The physical interpretation of the weak energy condition (WEC)

Question

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?

Answer 1

You have already noted that they are necessary conditions. That they are sufficient follows precisely because the pressure is isotropic. To see this consider a local Lorentz (orthonormal) frame (a.k.a tetrad). Any null vector can be written as $$ \ell^i = N\left(\delta^i_0 + \delta^i_1\right), $$ since we have specified no spacelike directions (really the 1 is just an arbitrarily chosen spacelike index), while $$ U^i = \delta^i_0. $$ The stress-energy tensor is $T_{ij} = \mathrm{diag}\left(\rho,p,p,p\right)$. Then $$T_{ij}U^iU^j = T_{00} = \rho \geq 0$$ and $$T_{ij}\ell^i\ell^j = N^2\left(T_{00} + T_{11}\right) = N^2\left(\rho + p\right) \geq 0$$ is equivalent to that $\rho + p \geq 0$. Finally, an arbitrary timelike vector can be decomposed, as you say $$ t^i = \ell^i + \lambda U^i, $$ with $\lambda^2 + 2N\lambda > 0$ (equal to 1 if normalized), since otherwise $t^i$ would be null or spacelike. Thus
\begin{align} T_{ij}t^it^j &= N^2(\rho + p) + (\lambda^2 + 2N\lambda)\rho \geq 0 \end{align} by the above results. Meaning that all observers observe the energy density to be non-negative.