Strong Subadditivity, Null Energy Condition and Charged Black Holes

TL;DR

This work investigates how bulk null energy condition (NEC) constraints in time-dependent AdS/CFT backgrounds with charge relate to strong sub-additivity (SSA) of boundary entanglement entropy via the HRT prescription. By studying AdS-RN-Vaidya spacetimes with time-dependent mass and charge, the authors identify a critical surface $z_c$ where NEC may be violated and analyze whether holographic entanglement surfaces can penetrate this region. They find SSA is violated only when the extremal surfaces reach beyond the NEC-violating region, which occurs if $z_c$ lies below the apparent horizon $z_{ m ah}$; otherwise SSA remains satisfied, indicating a non-trivial link between NEC and SSA during thermalization. The results extend across dimensions $d=3$ and $d=4$, suggesting SSA constrains the global time evolution during holographic thermalization and hints at a broader NEC-concavity connection in boundary theories.

Abstract

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.

Figures (14)

• Figure 1: Examples of two functions $m(v)$ where (a) NEC is obeyed, and (b) NEC is violated.
• Figure 2: Entropy function for the cases where (a) NEC is obeyed, and (b) NEC is violated. The different colors correspond to boundary times $t_b=0.3$ (red), $t_b=1.0$ (blue), $t_b=1.5$ (purple), and $t_b=2.0$ (black). Notice that the curves in (b) are not concave and, thus, SSA is violated.
• Figure 3: Left panel: $m(v)$ (black) and $q(v)$ (red). Right panel: $S(\ell)$, for $t_b=$ 0.01 (black), 0.5 (blue), 1 (purple), 1.5 (magenta), 2 (green), 3 (orange) and 5 (red).
• Figure 4: Profiles of a family of geodesics when SSA is obeyed. The dashed gray line represents the apparent horizon at $v=0$ and the red line represents the critical surface. This family is parametrized by $\ell$, the length of the entangling region at the boundary. Note that right panel shows that the geodesics do not intersect the critical surface as explained in text.
• Figure 5: Left panel: $m(v)$ (black) and $q(v)$ (red) given in (\ref{['exam2']}). Right panel: $S(\ell)$, for $t_b=$ 0.01 (black), 0.5 (blue), 1 (purple), 1.5 (magenta), 2 (green), 3 (orange) and 5 (red).