The equivalent statement of the dominant energy condition

Following Hawking and Ellis,section 4.3, page 91, the dominant energy condition stipulates that for every timelike vector $W_a$, the energy-momentum tensor $T^{ab}$ satisfies the inequality $T^{ab}W_aW_bge 0$, and $T^{ab}W_a$ is a non-spacelike vector. Below this definition, they give an equivalent statement of the dominant energy condition.

In any orthonormal basis the energy dominantes the other components of $T_{ab}$, i.e. $T^{00}ge |T^{ab}|$ for each a,b.

How to prove the equivalent statement?

I can only prove it when $T^{ab}$ is diagonal.

Is there any reference?