Comments on black hole interiors and modular inclusions

TL;DR

The paper reframes the traversable wormhole induced by a double-trace deformation as a modular inclusion between exterior operator algebras, revealing a nontrivial center that encodes interior data nonlocally. It demonstrates that state-dependent interior representations arise inevitably from the algebraic structure and discusses their relation to mirror operators and the Reeh–Schlieder theorem. By unifying entanglement-based emergence of spacetime with modular inclusions, the work provides a precise framework for studying how interior geometry relates to boundary data and hints at a nonlocal, QEC-like foundation for holography. It also outlines the limitations and future directions for formalizing emergent spacetime via AQFT concepts such as half-sided modular inclusions and relative entropy.

Abstract

We show how the traversable wormhole induced by a double-trace deformation of the thermofield double state can be understood as a modular inclusion of the algebras of exterior operators. The effect of this deformation is the creation of a new region of spacetime deep in the bulk, corresponding to a non-trivial center between the left and right algebras. This set-up provides a precise framework for investigating how black hole interiors are encoded in the CFT. In particular, we use modular theory to demonstrate that state dependence is an inevitable feature of any attempt to represent operators behind the horizon. Building on this geometrical structure, we propose that modular inclusions may provide a more precise means of investigating the nascent relationship between entanglement and geometry in the context of the emergent spacetime paradigm.

Figures (3)

• Figure 1: TFD under the double-trace deformation of Gao:2016bin. (Left) The deformation manifests in the bulk as two negative-energy shockwaves (red wavy lines). A null observer (dashed line) who falls into the black hole from the left suffers a time-advance upon crossing the shockwave, which shifts her to the exterior region on the right. A more physical way of understanding this is that the negative-energy shocks lower the mass of the black hole, thereby causing the future horizon to shrink (Right). The result is that the exterior wedge increases in size (from light to dark blue), which opens up a causeway through which the observer can safely travel. Note that the left and right exterior regions are no longer independent, but overlap in the central diamond (darkest blue). The algebraic structure that corresponds to these nested wedges is a modular inclusion.
• Figure 2: Same as the right panel of fig. \ref{['fig:tfd']}, but with a greatly exaggerated shift in the horizon for illustration. $\mathcal{N}_R$ and $\mathcal{N}_R'$ are shaded blue, while $\mathcal{M}_R$ and $\mathcal{M}_R'$ are filled with red lines. Note that $\mathcal{N}_R'=\mathcal{N}_L$, but $\mathcal{M}_R'\neq\mathcal{M}_L$ since the inclusion breaks the left-right symmetry. The figure also illustrates the fact that $\mathcal{N}_R\subset\mathcal{M}_R\implies\mathcal{M}_R'\subset\mathcal{N}_R'$. The behind-the-horizon region $\mathcal{D}_R\subset\mathcal{M}_R$, i.e., the portion of $\mathcal{M}_R$ not contained in $\mathcal{N}_R$, is shaded red.
• Figure 3: (Top row) TFD under sequential SS-type shocks, which move the horizons outwards. The original horizons are indicated in black, while their positions under the first three shocks are shown in orange (first shock, left), green (second shock, middle), and pink (third shock, right). The portion of the right exterior wedge $\mathcal{N}_R$ lost behind the horizon after each inclusion is shaded according to this same scheme, while the original interior is shaded grey. (Bottom row) TFD under sequential GJW-type shocks, using the same color scheme above. In this case, the horizons move inwards, so the interior of the black hole shrinks. Note the increasing overlap between the exterior wedges $\mathcal{N}_i$.