Energy Conditions

Energy conditions are pointwise inequalities on the stress-energy tensor T_{ab} that encode the physically reasonable demand that matter has non-negative energy density and that gravity, on the whole, focuses rather than defocuses geodesics. They are not derived from Einstein's equations; they are extra assumptions about the matter content, chosen to be weak enough to hold for all ordinary matter yet strong enough to drive the focusing arguments behind the singularity theorems and the area theorem.

For a perfect fluid of energy density ρ and isotropic pressure p, the standard conditions read: the null energy condition (NEC) requires ρ + p ≥ 0; the weak energy condition (WEC) adds ρ ≥ 0; the strong energy condition (SEC) requires ρ + p ≥ 0 and ρ + 3p ≥ 0; and the dominant energy condition (DEC) requires ρ ≥ |p|, ensuring energy never flows faster than light. Geometrically, the NEC is exactly the statement that the Raychaudhuri focusing term R_{ab}k^a k^b is non-negative for every null vector — which is what guarantees that a converging bundle of light rays cannot un-converge.

The conditions are also where the loopholes hide. A cosmological constant has p = −ρ: it obeys the null and weak conditions but violates the strong one, which is precisely why dark-energy-dominated space accelerates instead of recollapsing. Quantum fields routinely violate even the null condition — the Casimir vacuum and the negative-energy flux of Hawking radiation are real examples — so every proposed route around a singularity (bouncing cosmologies, traversable wormholes, evaporating black holes) works by breaking one of these inequalities.

Systematized through the 1960s–70s in the work of Penrose, Hawking and Ellis, the energy conditions became the standard hypotheses of the singularity and area theorems; their fragility under quantum effects was sharpened by Ford and Roman's quantum energy inequalities from the 1990s onward.