Energy Conditions

The Einstein field equations $G_{\mu \nu }=8\pi G{T}_{\mu \nu }$ will accept any symmetric stress–energy tensor on the right — including physically absurd ones. Given any metric you like, simply define $T_{\mu \nu }$ by the left-hand side and you have a "solution". To do physics, one restricts to reasonable matter by imposing energy conditions: inequalities demanding, roughly, that energy densities be non-negative and that pressures not be too negative. These conditions are the hypotheses of the great global theorems — the singularity theorems, the area theorem, positive-mass, and topological-censorship results.

We use $c=1$, mostly-plus signature; $u^{\mu }$ denotes an arbitrary future-directed timelike unit vector (an observer's four-velocity), $k^{\mu }$ an arbitrary null vector.

The conditions

For a perfect fluid with density $\rho$ and pressure $p$, each condition reduces to a simple statement about $\rho$ and $p$ (right column).

Logical relations: $\text{DEC}\Rightarrow \text{WEC}\Rightarrow \text{NEC}$, and $\text{SEC}\Rightarrow \text{NEC}$. The SEC and DEC are independent — neither implies the other (a cosmological constant satisfies DEC but violates SEC). The NEC is the weakest and hardest to violate with classical matter.

Why the strong condition is about attraction

The name "strong" is historical, not logical (it does not imply the weak condition). Its physical meaning comes from the Raychaudhuri equation, which governs the focusing of a bundle of geodesics: the term driving convergence is $R_{\mu \nu }{u}^{\mu }{u}^{\nu }$. Via the trace-reversed field equations,

$$R_{\mu \nu }{u}^{\mu }{u}^{\nu }=8\pi G(T_{\mu \nu }-\frac{1}{2}T{g}_{\mu \nu })u^{\mu }{u}^{\nu },$$

so the SEC is exactly the statement $R_{\mu \nu }{u}^{\mu }{u}^{\nu }\ge 0$ — that matter focuses geodesics, i.e. gravity attracts. In Newtonian terms the SEC is $\rho +3p\ge 0$, the relativistic source of gravity being $\rho +3p$ (pressure gravitates), not $\rho$ alone. This is why cosmic acceleration requires SEC violation: to push spacetime apart you need $\rho +3p<0$, which is precisely what a cosmological constant ($p=-\rho$, so $\rho +3p=-2\rho <0$) or inflation supplies.

What obeys and what violates them

Ordinary matter obeys them. Dust, radiation, and normal fluids with $\rho \ge 0$, $|p|\le \rho$ satisfy all four. For most of the 20th century the energy conditions were treated as near-axioms.

But important physics violates them:

• The cosmological constant / dark energy ($p=-\rho <0$): satisfies WEC, NEC, DEC, but violates the SEC. Since $\Lambda >0$ is observed (accelerating expansion), the SEC is false in our universe on large scales — a major reason it is now regarded as the least secure condition.
• Quantum fields. The vacuum energy of quantum fields can be locally negative — the Casimir effect has $T_{00}<0$ between plates, violating even the NEC/WEC. Hawking radiation requires a negative-energy flux across the horizon. Quantum field theory provides no local lower bound on energy density; only averaged/quantum-inequality versions survive.
• Inflation requires SEC violation (a scalar field with $V\gg {\dot{\varphi }}^2$, giving $p\approx -\rho$) to drive accelerated early expansion.

Because classical, physically realized matter can violate them, the energy conditions are best regarded as useful genericity assumptions rather than laws — sharp enough to prove theorems, but known to have exceptions.

What they buy: the global theorems

Energy conditions are the crucial physical input that lets local differential geometry yield global conclusions:

• Singularity theorems (Penrose 1965, Hawking–Penrose 1970). Given the NEC (Penrose) or SEC (Hawking–Penrose) plus a trapped surface or cosmological expansion and reasonable causality, spacetime is geodesically incomplete — singularities are generic, not artifacts of symmetry. This is what makes the Big Bang and black-hole singularities unavoidable predictions rather than special-case coincidences. (Penrose's use of the NEC in this theorem was cited in his 2020 Nobel Prize.)
• Black-hole area theorem (Hawking). Assuming the NEC, the total area of event horizons never decreases — the classical precursor of the second law of black-hole thermodynamics.
• Positive-mass theorem (Schoen–Yau, Witten). Assuming the DEC, the total ADM mass of an isolated system is non-negative, vanishing only for flat space. Gravitating matter cannot have negative total energy.
• Censorship results. Topological censorship and (conjecturally) cosmic censorship lean on the NEC to forbid traversable wormholes and naked singularities built from ordinary matter.

The pattern is uniform: NEC/SEC ⇒ focusing ⇒ singularities and horizons; DEC ⇒ positive mass and causal energy flow. Exotic phenomena that evade these conclusions — traversable wormholes, warp drives, time machines — all require energy-condition-violating "exotic matter", which is exactly why they remain speculative.

Summary

• Energy conditions restrict $T_{\mu \nu }$ to physically reasonable matter: WEC ($\rho \ge 0$, $\rho +p\ge 0$), NEC ($\rho +p\ge 0$, weakest), DEC ($\rho \ge |p|$, causal energy flow), SEC ($\rho +3p\ge 0$, gravity attracts).
• Implications: $\text{DEC}\Rightarrow \text{WEC}\Rightarrow \text{NEC}$; SEC $\Rightarrow$ NEC; SEC and DEC are independent.
• The SEC = geodesic focusing ($R_{\mu \nu }{u}^{\mu }{u}^{\nu }\ge 0$); cosmic acceleration and inflation violate it, and quantum effects (Casimir, Hawking) violate even the NEC.
• They are the hypotheses of the singularity theorems, area theorem, positive-mass theorem, and censorship — the bridge from local curvature to global structure.

Next

With sources and their constraints in hand, we solve the field equations in the highest-symmetry cases. The Schwarzschild solution gives the vacuum field of a spherical mass — the classical tests and the first black hole; FLRW gives the homogeneous, isotropic cosmologies.