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Perfect-fluid exchange projections and conduction heating

Statement

Let $Ω$ be an open subset of Smooth formal matter, EM tensors and observer quantities. Let $ϵ, p, σ \in C^{1} (Ω, R)$ and let $U \in C^{2} (Ω, T Ω)$ be future timelike with $g (U, U) = - c^{2}$ . Set

$T_{m}^{a b} = \frac{ϵ + p}{c^{2}} U^{a} U^{b} + p g^{a b}, a^{b} = U^{a} \nabla_{a} U^{b}, D = U^{a} \nabla_{a}, θ = \nabla_{a} U^{a}, h^{b}_{c} = δ^{b}_{c} + \frac{U^{b} U_{c}}{c^{2}} .$

For a current $j^{a} = σ U^{a} + q^{a}$ with $q^{a} U_{a} = 0$ , define $f^{b} = F^{b}_{c} j^{c}$ and assume the mathematical exchange equation $\nabla_{a} T_{m}^{a b} = f^{b}$ . Then it is equivalent to the longitudinal and rest-space equations

$D ϵ + (ϵ + p) θ = - U_{b} f^{b}, \frac{ϵ + p}{c^{2}} a^{b} + h^{b a} \nabla_{a} p = h^{b}_{c} f^{c} .$

The rest electric field of $U$ is $E_{U}^{a} = F^{a}_{b} U^{b}$ . If $q^{a} = χ E_{U}^{a}$ for a specified scalar $χ \geq 0$ , then the electromagnetic contribution to the comoving energy balance is $- U_{b} f^{b} = χ g (E_{U}, E_{U}) \geq 0$ . For $q = 0$ it is zero. The Ohmic relation is an additional constitutive hypothesis; no equation of state, entropy law, or fluid existence result follows.

Facts & Assumptions

Given: The fields and regularity in the Statement, $g (U, U) = - c^{2}$ , $q^{a} U_{a} = 0$ , and $\nabla_{a} T_{m}^{a b} = f^{b}$ .

[F1]

Smooth formal matter, EM tensors and observer quantities: $T_{m}^{a b} = (ϵ + p) U^{a} U^{b} / c^{2} + p g^{a b}$ and $E_{U}^{a} = F^{a}_{b} U^{b}$ .

[F2]

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: $\nabla$ is metric-compatible and its induced tensor derivative commutes with contractions and annihilates $g$ .

[F3]

Product connection on tensor and hom bundles: the induced tensor connection obeys the Leibniz rule on tensor products and on scalar multiples of tensor fields.

[F4]

The observer rest-space decomposition and the spatial projection: for every future unit timelike vector $n$ , the rest space $n^{⊥}$ has positive-definite restricted metric.

[A1]

The current is the stated sum $j^{a} = σ U^{a} + q^{a}$ with $q^{a} U_{a} = 0$ ; no constitutive law is assumed until the final conditional clause.

Proof

1.1F1F2F3algebra

Apply the product rule to the three factors of $T_{m}^{a b}$ and use $\nabla_{a} g^{a b} = 0$ . The result is $\nabla_{a} T_{m}^{a b} = \frac{U^{b}}{c^{2}} [D (ϵ + p) + (ϵ + p) θ] + \frac{ϵ + p}{c^{2}} a^{b} + \nabla^{b} p$ .

2.1F2step 1.1algebra

Contract this expression with $U_{b}$ . The normalization gives $U_{b} U^{b} = - c^{2}$ and $U_{b} a^{b} = \frac{1}{2} D (g (U, U)) = 0$ ; also $U_{b} \nabla^{b} p = D p$ . Hence $U_{b} \nabla_{a} T_{m}^{a b} = - D ϵ - (ϵ + p) θ$ . Equating this with $U_{b} f^{b}$ proves the longitudinal equation.

3.1F1F2step 1.1step 2.1algebra

The map $h^{b}_{c} = δ^{b}_{c} + U^{b} U_{c} / c^{2}$ satisfies $h^{b}_{c} U^{c} = 0$ and $h^{b}_{c} a^{c} = a^{b}$ , by $U \cdot U = - c^{2}$ and $U \cdot a = 0$ . Applying it to step 1.1 removes the term proportional to $U^{b}$ and gives the stated rest-space equation.

4.1F1F2F3step 1.1step 2.1step 3.1algebra

For any vector $w^{b}$ , $w^{b} = h^{b}_{c} w^{c} - U^{b} (U_{c} w^{c}) / c^{2}$ . Thus if both displayed projected equations hold, the projections of $\nabla_{a} T_{m}^{a b} - f^{b}$ onto $U$ and $U^{⊥}$ vanish, so that vector is zero. This proves the converse.

5.1F1F4A1step 2.1algebra∎

Antisymmetry of $F$ gives $U_{b} F^{b}_{c} U^{c} = 0$ , so $E_{U}^{a} = F^{a}_{b} U^{b}$ is orthogonal to $U$ . Since $c > 0$ , $n : = U / c$ is future unit timelike and $E_{U} \in n^{⊥}$ ; [F4] makes $g (E_{U}, E_{U}) \geq 0$ . Therefore $- U_{b} f^{b} = - U_{b} F^{b}_{c} q^{c} = g (E_{U}, q)$ . If $q = χ E_{U}$ with $χ \geq 0$ , this is $χ g (E_{U}, E_{U}) \geq 0$ ; if $q = 0$ it is zero. No sign or heating claim has been used without the stated current hypothesis.

Remarks

The source-free and charged cases share the same projection identities. A general conduction current is not assumed to obey Ohm's law; Example Explicit local conduction-energy calculation ↗ supplies a local explicit instance of the optional $q = χ E_{U}$ branch.
These equations are algebraic projections of a supplied balance law, not a global fluid evolution theorem.

Perfect-fluid exchange projections and conduction heating

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources