Selected stationary thermal equilibrium

On a nonempty connected open set $O$ in the smooth, time-oriented Lorentzian four-manifold of Thermodynamics-GR fields, currents, reference normalization and units, select one smooth future timelike Killing field $\xi$ and one smooth scalar $T:O\to (0,\mathrm{\infty })$ in kelvins. Its dimensionless lapse and selected observer congruence are

$N_{\xi }:=\sqrt{-g(\xi ,\xi )}>0,u:=\xi \mathrm{/}{N}_{\xi }\mathrm{.}$

Thus $g(u,u)=-1$ and the physical four-velocity is $cu$ in the length-coordinate convention $x^0=ct$. A reference normalization of $\xi$ is part of the selected model. The local temperature $T$ is assigned to this comoving observer congruence; it is not a coordinate temperature or a value assigned to every observer.

Adopt the inverse-temperature vector

$\beta :=\frac{u}{k_{B}T},$

where $k_B>0$ is Boltzmann's constant in joules per kelvin, and assume that this field is Killing on $O$. In this item, a selected stationary thermal equilibrium means precisely this adopted bath, observer congruence, positive proper-temperature field, and Killing condition on $\beta$. The field $N_{\xi }T$ is not assumed constant here; its constancy is the conclusion of Conditional Tolman–Ehrenfest relation.

This is a model assumption, not a statement that every stationary geometry admits a thermal state or that every material realizes this bath. It asserts no complete Killing flow, KMS state, trace-class stress exponential, collision or thermalization law, Hawking or Unruh temperature, or empirical thermometer reading.