Smooth Causal Patches for AdS Black Holes

TL;DR

The paper resolves a tension between low-energy boundary excitations in AdS/CFT and the statistical mechanics bound that such excitations should not drastically affect observables. By focusing on position-space correlators confined to single causal patches and proving a explicit bound $\delta\langle A_{\alpha} \rangle \\le 2\sqrt{\beta\delta E}\,\sigma_{\alpha}$, it shows that bulk observables in a patch remain stable even when the boundary state is excited with small energy. It then analyzes the Born-rule paradox using explicit near-horizon calculations and introduces causal patch complementarity, proposing patch-local field operators $\phi_{\cal C}$ that reproduce consistent physics within each patch but reflect complementary global descriptions. The results imply that a smooth interior and consistent bulk reconstruction can coexist with thermodynamic expectations, provided one respects causality and patchwise descriptions; this offers a refined view of black hole interiors in AdS/CFT and prompts future work on observer models and $1/N$ corrections.

Abstract

We review the paradox of low energy excitations about an AdS black hole. An appropriately chosen unitary operator in the boundary theory can create a locally strong excitation near the black hole horizon, whose global energy is small as a result of the gravitational redshift. The paradox is that this seems to violate a general rule of statistical mechanics, which states that an operator with energy parametrically smaller than $k T$ cannot create a significant excitation in a thermal system. When we carefully examine the position dependence of the boundary unitary operator that produces the excitation and the bulk observable necessary to detect the anomalously large effect, we find that they do not both fit in a single causal patch. This follows from a remarkable property of position space AdS correlators that we establish explicitly, and resolves the paradox in a generic state of the system, since no combination of observers can both create the excitation and observe its effect. As a special case of our analysis, we show how this resolves the "Born rule" paradox of arXiv:1506.01337 and we verify our solution using an independent calculation. We then consider boundary states that are finely tuned to display a spontaneous excitation outside the causal patch of the infalling observer, and we propose a version of causal patch complementarity in AdS/CFT that resolves the paradox for such states as well.

Figures (5)

• Figure 1: Two views of a causal patch. On the left, we show a causal patch in a Datt-Oppenheimer-Snyder collapse in AdS. The collapsing star is shaded in brown. On the right, we show a causal patch in an eternal single-sided geometry and we also show the early time cutoff at $\vartheta$. In both subfigures, the patch is demarcated in blue-gray and the future horizon is shown by a dashed magenta line.
• Figure 2: The causal wedge, $\wedge_{\cal C}$ is shaded in blue outside the horizon; the intersection of the causal patch with the interior, denoted by $\vee_{\cal C}$ is shaded pink inside the horizon.
• Figure 3: A boundary excitation at $x'$ commutes with the right moving hatted field at $x$ even though $x$ is in the causal future of $x'$. The boundaries of the causal patch are in blue. The light cone from $x'$ is marked off in brown.
• Figure 4: The slice ABC gives a Cauchy slice for the entire geometry. BC provides complete initial data for the causal wedge $\wedge_{\cal C}$. In the large-N limit, operators on AB and BC commute.
• Figure 5: The overlap of two causal patches