Conditional Tolman–Ehrenfest relation

Use the connected stationary thermal carrier of Selected stationary thermal equilibrium on $O$. Let $N_{\xi }=\sqrt{-g(\xi ,\xi )}>0$, $u=\xi \mathrm{/}{N}_{\xi }$, and $\beta =u\mathrm{/}(k_{B}T)$ as there, and assume the postulated Killing condition on $\beta$. Then a positive constant $T_K$ exists such that

$N_{\xi }(x)T(x)=T_K\text{and}\beta =\frac{\xi }{k_B{T}_K}(x\in O)\mathrm{.}$

In the same convention, $\xi (N_{\xi })=0$, the observer acceleration satisfies $a=\mathrm{\nabla }\log N_{\xi }$, the expansion and shear vanish, and

$\mathrm{\nabla }\log T=-a\mathrm{.}$

For any chosen reference event $x_{\mathrm{r}\mathrm{e}\mathrm{f}}\in O$, $T_K=N_{\xi }(x_{\mathrm{r}\mathrm{e}\mathrm{f}})T(x_{\mathrm{r}\mathrm{e}\mathrm{f}})$. If the reference normalization has $N_{\xi }(x_{\mathrm{r}\mathrm{e}\mathrm{f}})=1$, then $T_K$ is the local temperature at that event. Rescaling $\xi$ by a constant $b>0$ rescales $N_{\xi }$ and $T_K$ by $b$ while leaving the local scalar $T$ unchanged. The result is a conditional deduction from the selected physical postulate and the stated mathematical supplier; it is not a thermometer law for arbitrary observers or a KMS-existence theorem.

The mathematical supplier is a conditional local field argument. The cited treatments provide independent context; none supplies an empirical observation or proves that a selected bath describes every material.