Holographic Timelike Entanglement and Subregion Complexity in Localized AdS3×S3×T4 Black Holes

This paper investigates how timelike holographic observables—specifically timelike entanglement entropy and timelike subregion complexity—behave in localized black-pole geometries. Unlike the standard BTZ black hole, which is homogeneous across the internal $S^3 \times T^4$ space, localized black-pole solutions exhibit nontrivial dependence on internal angular coordinates. The authors aim to determine if these Lorentzian observables can capture the ten-dimensional structure of such localized black holes.

The authors employ a localized timelike prescription to construct bulk surfaces. Because the geometry varies over the internal sphere, they first define reduced Lorentzian branch profiles at a specific angular label $\theta_0$ and subsequently lift these results to the full ten-dimensional geometry. This method allows them to compare the exact black-pole results against the BTZ benchmark and a large-$r$ asymptotic limit. A critical aspect of their methodology is the use of a fixed-boundary-interval minimization, which is necessary because the temporal families in the exact black-pole geometry are non-monotonic.

The study demonstrates that timelike Lorentzian observables are sensitive to the internal angular structure of the black-pole geometry. While the large-$r$ regime recovers the expected BTZ-like behavior, the exact geometry introduces significant differences: the boundary-interval map becomes non-monotonic, and the allowed angular range depends on the target interval. As the boundary interval grows, the selected Lorentzian branches move inward, becoming increasingly sensitive to the transition region between the horizon and the smooth cap. Timelike entanglement entropy is recorded as a complex lifted area, while timelike subregion complexity is captured as a real, finite renormalized volume, providing complementary probes of the same localized geometry.