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Constructed static gas redshift and hydrostatics

Physical scope

Exact classical ideal-gas normalization, redshifted local thermodynamic parameters, pressure and hydrostatic identity on the prescribed bounded-lapse static background, plus the separate finite-mode trace identity. No Einstein backreaction, generic curved-space mode spectrum, self-gravitating star, or empirical equilibrium assertion.

Statement

Assume the prescribed static background in Fixed static ideal gas or finite fermion regulator preparation and consider its two alternative classical branches. Let $Z_{1} (β_{K}) = \int_{B} \int_{R^{3}} e^{- β_{K} N (x) E (\hat{p})} \frac{d V_{h} (x) d^{3} \hat{p}}{h_{*}^{3}}, d V_{h} = \sqrt{\det (h_{i j} (x))} d^{3} x .$ Then $0 < Z_{1} < \infty$ . In the fixed-particle canonical branch, for the selected integer $M \geq 1$ , the normalizer is $Z_{M} = Z_{1}^{M} / M!$ . In the grand-canonical branch, for dimensionless fugacity $z > 0$ , the normalizer is $Ξ = e^{z Z_{1}}$ .

For either classical branch, define its coefficient $α$ by $α = M / Z_{1}$ in the fixed-particle branch and $α = z$ in the grand-canonical branch. Its proper local expected particle phase-space density is $f_{α} (x, \hat{p}) = α h_{*}^{- 3} e^{- β_{K} N (x) E (\hat{p})} .$ Define proper local number density $n = \int f_{α} d^{3} \hat{p}$ , energy density $ε = \int E f_{α} d^{3} \hat{p}$ , and isotropic pressure $P = \frac{1}{3} \int \hat{p} \cdot \nabla_{\hat{p}} E f_{α} d^{3} \hat{p}$ . With $T_{K} = (k_{B} β_{K})^{- 1}$ , both branches satisfy $T_{l o c} (x) = \frac{T_{K}}{N (x)}, P = n k_{B} T_{l o c}, \partial_{i} P = - (ε + P) \partial_{i} \log N .$ Only in the grand-canonical branch, if $z = e^{β_{K} μ_{K}}$ for a Killing-normalized chemical parameter $μ_{K} \in R$ measured in joules, rewriting the local weight as $e^{- β_{K} N (E - μ_{l o c})}$ gives $μ_{l o c} = μ_{K} / N$ .

In the finite fermion-mode branch, let $E_{j} = N_{j} ϵ_{j}$ with $ϵ_{j} = \sqrt{m^{2} c^{4} + c^{2} ∣ {\hat{p}}_{j} ∣^{2}}$ , $N_{j} > 0$ , $β_{K} > 0$ , and $μ_{K} \in R$ . The normalized diagonal state has $⟨ n_{j} ⟩ = \frac{1}{e^{β_{K} (E_{j} - μ_{K})} + 1} = \frac{1}{e^{β_{j} (ϵ_{j} - μ_{j})} + 1}, β_{j} = β_{K} N_{j}, μ_{j} = \frac{μ_{K}}{N_{j}} .$ Both conclusions are conditional model deductions on the fixed background or finite supplied mode set. They assert no Einstein equation, gas backreaction, arbitrary curved-space spectrum, or observed equilibrium.

Facts & Assumptions

Given: The compact finite spatial region, smooth static metric/lapse bounds, positive particle mass and canonical parameters, and exactly one selected preparation: fixed-particle canonical, grand-canonical, or the separate finite-mode branch.

[A1]

The Postulate supplies $g = - N^{2} (d x^{0})^{2} + h_{i j} d x^{i} d x^{j}$ , $0 < N_{0} \leq N \leq N_{1}$ , the classical local energy $E (\hat{p})$ and either fixed particle count $M$ or grand fugacity $z$ , or the separate finite mode energies and occupancy basis. (Fixed static ideal gas or finite fermion regulator preparation)

[F1]

For the bounded-lapse static metric and classical gas, the one-particle integral is finite and positive; the fixed-particle branch has $Z_{M} = Z_{1}^{M} / M!$ , and the grand-canonical branch has $Ξ = e^{z Z_{1}}$ with $f_{z} = z h_{*}^{- 3} e^{- β_{K} N E}$ . Its local identities include $T_{l o c} = T_{K} / N$ , $P = n k_{B} T_{l o c}$ , $\partial_{i} P = - (ε + P) \partial_{i} \log N$ , and, when $z = e^{β_{K} μ_{K}}$ , $μ_{l o c} = μ_{K} / N$ . (Static relativistic finite gas normalization and hydrostatics)

[F2]

For a finite bit-string occupation basis and finite real energies, the trace is $\prod_{j} (1 + e^{- β (E_{j} - μ)})$ and each occupation marginal is $(e^{β (E_{j} - μ)} + 1)^{- 1}$ . (Finite static fermion-mode trace and occupation identity)

Proof

1.1givenA1F1algebra

The metric, positive lapse bounds, compact finite-volume region, and positive parameters in [A1] meet the hypotheses of [F1], which gives $0 < Z_{1} < \infty$ and the normalizer for whichever classical branch is selected. In the fixed-particle branch, put $d λ_{1} = d V_{h} d^{3} \hat{p} / h_{*}^{3}$ . The normalized $M$ -particle Gibbs density relative to $d λ_{1}^{\otimes M} / M!$ is $M! Z_{1}^{- M} \prod_{a = 1}^{M} e^{- β_{K} N (x_{a}) E ({\hat{p}}_{a})}$ . Its one-particle marginals are $Z_{1}^{- 1} e^{- β_{K} N E} d λ_{1}$ , so summing their densities gives $f_{M} = (M / Z_{1}) h_{*}^{- 3} e^{- β_{K} N E}$ . For comparison only, choose any $z_{0} > 0$ in the grand-canonical result [F1] and set $f_{z_{0}} = z_{0} h_{*}^{- 3} e^{- β_{K} N E}$ ; then $f_{M} = (M / (z_{0} Z_{1})) f_{z_{0}}$ . This positive constant scales $n$ , $ε$ , $P$ , and $\partial_{i} P$ equally, while $T_{l o c}$ is unchanged because it depends only on $β_{K} N$ . Therefore the pressure and hydrostatic identities from [F1] hold for $f_{M}$ too. The comparison does not introduce fugacity into the fixed-particle preparation.

1.2A1F2

For the finite mode set, apply [F2] with the finite real values $E_{j} = N_{j} ϵ_{j}$ , $β = β_{K}$ , and $μ = μ_{K}$ . This gives a positive finite trace, a normalized diagonal state, and $⟨ n_{j} ⟩ = (e^{β_{K} (N_{j} ϵ_{j} - μ_{K})} + 1)^{- 1}$ .

2.1A1F1step 1.1algebra

In the grand-canonical branch only, $z = e^{β_{K} μ_{K}}$ makes $z e^{- β_{K} N E} = e^{- β_{K} N (E - μ_{K} / N)}$ , so the local chemical parameter is $μ_{l o c} = μ_{K} / N$ .

3.1step 1.2algebra∎

The definitions $β_{j} = β_{K} N_{j}$ and $μ_{j} = μ_{K} / N_{j}$ imply $β_{j} (ϵ_{j} - μ_{j}) = β_{K} (N_{j} ϵ_{j} - μ_{K})$ . Substitution yields the stated local-parameter occupation exactly for each supplied mode.

Constructed static gas redshift and hydrostatics

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources