DefinitionDefinition: AI-adaptedProof: AI-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The pointwise energy conditions WEC, NEC, DEC and SEC

Physical scope

Pointwise inequalities that a model may adopt for its stress-energy tensor: weak, null, dominant and strong energy conditions, stated for the smooth tensor fields of this framework. They are additional assumptions available to a model; no condition is automatic, the actual pointwise implications are proved here, and their measured relevance is a separate empirical question.

Definition

Let $(M, g, N)$ be a time-oriented smooth Lorentzian four-manifold, let $T$ be a stress-energy tensor field on its matter domain $U \subseteq M$ (Stress-energy tensor field of a model), and let $♯_{p}$ and ${tr}_{g}$ be the musical isomorphism and the metric trace of The inverse metric, musical isomorphisms and index raising. For $p \in U$ and $v \in T_{p} M$ write $T_{p} (v, \cdot) \in T_{p}^{*} M$ for the covector $w \mapsto T_{p} (v, w)$ , and recall that, by Classification of future and past causal vectors by a timelike field, a vector $w \in T_{p} M$ lies in $cl (C_{p}^{+})$ exactly when $w = 0$ or $w$ is future-directed and causal, where $C_{p}^{+}$ is the component of the timelike set containing $N_{p}$ .

The model satisfies at $p$ :

the weak energy condition (WEC) when $T_{p} (v, v) \geq 0$ for every timelike $v \in T_{p} M$ ;
the null energy condition (NEC) when $T_{p} (v, v) \geq 0$ for every null $v \in T_{p} M$ ;
the dominant energy condition (DEC) when $- ♯_{p} (T_{p} (v, \cdot)) \in cl (C_{p}^{+})$ for every future-directed timelike $v \in T_{p} M$ ;
the strong energy condition (SEC) when $(T_{p} - \frac{1}{2} {tr}_{g} (T) (p) g_{p}) (v, v) \geq 0$ for every timelike $v \in T_{p} M$ , that is, when $T_{p} (v, v) - \frac{1}{2} {tr}_{g} (T) (p) g_{p} (v, v) \geq 0$ .

The model satisfies the condition (on $U$ ) when the corresponding inequality holds at every $p \in U$ .

Each is a pointwise condition on the pair $(g_{p}, T_{p})$ alone, evaluated on timelike or null vectors as displayed; the expressions use only the metric pairing $g_{p}$ , its inverse pairing on covectors, and the endomorphism $T_{p}^{♯} : v \mapsto ♯_{p} (T_{p} (v, \cdot))$ of The inverse metric, musical isomorphisms and index raising, whose trace is ${tr}_{g} (T) (p)$ . The timelike and null vector sets are those of the Lorentz vector space $(T_{p} M, g_{p})$ (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model).

Remarks

Adopted, not automatic. None of the four conditions is a consequence of the definition of $T$ or of the framework postulates; from a chosen matter model some may hold, fail, or be undefined (for instance, a sign of the pressure that is not fixed by the model's definition). A model that uses one of them adopts it as an additional assumption, and the framework adopts none of them automatically.
Implications. Pointwise, DEC implies WEC, WEC implies NEC, and SEC also implies NEC, by the proof below. No converse or implication between WEC and SEC is assumed. These are consequences of the definitions and Lorentzian algebra, not extra restrictions on the chosen matter model.
Timelike formulation. WEC and SEC are stated on timelike vectors; continuity extends the corresponding quadratic inequalities to null limits. Consumers may use these proved consequences without a further physical assumption.

Facts & Assumptions

Given: At $p \in U$ , the Lorentz form $g_{p}$ , symmetric bilinear $T_{p}$ , and the definitions of WEC, NEC, DEC and SEC above.

[F1]

Future null vectors lie in the closure of the future timelike component (Classification of future and past causal vectors by a timelike field). The closed future cone also contains zero.

[F2]

The inverse metric satisfies $g (♯ α, v) = α (v)$ (The inverse metric, musical isomorphisms and index raising); bilinear forms in a finite basis are finite quadratic polynomials, so their values are continuous under coordinate convergence.

[F3]

In an orthonormal basis oriented with the future cone, a future causal vector has $j^{0} \geq ∣ j_{v e c} ∣$ and a future timelike vector has $v^{0} > ∣ v_{v e c} ∣$ (The timelike set has exactly two convex components, described in orthonormal coordinates). The spatial Euclidean pairing obeys Cauchy–Schwarz (Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors).

Proof

1.1givenF1F2F3algebra

Suppose DEC holds and let $v$ be future timelike. Put $j = - ♯ (T_{p} (v, \cdot))$ . DEC puts $j$ in the closed future cone. In the future-oriented orthonormal coordinates of [F3], $g (j, v) = - j^{0} v^{0} + j_{v e c} \cdot v_{v e c} \leq - j^{0} v^{0} + ∣ j_{v e c} ∣ ∣ v_{v e c} ∣ \leq 0$ , including $j = 0$ . By [F2], $T_{p} (v, v) = - g (j, v) \geq 0$ . Every past timelike vector is the negative of a future timelike vector, and its quadratic value is the same. Hence DEC implies WEC on all timelike vectors.

2.1step 1.1F1F2F3algebra

For a future null vector $k$ and future timelike $n$ , The same estimate as in step 1.1, using $k^{0} = ∣ k_{v e c} ∣ > 0$ and $n^{0} > ∣ n_{v e c} ∣$ , gives $g (k, n) < 0$ . For every $ϵ > 0$ , $k_{ϵ} = k + ϵ n$ has $g (k_{ϵ}, k_{ϵ}) = 2 ϵ g (k, n) + ϵ^{2} g (n, n) < 0$ and $g (k_{ϵ}, n) < 0$ , so it is future timelike and tends to $k$ . If WEC holds, $T_{p} (k_{ϵ}, k_{ϵ}) \geq 0$ ; [F2] and the limit give $T_{p} (k, k) \geq 0$ . Negating past null vectors gives the same conclusion. Thus WEC implies NEC.

3.1step 2.1F2algebra∎

If SEC holds, apply the same timelike limit to $B = T_{p} - \frac{1}{2} {tr}_{g} (T) (p) g_{p}$ . Continuity gives $B (k, k) \geq 0$ at every null $k$ , and $g_{p} (k, k) = 0$ makes this $T_{p} (k, k) \geq 0$ . Thus SEC implies NEC as well. These deductions preserve the stated pointwise conditions and impose none automatically.

The pointwise energy conditions WEC, NEC, DEC and SEC

Definition

Remarks

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources