Holographic Screen Sequestration

Abstract

Holographic screens are codimension-one hypersurfaces that extend the notion of apparent horizons to general (non-black hole) spacetimes and that display interesting thermodynamic properties. We show that if a spacetime contains a codimension-two, boundary-homologous, minimal extremal spacelike surface $X$ (known as an HRT surface in AdS/CFT), then any holographic screens are sequestered to the causal wedges of $X$. That is, any single connected component of a holographic screen can be located in at most one of the causal future, causal past, inner wedge, or outer wedge of $X$. We comment on how this result informs possible coarse grained entropic interpretations of generic holographic screens, as well as on connections to semiclassical objects such as quantum extremal surfaces.

Figures (11)

• Figure 1: The causal wedges and null light sheets defined by a Cauchy hypersurface $\Sigma$, a Cauchy-splitting surface $\nu$, and the two null directions $k$ and $\ell$.
• Figure 2: A configuration of $H$ and $X$ showing how rules \ref{['rule1']}-\ref{['rule3']} apply in practice. Arrows on hypersurfaces represent the direction of increasing cross-sectional area. The double arrow on $\Sigma$ indicates that a notion of increasing area is only true when comparing a Cauchy-splitting surface $\nu \in \Sigma$ with the HRT surface $X$. The purple dashed line highlights the fact that in this configuration a closed, directed path can be formed that is inconsistent with the rules. This is an example of what we call a forbidden loop.
• Figure 3: Forbidden loops (purple dashed lines) for spacelike segments of a holographic screen (green solid line) passing through each of the four null congruences $N_{\pm k}[X]$ and $N_{\pm \ell}[X]$ fired from $X$ (red and blue solid lines respectively). In each case, in addition to the screen segment and one of the four null congruences, the forbidden loop is closed by $N_{\pm k}[\sigma]$, a null congruence of $\sigma$ in the marginal null direction $k$, and possibly the Cauchy slice $\Sigma$. Analogous results hold for timelike or mixed signature holographic screen segments.
• Figure 4: Representative trajectories of future holographic screens that are consistent with the rules outlined above. The left figure shows all allowed trajectories with early timelike flows while the right figure shows all allowed trajectories with early spacelike flows. In both cases, sequestration stipulates that no screens are allowed to cross the null congruences $N_{\pm k}[X]$ (red line) and $N_{\pm \ell}[X]$ (blue line) fired from the HRT surface $X$. Note that although screens may start on $X$ in some cases, they can never end on $X$. Here, we are depicting the allowed early and late asymptotic behavior along the screen; any number of intermediate transitions between spacelike and timelike regions is allowed.
• Figure 5: Forbidden loops (purple dotted lines) that appear when a holographic screen (green line) in the interior $I_W[X]$ (left panel) or future $I^+[X]$ (right panel) ends at the HRT surface $X$. Analogous loops can be found when the screen is to the past of the Cauchy slice $\Sigma$ in the interior.

Theorems & Definitions (11)

• Theorem 3.1: Poincaré-Hopf H1886Hopf1927
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Corollary 3.4
• proof
• Proposition 3.5
• proof
• Theorem 3.6: Sequestration