  Using the Hubeny-Rangamani-Takayanagi (HRT) conjectured formula for
entanglement entropy in the context of the AdS/CFT correspondence with
time-dependent backgrounds, we investigate the relation between the bulk null
energy condition (NEC) of the stress-energy tensor with the strong
sub-additivity (SSA) property of entanglement entropy in the boundary theory.
In a background that interpolates between an AdS to an
AdS-Reissner-Nordstrom-type geometry, we find that generically there always
exists a critical surface beyond which the violation of NEC would naively
occur. However, the extremal area surfaces that determine the entanglement
entropy for the boundary theory, can penetrate into this forbidden region only
for certain choices for the mass and the charge functions in the background.
This penetration is then perceived as the violation of SSA in the boundary
theory. We also find that this happens only when the critical surface lies
above the apparent horizon, but not otherwise. We conjecture that SSA, which is
thus non-trivially related to NEC, also characterizes the entire time-evolution
process along which the dual field theory may thermalize.
