AI 中文总结
研究局部 AdS3×S3×T4 黑洞的类时纠缠熵和子区域复杂度，重点是黑极解。通过特定处方构建可观测量，发现其探测几何方式不同，在大 $r$ 区域有相应行为，精确几何中时间族非单调，结果展示了局部几何效应。
AI 中文摘要
我们研究了具有渐近 AdS3×S3×T4 几何的局部黑洞中的类时纠缠熵和类时子区域复杂度，重点关注黑极解。与 BTZ 解不同，黑极通过函数 $K_y(r,\theta)$ 和 $G(r,\theta)$ 对内部球面呈现非平凡依赖。这两个可观测量由类空和类时洛伦兹分支构建，但探测几何的方式不同：类时纠缠产生一个复提升面积，类时复杂度给出一个实的、有限的重整化体积。我们采用局部类时处方，在角标 $\theta_0$ 处构建分支轮廓并在物理内角 $\theta$ 上提升。在大 $r$ 区域，主导角依赖消失，恢复预期的短区间行为。在精确的黑极几何中，时间族变得非单调，使得固定边界区间选择至关重要。随着边界区间增加，所选分支向内移动并对局部帽视界过渡区域敏感。这些结果表明类时洛伦兹可观测量探测了 BTZ 和主导大 $r$ 描述中不存在的局部几何效应。
英文摘要
We study timelike entanglement entropy and timelike subregion complexity in localized black holes with asymptotic AdS3*S3*T4 geometry, focusing on the black-pole solution. Unlike the BTZ solution, the black pole exhibits a nontrivial dependence on the internal sphere through the functions $K_y(r,θ)$ and $G(r,θ)$. Both observables are constructed from spacelike and timelike Lorentzian branches, but they probe the geometry in different ways: timelike entanglement yields a complex lifted area, while timelike complexity gives a real, finite renormalized volume. We employ a localized timelike prescription in which the branch profile is built at an angular label $θ_0$ and subsequently lifted over the physical internal angle $θ$. In the large-$r$ regime, the leading angular dependence drops out, recovering the expected short-interval behaviour. In the exact black-pole geometry, the temporal families become non-monotonic, making a fixed-boundary-interval selection essential. As the boundary interval increases, the selected branches move inward and become sensitive to the localized cap-horizon transition region. These results demonstrate that timelike Lorentzian observables probe localized-geometry effects that are absent in BTZ and in the leading large-$r$ description.
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