How to see that the electromagnetic stress-energy tensor satisfies the null energy condition? I am trying to show that the Maxwell stress-energy tensor,
$$T_{\mu\nu} = \frac{1}{4\pi}\left( F_{\mu\rho} F^{\rho}{}_{\nu} - \frac{1}{4}\eta_{\mu\nu}F_{\rho \sigma} F^{\rho\sigma} \right),$$
satisfies the null energy condition, i.e., that
$$T_{\mu \nu}k^\mu k^\nu \geq 0$$
for all null vectors $k^\mu$. I see that the second term vanishes on contraction with $k^\mu k^\nu$, but I'm struggling to see how to manipulate the first term.
 A: Notice that what you are trying to show is that $k^\mu F_{\mu}{}^\nu$ is a spacelike vector (in this answer, I'm assuming the $-+++$ metric convention). Hence, let us focus on this particular vector.
Given $k^\mu$ at some point, pick a choice of Cartesian coordinates such that $k^\mu = (1,1,0,0)^\intercal$, which is always possible. In this choice of coordinates, the field strength tensor reads (units with $c=1$)
$$F_{\mu\nu} = \begin{pmatrix} 0 & -E_1 & -E_2 & -E_3 \\ E_1 & 0 & B_3 & -B_2 \\ E_2 & -B_3 & 0 & B_1 \\ E_3 & B_2 & -B_1 & 0 \end{pmatrix}.$$
Notice then that
$$k^\mu F_{\mu}{}^{\nu} = \begin{pmatrix} -E_1 \\ -E_1 \\ -E_2 + B_3 \\ -E_3 - B_2 \end{pmatrix}.$$
A straighforward computation then shows that $k^\mu F_{\mu}{}^{\nu} k^\rho F_{\rho}{}_{\nu}$ is the sum of two explicitly non-negative terms.
A: While the answer provided by Nickolas Alves should suffice, here is an alternate proof of NEC satisfied by free Electromagnetic field using the idea of 2-spinor formalism (and hence, this proof is very particular to 3+1 dim space-time, see [1] )
The idea is that a real null vector $k^a$ can be written as tensor product of two 2-spinors, one being the conjugate of other:
$k^a\leftrightarrow k^A\bar{k}^{A'}$
where $k^A$ is defined upto an overall phase factor. Note that $k^ak_a\leftrightarrow k^Ak_A\bar{k}^{A'}\bar{k}_{A'}=0$ which follows form the fact that $k^Ak_A=\epsilon_{AB}k^Ak^B= \epsilon_{[AB]}k^Ak^B=0$
The Maxwell tensor $F_{ab}$ in this formalism can be written in terms of a symmetric 2-spinors $\phi_{AB}$ as follows:
$F_{ab}\leftrightarrow \phi_{AB}\epsilon_{A'B'}+c.c.$
It turns out that the stress energy tensor $T_{ab}$ of the EM field can be simply written as (see chapter 3 & 5 from [1])
$T_{ab}\leftrightarrow\phi_{AB}\bar{\phi}_{A'B'}$
The Null energy condition follows naturally: $T_{ab}k^ak^b\leftrightarrow |\phi_{AB}k^Ak^B|^2\geq 0$
[1] R. Penrose, W. Rindler, "Spinors and Space-Time. Volume-I: Two-Spinor Calculus and Relativistic Fields", Cambridge University Press (1984)
