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Energy-condition verdicts for dust, radiation and the cosmological effective fluid

Physical scope

Pointwise algebraic reading of the four energy conditions for prescribed perfect-fluid models in a rest frame, including the effective fluid associated with a cosmological parameter in the adopted field law. No dynamics, no solution of any field equation, no measurement and no claim about nature is asserted; the example records the inequalities alone.

Example

Adopt a perfect-fluid model $(ε, p, n)$ in the classical framework (The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy) with derived stress-energy tensor $T_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ , and use the rest-frame equivalences of Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid clause (f): at a point with rest frame $(e_{0}, e_{1}, e_{2}, e_{3})$ , $e_{0} = n$ , the four conditions of The pointwise energy conditions WEC, NEC, DEC and SEC read

$W E C ⟺ ε \geq 0 and ε + p \geq 0, N E C ⟺ ε + p \geq 0, D E C ⟺ ε \geq ∣ p ∣, S E C ⟺ ε + 3 p \geq 0 and ε + p \geq 0.$

Then:

Dust ( $p = 0$ , $ε > 0$ ): WEC, NEC, DEC and SEC all hold.
Radiation ( $p = ε / 3$ , $ε > 0$ ): WEC, NEC, DEC and SEC all hold.
Positive cosmological effective fluid ( $ε > 0$ , $p = - ε$ , as for the term $- Λ g_{a b} / κ$ with $Λ > 0$ in Adopted Einstein field equation in the direct-equation presentation): WEC, NEC and DEC hold (with equality in the non-strict inequalities) and SEC fails.
Negative cosmological effective fluid ( $ε < 0$ , $p = - ε$ , $Λ < 0$ ): WEC and DEC fail, NEC holds with equality and SEC holds. Together with clause 3, this shows that WEC and SEC are incomparable by implication.

Facts & Assumptions

Given: A perfect-fluid model with rest energy density $ε$ and pressure $p$ , and the four equivalences displayed above.

[L1]

The rest-frame equivalences for the four conditions are the equivalences of Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid clause (f), derived there from the tensor form $T_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ .

[L2]

The stress-energy tensor of the field-law term $- Λ g_{a b} / κ$ has the perfect-fluid form with $ε = Λ / κ$ and $p = - Λ / κ$ : $T_{a b} = - (Λ / κ) g_{a b} = (ε + p) n_{a} n_{b} + p g_{a b}$ with $ε + p = 0$ (Adopted Einstein field equation in the direct-equation presentation clause (i), The perfect-fluid matter model: rest energy density, pressure, four-velocity and stress-energy, SI unit and coordinate conventions of the classical gravitational framework).

Verification

technique · direct substitution into the four equivalences

1.1L1givenalgebra

(Dust and radiation.) By [L1] the four conditions are read off from the displayed inequalities. For $p = 0$ and $ε > 0$ : $ε \geq 0$ , $ε + p = ε > 0$ , $ε \geq ∣ p ∣ = 0$ and $ε + 3 p = ε > 0$ , so all four right-hand sides hold. For $p = ε / 3$ and $ε > 0$ : $ε \geq 0$ , $ε + p = \frac{4}{3} ε > 0$ , $ε \geq ∣ p ∣ = ε / 3$ , and $ε + 3 p = 2 ε > 0$ , so all four hold. These are clauses 1 and 2.

1.2L1L2algebra

(Positive effective fluid.) By [L1] and [L2], let $ε > 0$ , $p = - ε$ . Then $ε \geq 0$ and $ε + p = 0 \geq 0$ : WEC holds; $ε + p = 0$ : NEC holds; $ε \geq ∣ p ∣ = ε$ : DEC holds; but $ε + 3 p = - 2 ε < 0$ , so SEC fails. This is clause 3.

2.1L1L2algebra∎

(Negative effective fluid.) By [L1] and [L2], let $ε < 0$ , $p = - ε > 0$ . Then $ε \geq 0$ fails, so WEC fails; $ε \geq ∣ p ∣ = - ε$ fails (an inequality $t \geq ∣ t ∣$ with $t < 0$ is false), so DEC fails; $ε + p = 0 \geq 0$ , so NEC holds; and $ε + 3 p = - 2 ε > 0$ together with $ε + p = 0$ gives SEC. This is clause 4 and shows that SEC need not imply WEC or DEC. Clause 3 shows that WEC need not imply SEC, so WEC and SEC are incomparable. These examples are consistent with the implications DEC $\Rightarrow$ WEC $\Rightarrow$ NEC and SEC $\Rightarrow$ NEC proved in The pointwise energy conditions WEC, NEC, DEC and SEC.

Remarks

Not a solution claim. The verdicts concern the pointwise inequalities for prescribed values of $ε$ and $p$ ; they do not assert that the corresponding fluid model solves the field equation or that the cosmological parameter has either sign.
Convention dependence. The equivalences use the strict-timelike convention of The pointwise energy conditions WEC, NEC, DEC and SEC; the proof obtains boundary inequalities by limits of timelike vectors, using continuity of this tensor.

Energy-condition verdicts for dust, radiation and the cosmological effective fluid

Example

Facts & Assumptions

Verification

Remarks

Depends on

Dependency tree · 2 levels

Sources