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The weak energy condition does not imply conservation of a prescribed stress-energy tensor

Physical scope

Pointwise algebraic statement about a prescribed smooth symmetric stress-energy tensor field on an open region of the flat classical spacetime model: the weak energy condition is a pointwise positivity condition on the tensor and carries no conservation content. No dynamical model, no solution of the field equation, no measurement and no claim about nature is asserted; the example shows only that the named condition is not by itself the conservation identity.

Statement refuted

Statement refuted: "Let $M = R^{4}$ carry the flat metric $g = diag (- 1, 1, 1, 1)$ and the identity chart, and let $U$ be an open subset. Let $T$ be a smooth symmetric covariant $2$ -tensor field on $U$ which satisfies the weak energy condition of The pointwise energy conditions WEC, NEC, DEC and SEC: $T_{p} (v, v) \geq 0$ for every timelike $v \in T_{p} M$ and every $p \in U$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space). Then $T$ is conserved, $\nabla^{a} T_{a b} = 0$ on $U$ ."

Facts & Assumptions

Given: The flat Lorentzian four-manifold $(R^{4}, g)$ with identity chart, $g = diag (- 1, 1, 1, 1)$ , time orientation $N = \partial_{0}$ , metric-selected connection $\nabla$ ; length coordinates $x^{a}$ in metres ( $x^{0} = c t$ ), fixed constants $L > 0$ in $m$ and $E_{*} > 0$ in $J m^{- 3}$ , the open set $U : = {x \in R^{4} : ∣ x^{1} ∣ < L}$ ; and the smooth symmetric covariant $2$ -tensor field $T$ on $U$ with components $T_{00} = 3 E_{*}, T_{01} = T_{10} = E_{*} x^{1} / L, T_{i j} = 0 otherwise .$

[L1]

On the identity chart $Γ \equiv 0$ , so $\nabla$ is the coordinate connection and $\nabla^{a} T_{a b} = g^{a c} \partial_{c} T_{a b} = η^{a c} \partial_{c} T_{a b}$ with $η = diag (- 1, 1, 1, 1)$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields, The inverse metric, musical isomorphisms and index raising).

[L2]

The weak energy condition at $p$ is $T_{p} (v, v) \geq 0$ for every timelike $v \in T_{p} M$ ; a vector $v = (v^{0}, v^{1}, v^{2}, v^{3})$ is timelike exactly when $- (v^{0})^{2} + (v^{1})^{2} + (v^{2})^{2} + (v^{3})^{2} < 0$ , which implies $∣ v^{1} ∣ < ∣ v^{0} ∣$ (The pointwise energy conditions WEC, NEC, DEC and SEC, Timelike, null, spacelike and causal vectors of a Lorentz vector space).

[L3]

$U$ is an open subset of the smooth manifold $R^{4}$ , hence carries a restricted smooth structure and the field with the displayed components is a smooth symmetric covariant $2$ -tensor field on $U$ (An open subset of a smooth manifold has a canonical restricted smooth structure, Smooth covariant 2-tensor fields on a smooth manifold, $C^{r}$ and smooth maps between smooth manifolds).

Counterexample

1.1L2L3algebra

(Weak energy condition.) Let $p \in U$ and let $v = (v^{0}, v^{1}, v^{2}, v^{3})$ be timelike. By [L2], $∣ v^{1} ∣ < ∣ v^{0} ∣$ , so in particular $v \neq 0$ and $v^{0} \neq 0$ . By the components of $T$ , $T_{p} (v, v) = E_{*} [3 (v^{0})^{2} + 2 (x^{1} / L) v^{0} v^{1}] \geq E_{*} [3 (v^{0})^{2} - 2 (∣ x^{1} ∣ / L) ∣ v^{0} ∣ ∣ v^{1} ∣] \geq E_{*} (3 - 2 ∣ x^{1} ∣ / L) (v^{0})^{2} > E_{*} (v^{0})^{2} > 0,$ using $∣ v^{1} ∣ < ∣ v^{0} ∣$ and $∣ x^{1} ∣ / L < 1$ on $U$ , with $E_{*} > 0$ . Thus the weak energy condition of [L2] holds at every point of $U$ .

1.2L1algebra

(Nonconservation.) By [L1] and the components of $T$ (which are independent of $x^{0}$ ), $\nabla^{a} T_{a 0} = η^{00} \partial_{0} T_{00} + η^{11} \partial_{1} T_{10} = (- 1) \cdot 0 + 1 \cdot (E_{*} / L) = E_{*} / L \neq 0.$ Hence $T$ is not conserved on $U$ .

2.1step 1.1step 1.2algebra∎

(Conclusion.) The tensor field $T$ of the Facts satisfies the weak energy condition on $U$ by step 1.1 and is not conserved by step 1.2, so the displayed statement is refuted: the weak energy condition is a pointwise positivity condition and does not imply the conservation identity.

Remarks

Source scope. Carroll (4.72) and Tong section 4.5.7 supply the weak-energy-condition definition. The prescribed tensor and its nonconservation are established by the local calculation above; no source passage is cited as asserting this counterexample or the nonimplication.
No field-equation claim. The counterexample does not assert that $T$ solves the Einstein equation with the flat metric; a model whose field law holds must additionally satisfy the conservation derived in Consistency conditions of the adopted field law: conservation, trace, parameter variation and the energy conditions of a perfect fluid, which this $T$ fails. The point is exactly that the energy condition alone does not supply that consistency.
Sign conventions. The condition $T (v, v) \geq 0$ is evaluated with the metric convention $g = diag (- 1, 1, 1, 1)$ ; the inequality $∣ v^{1} ∣ < ∣ v^{0} ∣$ used in the proof is the timelike condition of Timelike, null, spacelike and causal vectors of a Lorentz vector space for this convention.
No empirical content. No measurement or material realization is asserted. The constants $E_{*}$ and $L$ set the units of the prescribed test field: every displayed tensor entry has energy-density units, and its nonzero divergence has units $J m^{- 4}$ . No numerical measured values are assigned (SI unit and coordinate conventions of the classical gravitational framework).

The weak energy condition does not imply conservation of a prescribed stress-energy tensor

Statement refuted

Facts & Assumptions

Counterexample

Remarks

Depends on

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