Energy conditions

Remarks

The weak and the strong conditions each imply the null condition, by continuity, since null vectors are limits of timelike vectors; the dominant condition implies the weak one. The strong condition does not imply the weak one. If $g$ solves the Einstein field equations with zero cosmological constant, the strong condition is equivalent to $R_{ab} X^a X^b \ge 0$ for all timelike $X$, where $R_{ab}$ is the Ricci tensor.

For a perfect fluid with energy density $\rho$ and pressure $p$, the conditions read: null, $\rho + p \ge 0$; weak, $\rho \ge 0$ and $\rho + p \ge 0$; strong, $\rho + 3p \ge 0$ and $\rho + p \ge 0$; dominant, $\rho \ge \abs{p}$.

Example: dust, a perfect fluid with $\rho \ge 0$ and $p = 0$, satisfies all four conditions.

Non-example: a positive cosmological constant acts like a perfect fluid with $p = -\rho$ and $\rho > 0$. It satisfies the null, weak and dominant conditions, but $\rho + 3p = -2\rho < 0$, so it fails the strong energy condition.