Stationary thermal geometric data

Let $(M,g,N_0)$ be a smooth time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$ in the sense of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, and let $\mathrm{\nabla }$ be its unique torsion-free $g$-compatible connection supplied by Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Fix a nonempty connected open set $O\subseteq M$. A selected stationary thermal carrier on $O$ consists of a smooth future timelike vector field $\xi$ and a smooth positive scalar field $T:O\to (0,\mathrm{\infty })$, where $T$ is measured in kelvins, such that $\xi$ is a Killing field of $g$ in the sense of Killing fields of a Lorentzian metric. Equivalently, the criterion is ${\mathrm{\nabla }}_{(a}{\xi }_{b)}=0$ by Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor. The round brackets mean symmetrization with factor $1\mathrm{/}2$. The choice of $\xi$ specifies the thermal congruence; no coordinate chart or universal observer field is selected.

Set $N_{\xi }:=\sqrt{-g(\xi ,\xi )}>0$ and $u:=\xi \mathrm{/}{N}_{\xi }$. Then $g(u,u)=-1$. In the length-coordinate convention $x^0=ct$, $u$ is dimensionless and $cu$ is the physical four-velocity, with $c>0$ in $\mathrm{m}{\mathrm{s}}^{-1}$. For a fixed Boltzmann constant $k_B>0$ in $\mathrm{J}{\mathrm{K}}^{-1}$, define the inverse-temperature vector $\beta :=u\mathrm{/}(k_{B}T)$, with units ${\mathrm{J}}^{-1}$, and the Killing-normalized temperature field $T_K:=N_{\xi }T$. Thus $\beta =\xi \mathrm{/}(k_B{T}_K)$; this definition does not assert that $T_K$ is constant.

For a smooth chemical-potential scalar $\mu :O\to \mathbb{R}$ in joules, put $\zeta :=\mu \mathrm{/}(k_{B}T)$, a dimensionless scalar. Define the rest-space metric, derivative, acceleration, expansion and shear by $h_{ab}:=g_{ab}+u_a{u}_b,D:=u^a{\mathrm{\nabla }}_a,a_b:=u^a{\mathrm{\nabla }}_a{u}_b,\theta :={\mathrm{\nabla }}_a{u}^a,{\sigma }_{ab}:=h_a{}^c{h}_b{}^d{\mathrm{\nabla }}_{(c}{u}_{d)}-\frac{1}{3}\theta h_{ab}\mathrm{.}$ Here $a$, $\theta$ and $\sigma$ have inverse-length units; the physical proper acceleration is $c^{2}a$. The smooth tangent-bundle structure used by the cited GR suppliers assumes the Axiom of Countable Choice The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$); no stronger choice principle is part of this definition.

The Killing characterization cited in the Definition is applied locally. For each $p\in O$, take a chart domain $U\subseteq O$ containing $p$ and orient $U$ by its ordered chart coordinates. The restrictions of $g$ and $N_0$ make $U$ an oriented time-oriented Lorentzian four-manifold, and $\mathrm{\nabla }{\mathrm{\mid }}_U$ is its metric-selected connection by uniqueness. Clause (a) of Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor therefore gives ${\mathcal{L}}_{\xi {\mathrm{\mid }}_U}(g{\mathrm{\mid }}_U)=0$ if and only if ${\mathrm{\nabla }}_{(a}{\xi }_{b)}{\mathrm{\mid }}_U=0$. Both tensors are restrictions of the corresponding tensors on $O$, so their vanishing is equivalent on $O$. No global orientation, or simultaneous choice of chart orientations, is required.

The data are mathematical fields on a connected open Lorentzian domain. No complete Killing flow, thermal state, KMS condition, material realization or empirical thermometer law is asserted.

For the adopted length-coordinate convention, take the spacetime coordinates (including $x^0=ct$) and the local flow parameter of $\xi$ in length units, and take the metric components to be dimensionless. Then the coordinate components of $\xi$ and $N_{\xi }=\sqrt{-g(\xi ,\xi )}$ are dimensionless, so $T_K=N_{\xi }T$ is measured in kelvins. A constant rescaling $\xi \mapsto b\xi$ uses a dimensionless $b>0$.