I am confused by a statement in Sean Carroll's "Spacetime and geometry".
On page 174-175, he makes the following statement about the physical interpretation of the WEC in the case of a perfect fluid:
"Because the pressure is isotropic, $T_{\mu\nu}t^\mu t^\nu$ will be non-negative for all timelike vectors $t^\mu$ if both $T_{\mu\nu}U^\mu U^\nu \geq 0$ and $T_{\mu\nu}l^\mu l^\nu \geq 0$ for some nullvector $l^\mu$ ".
$T_{\mu\nu}$ is the energy momentum tensor and $U^\mu$ is the four velocity. For a perfect fluid, the two last inequalities respectively reduce to $\rho \geq 0$ and $\rho + p \geq 0$.
I do understand that both inequalities are necessary conditions for the WEC ( $T_{\mu\nu}t^\mu t^\nu \geq 0$) to hold, but they are not obviously sufficient conditions.
They are necessary conditions as $U^\mu$ is a timelike vector and, by continuity, the inequality involving $l^\mu$ should hold as well. However, they are treated as sufficient conditions, i.e. the WEC is used as a synonym for "$\rho \geq 0$ and $\rho + p \geq 0$".
In a metric that does not differ too much from a Minkowskian metric (weak field limit), we can decompose any timelike vector $t^\mu$ as a sum of a well chosen nullvector $l^\mu$ and a multiple of $U^\mu$ or $$t^\mu = l^\mu+ \lambda U^\mu$$ with $\lambda$ a real number. This directly leads to $$T_{\mu\nu}t^\mu t^\nu = \rho (\lambda -U_\mu l^\mu )^2 +p(U_\mu l^\mu)^2$$. This last equation implies that in the weak field limit, the WEC is equivalent to $\rho \geq 0$ and $p \geq 0$. Can somebody help me understanding where is the mistake I am making and why the conditions are sufficient conditions?