Seeing behind black hole horizons in SYK

TL;DR

This work demonstrates explicit interior reconstruction of an AdS$_2$ black hole in Jackiw–Teitelboim gravity using only boundary SYK data by introducing a probe entangled with a reference and evolving operators via modular flow. A replica-trick on a boundary necklace diagram yields the modular-flow correlator $W(s,t)$, whose bulk interpretation as a proper-time translation along the infalling worldline reproduces the expected light-cone crossing and fixes the probe's clock, $eta_{probe}$. The analysis connects boundary SYK dynamics to bulk JT gravity through both a boundary replica calculation and a bulk Euclidean wormhole saddle, showing a precise match between the SYK result and the gravitational computation in the intermediate-probe-entropy regime. The results illuminate a universal, chaos-driven mechanism for interior reconstruction and suggest avenues for extending the approach to higher dimensions and scattering behind horizons, with potential applications to single-sided black holes and holographic information recovery.

Abstract

We present an explicit reconstruction of the interior of an AdS$_2$ black hole in Jackiw-Teitelboim gravity, that is entirely formulated in the dual SYK model and makes no direct reference to the gravitational bulk. We do this by introducing a probe "observer" in the right wormhole exterior and using the prescription of [arXiv:2009.04476] to transport SYK operators along the probe's infalling worldline and into the black hole interior, using an appropriate SYK modular Hamiltonian. Our SYK computation recovers the precise proper time at which signals sent from the left boundary are registered by our observer's apparatus inside the wormhole. The success of the computation relies on the universal properties of SYK and we outline a promising avenue for extending it to higher dimensions and applying it to the computation of scattering amplitudes behind the horizon.

Figures (12)

• Figure 1: (a) Euclidean path integral preparation of the thermofield double state. The blue half disk is Euclidean path integral and the green strip is the Lorentzian continuation. (b) Euclidean path integral preparation of the thermofield double state with a probe following geodesic (\ref{['eq:geod']}), which is plotted as the red curve. The purple curve is the Euclidean geodesic of the probe. (c) HKLL reconstruction of a bulk spinor field $\chi$ (black dot) with $\ell$ distance from the probe (red curve). Its boundary representation involves an integral of the HKLL kernel over the boundary region $D(t_*)=[-t_{*},t_{*}]$ which is spacelike separated from $\chi$. Translating the bulk field $\chi$, originally located outside the horizon, along the proper time of the red geodesic, while keeping its geodesic distance from this geodesic fixed (purple curve), allows us to probe the $AdS_2$ wormhole interior. In the dual SYK model, this proper time translation is generated by the modular flow $\rho^{is}$ of the red probe, after tracing out the reference system it is entangled with.
• Figure 2: (a) The "necklace" SYK diagram, summarizing the replica manifold for $k=4$ replicas. The green dotted lines connecting SYK$_l$ and SYK$_r$ correspond to local insertions of $\rho_0=e^{-\mu S}$ where $S$ is the "size" operator (\ref{['size']}) and $\mu$ a parameter related to the entropy of the probe $S_{probe}$ and defined in Section \ref{['sec:prepare']}. This coupling between the two boundary quantum systems appears after we trace out the reference and is a consequence of the entanglement between the probe and the reference. The modular flowed anticommutator (\ref{['Wintro']}) is obtained by an analytic continuation of the SYK propagator on this "necklace" diagram. (b) The SYK "necklace" diagram serves as the boundary condition for the Euclidean path integral of the dual JT gravity. In the limit of probe entropy the dominant saddle is a pair of disconnected geometries with disk topology, leading to trivial modular flow. (c) At intermediate values of the probe entropy for $\mu$ greater than a critical value $\mu_{cr}$, the Euclidean wormhole saddle with cylindrical topology dominates, supported by the $\rho_0$ path integral insertions. The modular flowed commutator $W$ becomes non-trivial in this regime, allowing us to propagate into the black hole interior and detect signals sent from the other side. (d) At very small probe entropies, the backreaction of the $\rho_0$ becomes large, squeezing the wormhole at the insertion points, and causing it to "pinch off" into a product of $k=4$ disconnected disks with perimeter $\beta_l+\beta_r$. Modular flow becomes trivial in this limit.
• Figure 3: (a)(b) The worldline of probe (red curve) and spatial geodesics with equal $s_{p}$ separation and orthogonal to it (blue curves) in the two sided big black holes spacetime. The two shaded regions are left and right wedge respectively. The yellow dashed line is null and shot from left boundary from $T=-1$. We see clearly that the probe takes more proper time in (b) than (a) to reach the lightcone of the yellow line. (c) The location of past lightcone location $T_{LC}$ on left boundary of an atmosphere operator on right boundary after proper time $s_{p}$ evolution. Blue, yellow and green curves are for $\xi=2,0,-2$.
• Figure 4: Necklace diagram. Splitting every circle into $l$ and $r$ on which the system evolves with SYK Hamiltonian $H_{l,r}$ respectively. Each green dot means insertion of $\rho_{0}$.
• Figure 5: The plot of $\sigma_{rl}^{s}(\tau,\beta_{l}/2)/q$, where different $s$ are joined together in order. Blue, yellow and green curves are for $\mathcal{J}=20,200,2000$ respectively. We see the correlation decays exponentially as $s$ increases and the decay is stronger when we increase $\mathcal{J}$. Here other parameters are $\beta_{l}=1$, $\beta_{r}=4$, $\alpha=1/500$, $q=20$ and $k=9$.