Fixed static ideal gas or finite fermion regulator preparation

For the classical branch, prescribe a smooth static metric on a neighborhood of $I_{x^0}\times \bar{B}$, where $I_{x^0}\subset \mathbb{R}$ is an open interval and $B\subset {\mathbb{R}}^3$ is an open bounded coordinate box with compact closure and positive volume: $g=-\mathcal{N}(x{)}^2(d{x}^0{)}^2+h_{ij}(x)d{x}^{i}d{x}^j,x^0=ct\mathrm{.}$ Here $h_{ij}$ is a smooth positive-definite spatial metric whose eigenvalues are bounded above and away from zero, and the dimensionless lapse satisfies $0<{\mathcal{N}}_0\le \mathcal{N}(x)\le {\mathcal{N}}_1<\mathrm{\infty }$. The static chart and observer congruence are distinct objects: with $\xi ={\mathrm{\partial }}_{x^0}$, the static unit observer is $u={\mathcal{N}}^{-1}\xi$, satisfying $g(u,u)=-1$, under the conventions of Thermodynamics-GR fields, currents, reference normalization and units. This geometry is prescribed and is not assumed to solve an Einstein equation with gas stress as source.

Take particle mass $m>0$ in kilograms, $c>0$ in metres per second, orthonormal momentum $\widehat{p}\in {\mathbb{R}}^3$ in $\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-1}$, and adopt the massive-particle dispersion function $E(\widehat{p})=\sqrt{m^2{c}^4+c^2\mathrm{\mid }\widehat{p}{\mathrm{\mid }}^2}$ in joules as a model input. The Killing energy is the derived function $H_K(x,\widehat{p})=\mathcal{N}(x)E(\widehat{p})$, and the action reference $h_{\ast }>0$ is in joule-seconds. For a classical ideal gas with no interparticle potential and inverse Killing-energy parameter ${\beta }_K>0$ in inverse joules, choose exactly one of these alternatives: (i) a fixed-particle canonical branch with an integer $M\ge 1$ and the $1\mathrm{/}M!$ phase-counting convention; or (ii) a grand-canonical branch with dimensionless fugacity $z>0$, in which particle number is summed using the same $1\mathrm{/}M!$ convention. The finite proper-phase one-particle integral and the corresponding fixed-particle and grand-canonical normalizers are supplied by Static relativistic finite gas normalization and hydrostatics. If a Killing-normalized chemical parameter ${\mu }_K$ in joules is used for the classical gas, it belongs to the grand-canonical branch and is related by $z=e^{{\beta }_K{\mu }_K}$; the fixed-particle branch has no fugacity parameter. There is no interaction term or gas-induced change of the metric in either branch.

As a distinct finite quantum regulator, one may instead specify a finite mode set $J=\{1,\dots ,M\}$, $M\ge 0$, positive dimensionless lapse values ${\mathcal{N}}_j$, and momenta ${\widehat{p}}_j\in {\mathbb{R}}^3$. Adopt mode energies $E_j={\mathcal{N}}_j\sqrt{m^2{c}^4+c^2\mathrm{\mid }{\widehat{p}}_j{\mathrm{\mid }}^2}$, canonical parameter ${\beta }_K>0$ in inverse joules, and chemical-potential parameter ${\mu }_K\in \mathbb{R}$ in joules, together with the finite occupation basis $n_j\in \{0,1\}$. The associated finite trace and occupation identity are given by Finite static fermion-mode trace and occupation identity. This is an explicitly supplied spectral cutoff, not a claim that arbitrary curved spacetime has these modes. Neither branch reports an observation or asserts actual thermalization.

The metric $h_{ij}$, lapse $\mathcal{N}$, region, $m,c,h_{\ast }$, canonical parameters and (in the finite-mode branch) the mode list are supplied inputs. The local energy, Killing-energy weight, partition functions, normalized state, local densities and response identities are derived constructions under the stated mathematical results; no universal equation of state or metric backreaction is assumed.