It's a question on Sean Carroll's Spacetime and Geometry, where we are supposed to prove that the energy momentum tensor of scalar field theory satisfies Weak Energy Condition (WEC). The energy momentum tensor is
$$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(\nabla_{\lambda}\phi\nabla^{\lambda}\phi+V(\phi)\right),$$
and the condition for WEC is
$$T_{\mu\nu} U^\mu U^\nu \geq 0,$$
where $U^\mu$ is an arbitrary non-spacelike vector(=timelike or null).
But how can this be proved when there are no known properties about the scalar field variable $\phi$ and potential $V(\phi)$?
Best Answer
I don't have the book, so can't check out his assumptions, so this might not quite answer your question, since you're asking about arbitrary 4 vectors $U^{\mu}$, but I'll offer it in case some of it is useful. In proving the weak energy condition (which is part way to proving the dominant energy condition), the 4 vectors in question are timelike. If this is the case, I might try the following:
Assume a signature (- + + + )
Starting with $$T_{\mu\nu}=\nabla_{\mu}\phi\nabla_{\nu}\phi-\frac{1}{2}g_{\mu\nu}\left(\nabla_{\lambda}\phi\nabla^{\lambda}\phi+V(\phi)\right)$$
if $U^{\mu}$ is timelike and future pointing, then at any given point we can work in an orthonormal frame for which the components are $U^{\mu}=(1,0,0,0)$
If we then demonstrate the positivity of $T_{\mu\nu}U^{\mu}U^{\nu}$ in that frame, then it will hold in any frame since it's a scalar.
So, plugging in the components of $U^{\mu}$, we get
$$T_{\mu\nu}U^{\mu}U^{\nu}=(\nabla_{0}\phi)^{2}+\frac{1}{2}(g^{\lambda\rho}\nabla_{\lambda}\phi\nabla_{\rho}\phi+V(\phi))$$
$$=\frac{1}{2}(\nabla_{0}\phi)^{2}+\delta^{ij}\nabla_{i}\phi\nabla_{j}\phi+V(\phi)$$
So provided $V(\phi)$ is positive and $\phi$ is a real field (which it surely is otherwise you'd have had complex conjugates in the energy momentum tensor), then in that frame, at that point $T_{\mu\nu}U^{\mu}U^{\nu}$ is positive.
But this is just the weak energy condition you'd have to work a bit harder to prove the dominant energy condition.