }(q):={\mathcal{N}}_{\xi }(q)\vartheta (q)$ and $E_{\xi }(p;q):=-c{p}_q({\xi }_q)=-c{p}_a(q){\xi }^a(q)$. Rescale the generator by $\widehat{\xi }:=\xi \mathrm{/}{\mathcal{N}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$. The reference lapse is then ${\widehat{\mathcal{N}}}_{\xi }(q_{\mathrm{r}\mathrm{e}\mathrm{f}})=1$, while the normalized temperature and particle-energy ratios at every $q\in U$ are unchanged: $\frac{{\mathcal{T}}_{\widehat{\xi }}(x)}{{\widehat{\mathcal{N}}}_{\xi }(x)}=\frac{{\mathcal{T}}_{\xi }(x)}{{\mathcal{N}}_{\xi }(x)}=\vartheta (x),\frac{E_{\widehat{\xi }}(p;x)}{{\widehat{\mathcal{N}}}_{\xi }(x)}=\frac{E_{\xi }(p;x)}{{\mathcal{N}}_{\xi }(x)}=-c{p}_a(x)u^a(x)\mathrm{.}$ For each slice charge defined in Finite Killing energy current and reference normalization, $H_{\widehat{\xi }}=H_{\xi }\mathrm{/}{\mathcal{N}}_{\mathrm{r}\mathrm{e}\mathrm{f}}$. This choice is meaningful in a finite cavity without an asymptotic normalization; it does not make $\vartheta$ spatially constant or assert that distinct reference-wall choices produce equal local temperatures.