Null, Weak, Dominant, and Strong Energy Conditions

Energy conditions are quantified inequalities on a specified stress tensor. The null condition constrains null contractions, the weak condition constrains energy density measured by every observer, the dominant condition additionally constrains the causal direction of energy flux, and the strong condition constrains a trace-reversed timelike contraction. They are distinct classical pointwise assumptions, not universal laws of matter.

Required background. Stress-energy supplies observer energy density and flux. Special relativity supplies causal vectors and boosts for testing the inequalities on every observer.

Helpful background. Tensor fields clarify the contractions. Einstein’s equation relates them to curvature; incompleteness applications require separate global causal structure.

Pointwise definitions

Let $T_{ab}=T_{ba}$ in a time-oriented Lorentzian spacetime of dimension $D>2$.

• The null energy condition (NEC) is $$T_{ab}k^ak^b\ge0$$ for every null vector $k^a$.
• The weak energy condition (WEC) is $$T_{ab}u^au^b\ge0$$ for every timelike vector $u^a$. For unit future $u^a$, this is nonnegative measured energy density.
• The dominant energy condition (DEC) requires WEC and, for every future timelike $u^a$, the energy-flux vector $$j^a=-T^a{}_bu^b$$ to be future-directed causal or zero. The zero case includes vacuum and saturation.
• The strong energy condition (SEC) is $$\left(T_{ab}-\frac{T}{D-2}g_{ab}\right)u^au^b\ge0$$ for every timelike $u^a$, where $T=T^a{}_a$.

Kontou and Sanders give these definitions, their causal quantifiers, and their logical relations Kontou and Sanders 2020, § 2.1, pp. 8–11, Eqs. (18)–(27) and Table 1, PDF. The SEC does not assert positive energy density: it constrains the trace-reversed source that enters timelike Ricci focusing in Einstein gravity.

Perfect-fluid forms

For

$$T_{ab}=(\rho+p)v_av_b+pg_{ab},$$

evaluate the inequalities in the fluid rest frame and then allow arbitrary boosted causal vectors. For an isotropic fluid:

$$\begin{array}{c|l} \text{condition}&\text{equivalent inequalities}\\ \hline \text{NEC}&\rho+p\ge0,\\ \text{WEC}&\rho\ge0,\quad \rho+p\ge0,\\ \text{DEC}&\rho\ge0,\quad -\rho\le p\le\rho,\\ \text{SEC}&\rho+p\ge0,\quad (D-3)\rho+(D-1)p\ge0. \end{array}$$

In four dimensions the final SEC inequality is $\rho+3p\ge0$. The first SEC inequality arises from highly boosted observers; checking only the comoving vector misses it.

The table follows from a short calculation. In a fluid-rest orthonormal frame, take a unit observer $u^{\hat a}=\Gamma(1,\mathbf w)$, where $\Gamma=(1-|\mathbf w|^2)^{-1/2}$. The measured energy density and trace-reversed contraction are

$$\begin{aligned} T_{ab}u^au^b&=(\rho+p)\Gamma^2-p,\\ \left(T_{ab}-\frac{T}{D-2}g_{ab}\right)u^au^b &=(\rho+p)\Gamma^2+\frac{p-\rho}{D-2}. \end{aligned}$$

Each is linear in $\Gamma^2\ge1$. Nonnegativity for every observer requires a nonnegative slope, $\rho+p\ge0$, and a nonnegative value at $\Gamma=1$. Those endpoint values give the remaining WEC and SEC inequalities. For DEC, the same observer has $j^{\hat a}=\Gamma(\rho,-p\mathbf w)$, so

$$j_aj^a=\Gamma^2\left(-\rho^2+p^2|\mathbf w|^2\right).$$

Future orientation requires $\rho\ge0$, and causality for arbitrarily large boosts requires $|p|\le\rho$. Finally, a null vector $k^{\hat a}=\omega(1,\mathbf n)$, $|\mathbf n|=1$, gives $T_{ab}k^ak^b=\omega^2(\rho+p)$ directly. These calculations explain why rest-frame energy density alone cannot decide all four conditions.

For dust, $p=0$ and $\rho\ge0$, all four conditions hold. For a positive cosmological term represented as effective matter, $p=-\rho$ with $\rho>0$: NEC is saturated, WEC and DEC hold, but SEC fails because $\rho+3p=-2\rho<0$. This example decisively separates SEC from positive energy.

Proof of the logical implications

DEC includes WEC by definition in the convention above. WEC implies NEC under continuity: approach any null $k^a$ by timelike vectors $u^a_\epsilon=k^a+\epsilon t^a$ with suitable future timelike $t^a$, apply WEC, and take $\epsilon\to0$. SEC likewise implies NEC as a null limit because the trace term vanishes when $g_{ab}k^ak^b=0$.

The converses fail. NEC alone permits negative timelike energy density. WEC does not ensure causal energy flux and hence need not imply DEC. SEC can hold while WEC fails for suitable stress tensors, and WEC can hold while SEC fails, as positive vacuum energy shows. Kontou and Sanders summarize the implication structure in Kontou and Sanders 2020, § 2.1, Fig. 2, printed p. 9, PDF and give classical field-theory examples and counterexamples in Kontou and Sanders 2020, § 2.3, Table 2, printed pp. 14–15, PDF.

These implications assume the same symmetric pointwise stress tensor. Moving a curvature coupling between “geometry” and an effective $T_{ab}$ can change which effective tensor satisfies an inequality without changing the full field equation.

For anisotropic matter the perfect-fluid shortcuts no longer apply. In an orthonormal eigenframe with principal pressures $p_i$, a type-I stress tensor has NEC conditions $\rho+p_i\ge0$ for every $i$, WEC adds $\rho\ge0$, and DEC further requires $|p_i|\le\rho$. SEC adds the relevant trace-reversed combinations. These formulas require diagonalizability of the assumed algebraic type; the causal-vector definitions above remain primary in all cases.

DEC has several closely related formulations in the literature. Some define it by requiring $-T^a{}_bu^b$ causal and future for every future timelike $u^a$ and derive WEC; others state both explicitly. For a symmetric stress tensor under standard continuity assumptions they agree in the ordinary type-I cases considered here. Naming the chosen formulation avoids apparent disagreements caused by convention.

Physical interpretations and limits

WEC formalizes nonnegative local energy density for every timelike observer. DEC supplements it with causal energy-momentum flux, but it should not be identified with a theorem about signal propagation: characteristic speeds follow from the matter equations, and models can be hyperbolic even when a chosen stress tensor violates DEC.

NEC is the matter condition most directly paired with null Ricci convergence. SEC is paired with timelike convergence only after using the trace-reversed Einstein equation and treating $\Lambda$ consistently. None of the conditions follows from Lorentz symmetry, conservation, or the equivalence principle alone.

The definitions here are classical, pointwise, and on shell unless stated otherwise. Nonminimally coupled classical fields and renormalized quantum stress tensors can violate them. Averaged inequalities and quantum energy inequalities have different quantifiers and cannot be substituted into a pointwise proof without rebuilding that proof.

Conservation is logically separate. A symmetric tensor can obey $\nabla^aT_{ab}=0$ while violating every listed positivity condition, and a tensor satisfying a pointwise inequality at one event need not be conserved. Energy conditions constrain algebraic contractions; field equations and symmetries govern propagation and balance. Keeping these roles separate prevents an inequality from being treated as a dynamical equation.

Exercises

For a four-dimensional perfect fluid with equation of state $p=w\rho$ and $\rho>0$, find the ranges of $w$ satisfying each condition.

Solution

NEC and WEC require $1+w\ge0$, so $w\ge-1$. DEC requires $-1\le w\le1$. SEC requires both $w\ge-1$ and $1+3w\ge0$, hence $w\ge-1/3$. These ranges show explicitly that positive vacuum energy ($w=-1$) satisfies WEC and DEC but violates SEC.

References

• Kontou, Eleni-Alexandra, and Ko Sanders. “Energy Conditions in General Relativity and Quantum Field Theory.” Classical and Quantum Gravity 37 (2020): 193001. DOI; arXiv.