Weak energy condition

The weak energy condition requiresfor every timelike vector $X$, so every observer measures nonnegative energy density.

For a perfect fluid in general relativity in metric signature $(-,+,+,+)$, the weak energy condition is equivalent to $\rho \ge 0$ and $\rho +p\ge 0$. A normalized observer's four-velocity $W$ has Lorentz factor $\gamma =-u\cdot W\ge 1$ relative to the fluid, and $T_{ab}{W}^a{W}^b=\rho +(\rho +p)({\gamma }^2-1)$. The case $\gamma =1$ forces the first inequality and arbitrarily large $\gamma$ forces the second; conversely the same expression proves sufficiency.

For the general scalar ansatz in Conserved Klein-Gordon scalar stress-energy ansatz, work in a unit timelike frame. ThenThe weak energy condition for arbitrary local field data is therefore equivalent to